Some disturbing aspects of inland vessel stability rules

Abstract

Nonlinear rolling of typical inland container vessels subjected to severe beam wind was investigated in series of numerical tests. The mean wind speed was estimated from the stability rules, and the wind gusts from the appropriate wind spectrums. From the time history of vessel motion, the probability that the open container hold would be flooded, was obtained. Surprisingly, it was found that inland vessels are not as safe as are seagoing ships in beam wind and waves. In some extreme but realistic circumstances, the vessels, satisfying all the present stability rules, could be flooded and eventually capsized. Such results show not only the vulnerability of inland container vessels to severe beam winds, but also indicate some shortcomings of the present inland stability rules. A previous analysis of the same subject [1] indicated a question: why there are no accidents of inland container vessels in severe beam storms, as the obtained results anticipate. The present investigation offers a possible, but somewhat unexpected solution of such dilemma.

Full text

Vol. 2, No. 2, July 2006 Journal of Ship Technology 1
Some Disturbing Aspects of Inland Vessel Stability Rules
Milan Hofman, Ivan Maksic and Igor Backalov
ABSTRACT
Nonlinear rolling of typical inland container vessels subjected to severe beam wind was
investigated in series of numerical tests. The mean wind speed was estimated from the stability
rules, and the wind gusts from the appropriate wind spectrums. From the time history of vessel
motion, the probability that the open container hold would be flooded, was obtained. Surprisingly,
it was found that inland vessels are not as safe as are seagoing ships in beam wind and waves.
In some extreme but realistic circumstances, the vessels, satisfying all the present stability rules,
could be flooded and eventually capsized. Such results show not only the vulnerability of inland
container vessels to severe beam winds, but also indicate some shortcomings of the present
inland stability rules. A previous analysis of the same subject [1] indicated a question: why
there are no accidents of inland container vessels in severe beam storms, as the obtained results
anticipate. The present investigation offers a possible, but somewhat unexpected solution of
such dilemma.
Keywords : Inland container vessels, stability rules, beam wind, nonlinear rolling.
Journal of Ship Technology, Vol. 2, No.2, July 2006, pp 1-14
Milan Hofman
Department of Naval Architecture,
Faculty of Mech. Engineering
University of Belgrade
Ivan Maksic
Institute of Technical Sciences,
Serbian Academy of Science and Art,
Belgrade
Igor Backalov
Department of Naval Architecture
Faculty of Mech. Engineering
University of Belgrade
1. NOMENCLATURE
ALlateral vessel area exposed to wind (m2)
Anwind-gust amplitude (m/s)
Bvessel breadth (m)
cdrag coefficient (-)
FBvessel freeboard (m)
Fwwind force (kN)
d
w
Fdynamic wind force (kN)
st
w
Fstatic wind force (kN)
ggravitational acceleration (m/s2)
Hvessel depth (m)
htotal stability lever (m)
h’ additional stability lever (m)
hchatch coaming height (m)
Jxmoment of inertia for x axes (tm2)
jxradius of gyration for x axes (m)
Lvessel length (m)
lddynamic wind moment lever (m)
lsstatic wind moment lever (m)
m
ϕ
added mass of roll (tm2)
MG metacentric height (m)
MGmin minimal metacentric height (m)
Mddamping moment (kNm)
Mst stability moment (kNm)
Mwwind moment (kNm)
d
w
Mdynamic wind moment (kNm)
s
w
Mstatic wind moment (kNm)
Ncnumber of cycles (-)
pddynamic wind pressure (kPa)
psstatic wind pressure (kPa)
Pprobability (-)
Paacceptable probability (-)
Swind spectrum (m2/s)
s
ϕ
standard deviation of roll (rad)
ttime (s)
Tvessel draught (m)
'

2 Journal of Ship Technology Vol. 2, No. 2, July 2006
tsperiod - duration of storm (s)
Tmean period of roll (s)
vwind speed (m/s)
vmean wind speed (m/s)
v’ fluctuating wind speed (m/s)
xlongitudinal central axes
n
α
phase shift of n-th wind component (-)
β
quadratic damping coefficient (-)
Ävessel displacement (t)
φ
prescribed angle of heel (rad)
ϕ
roll angle, heel (rad,°)
ϕ
mean value of roll (rad)
c
ϕ
critical angle of heel (rad)
d
ϕ
dynamic angle of heel (rad)
f
ϕ
flooding angle (rad)
s
ϕ
static angle of heel (rad)
κ
coefficient of terrain roughness (-)
µ
linear damping coefficient (-)
ρ
air density (t/m3)
n
ω
frequency of n-th component (rad/s)
UNECE United Nations Economic
Commission for Europe
2. INTRODUCTION
In a recent paper on ship stability (Hofman &
Backalov [1]), the rolling of inland container vessels
due to gusts of severe beam wind was investigated.
Although usually believed that inland vessels are
reasonably safe under the action of such wind, it was
found (to some surprise) that the vessels satisfying all
the present stability standards, could be flooded in some
realistic circumstances. If true, the result indicates not
only the vulnerability of container vessels to the action
of severe beam wind, but also some serious shortcomings
of the present inland stability rules.
In spite of such result, there seem to be no accidents
of inland container vessels that could be connected to
their rolling in the severe beam storms. Is that just a
good fortune, or the obtained results represent only a
theoretical speculation involving some improper
assumptions and modeling? The previous paper, mainly
concerned with some other aspects of ship stability, left
these questions opened.
The detected dilemma concerning inland vessel
stability is investigated in detail in the present paper.
Typical European inland container vessels were subjected
to the beam wind of average speed prescribed by the
stability rules. The wind was, however, not constant,
and its fluctuations were obtained from the appropriate
wind spectrums. Nonlinear differential equation of motion
due to wind gusting is solved numerically, giving the
time history of vessel rolling. From it, by the usual
technique, the probability that the open container hold
would be flooded is found. Numerous numerical tests
have been carried out for different vessels, different
number of container layers, different terrain roughness,
etc. In these tests, the metacentric height and the hatch
coaming height were systematically varied. In contrast
to the paper [1], where Yugoslav inland stability rules
[2] were applied, the results of the present analysis were
compared to recent version of UNECE Recommendations
on Technical Requirements for Inland Navigation Vessels
[3] (referred here as UNECE Rules). The formal
recognition of these Rules is still in progress.
Although the UNECE Rules were found
considerably stricter than the previously applied
standards, the results still show that the flooding
probability of inland container vessel is, in some cases,
considerably higher than the appropriate value for
seagoing ships. So, the analysis proves that the wind
gusting plays much more dangerous role than considered
by the rules, and indicates that the vessel, fulfilling the
stability standards, could be flooded and eventually
capsized in some extreme but realistic circumstances!
The paper, finally, suggests some improvements of
the UNECE Rules and offers a possible alternative - a
probabilistic criterion which, we believe, has the ability
to predict and rank the vessel safety against flooding
much more consistently.
3. INLAND STABILITY RULES
Presently, there exist a variety of inland stability
rules, with an urgent need for their harmonization. The
most recent attempt in that direction, UNECE Rules,
were chosen here as a relevant representative of the
present stability standards, and compared to our new,
probabilistic approach.
Only a part of the Rules that apply to inland
container vessel stability, sailing in (so called) waterway
zones 2 and 3, would be presented here. Even this part
would not be given in its original form, implying (as in
every rule) strict definitions, with little technical

Vol. 2, No. 2, July 2006 Journal of Ship Technology 3
reasoning. An attempt was made to understand the ideas
hidden in the rules, and present their sense and meaning.
It should be noted, however, that we did not have any
role in preparing the Rules, nor have any additional
knowledge concerning the background of the text. So,
some remarks and criticism (that could not be avoided)
should be understood as our personal view, only.*
According to the UNECE Rules, an inland container
vessel sailing in the zones 2 and 3, would be sufficiently
stable if it satisfies:
– Inland Weather Criterion,
– Static Wind-Heel Criterion,
– Additional Container Vessels Criteria.
Inland Weather Criterion. This is the basic
criterion according to UNECE Rules, which should be
satisfied by all inland vessels. The vessel is exposed to
the front of beam wind of prescribed speed, and
appropriate dynamic angle of heel
ϕ
d is calculated (see
Fig. 1). Such dynamic angle should be smaller than the
critical angle of heel
ϕ
c , prescribed by the rules.
This criterion is very different from the Weather
Criterion for seagoing ships, not only in respect that the
Inland Weather Criterion (correctly) accounts for no
waves. The idea behind the Weather Criteria for seagoing
ships is to account (in addition to waves) the fluctuations
of steady wind - the wind gusts. In contrast to that, the
inland Weather Criteria, as defined in the Rules, considers
just the first wind blow on the previously upright and
steady vessel.
Static Wind-Heel Criterion. This stability criterion
should be satisfied by all cargo vessels having the centre
of exposed lateral area more than 2 m above the waterline.
So, it applies to all the vessels carrying containers. The
vessel is exposed to the prescribed (steady) beam wind,
and appropriate static angle of heel
ϕ
s calculated (see
again Fig. 1). This static angle should be less than 80%
of the critical angle of heel.
The fact that the Rules allow only 80% of the
critical heel due to the action of static wind, leaves the
20% margin for wind fluctuations. So, the account of
wind gusts is, actually, hidden in this “static” criterion,
not (as could been expected) in the previously defined
Inland Weather Criterion. The sufficiency of such 20%
margin would be thoroughly analyzed throughout this
paper.
Critical angle. The critical angle prescribed in the
two former criteria is: angle of heel at which water begins
to fill the vessel through unsecured openings, but not
exceeding the angle at which the edge of the freeboard
deck is submerged, or the middle of the bilge leaves the
water. We, deliberately, gave here the whole original
definition. In the context of loaded container vessels, it
is, however, always the angle at which the freeboard
deck submerges.
It is important to note that the critical angle given
in UNECE Rules is considerably stricter than the
permissible angle in number of other standards. For
instance, in Yugoslav [2], and other “East-European”
Rules, as well as in ADN Rules [4], the permissible
angle is the angle of unsecured openings (angle of
flooding), which is, at least for container vessels, much
larger than the angle of freeboard deck. Although the
authors do not know the reasons that lead UNECE Rules
to accept such stricter definition, it turned to be very
much in line with the results of the present analysis.
Wind Moments. The UNECE Rules define two
different wind moments: the static wind moment,
applicable in Static Wind-Heel Criterion, and the dynamic
wind moment, applicable in Inland Weather Criterion.
Both moments could be expressed as
,
,,
sd
wsdLsd
MpAl
=
,
where p is wind pressure, AL is exposed lateral area, l
Fig. 1. UNECE Rules for inland vessel stability
* The reader is strongly advised to get acquainted with original text of
UNECE Stability Rules (see website www.unece.org/trans). In that
way he could not only follow this paper more easily, but also judge
independently our critical remarks.

4 Journal of Ship Technology Vol. 2, No. 2, July 2006
moment lever, while indices s, d refer to static and
dynamic condition, respectively (see again Fig. 1). The
static wind pressure is given as ps = 0.25 kPa for zone
2, and ps = 0.15 kPa for zone 3. These pressures
correspond (approximately) to the mean wind speeds of
18 m/s and 15 m/s, respectively. The dynamic pressure,
however, is supposed to depend upon the height above
the water level. It increases with height from 0.232-
0.388 kPa in zone 2, and from 0.178-0.302 kPa in
zone 3.
It is not fully clear why the dependence on the
height is considered in the case of dynamic wind, only.
It seems logical that the change of wind speed with
height should be accounted in both criterions, or in none.
The prescribed difference, therefore, seems to be an
unnecessary complication of the rules.
Additional Container Vessels Criteria. The
UNECE Rules give two alternatives: Method A and
Method B. The Method A is, actually, the same as
container vessel stability criterion of ADN Rules [4].
Both the methods strictly distinguish the case of fixed
and non-fixed containers. For non-fixed containers, both
the methods define explicitly minimal metacentric height
as MGmin = 1 m. Both take the combined effect of wind
and turning of the vessel, and limit the heel to 5°. The
wind speed in the two methods is the same, and the
differences are only in details concerning the action of
the turning force and the effects of free surfaces. For
fixed containers, Method A prescribes minimal
metacentric height MGmin = 0.5 m, and takes into account
(as for non-fixed containers) combined effect of wind,
turning and the free surfaces. It permits, however, the
vessel heel up to the angle of flooding, not to the critical
angle defined previously. Such change of permissible
angle of heel seems unclear, and could be understood as
an inconsequence of the Rules. The Method B gives no
additional criteria for fixed containers: the vessel just
has to meet the Inland Weather Criterion and Static Wind-
Heel Criterion, prescribed previously.
The two alternatives left in the text indicate that,
the UNECE Rules in the context of container vessel
stability, fail to fulfill their main task: to harmonize
different technical requirements for inland navigation
vessels. There must be a good reason for that, which
one can only guess. We believe, however, that some
additional effort should be made to overcome this
insufficiency, especially as the two methods do not
principally differ. For instance, in the case of fixed
containers, the requirement for minimal MGmin = 0.5 m
could be added to the Method B and, in Method A, the
doubts concerning the permissible angle of heel (to deck,
or to hatch coaming edge), clarified. Also, in the case of
non-fixed containers, the differences in centrifugal force
treatment do not seem (at least to us) so significant that
could not be matched.
Together with direct stability requirements
discussed above, UNECE Rules prescribe additional
demands concerning vessel freeboard FB and hatch
coaming height hC, which indirectly affect the vessel
stability. It is requested, for open hold vessels operating
in zone 2,
FB > 0.6 m , FB + hC > 1 m ,
leaving 0.4 m for the minimal hatch coaming height. It
would be shown in the later analysis that, actually, these
indirect demands are not strict enough, and (in some
extreme cases) jeopardize vessel safety against flooding.
4. PROBABILISTIC APPROACH
In the probabilistic approach to be used in the
present analysis, two different phases could be
distinguished. In the first, vessel rolling under the gusting
beam wind is estimated by solving the appropriate
differential equation(s) of motion. Then, in the second
phase, the statistical analysis of the obtained motion is
performed, and the probability that the vessel heels to
some given angle (e.g. the angle of flooding) is found.
In the present attempt, we consider the motion with
a single degree of freedom, only. So the differential
equation of rolling is
( ) () () ()
xdstw
Jm M M Mt
ϕ
ϕϕ ϕ
++ + =
 
,(1)
where Jx is moment of inertia for longitudinal central
axes, m
ϕ
is additional moment of inertia (added mass of
roll) and Md , Mst , Mw are damping, stability and wind
moments, respectively.
The main feature of the method is the way it treats
the wind moment on the right hand side of the equation.
Unlike the classical approach, this moment is calculated
from the wind spectrum that corresponds to the mean
wind speed prescribed by the stability rules. Namely, if
the wind spectrum S(
ω
n) is known, one can obtain the
appropriate wind-gust amplitudes as

Vol. 2, No. 2, July 2006 Journal of Ship Technology 5
()
nn
A2S d
ωω
=⋅
, (2)
and find the fluctuating wind speed as
()
() () cos
N
nnn
n1
vt v v t v A t
ωα
=
′
=+ =+ +
∑, (3)
where vis the mean wind speed, v’ is the fluctuation
speed (the gust) and
ω
n ,
α
n are frequency and random
phase shift of n-th gusting wind component. The wind
force is then
2
wL
1
FAcv
2
ρ
=⋅
,
where
ρ
is the air density and c the drag coefficient. In
order to apply the appropriate wind moment lever, we
break the force in its steady (static) and fluctuating
(dynamic) part
()
2
wL L
st d
ww
11
FcAvcAv2vv
22
FF
ρρ
′′
=+ +
"
"!"""
"""!, (4)
and calculate wind moment as
st d
wwswd
MFlFl
=⋅+⋅
. (5)
The static and dynamic levers ls , ld in this formula
are taken as prescribed by the UNECE (and many other)
inland stability rules (see Fig. 1).
There are various wind spectra recommended by
different authors (see. e.g. [5]). In the present analysis
we use, as the most relevant, Davenport spectrum
()
/
()
22
n
n34
2
nn
4vX
S
1X
κ
ωω
⋅
=+,n
n
600
Xv
ω
π
=⋅ , (6)
where k is the coefficient of terrain roughness.
Concerning the terms on the left hand side of the
equation (1), the added mass m
ϕ
is, in the present attempt,
calculated by the usual strip-theory technique. As that
result depends on the frequency, an approximate mean
value of roll frequency is applied in nonlinear
calculations. Damping moment is assumed to be
nonlinear, with linear and quadratic terms :
d
M
µϕ β ϕ ϕ
=+⋅

. (7)
The assessment of damping coefficients
µ
and
β
,
as a complicated and delicate matter, will be discussed
in more detail, later in the text.
So, if the wind spectrum is known and all the
coefficients of nonlinear differential equation (1)
estimated, the equation could be solved numerically. In
present calculations, Runge-Kutta method was used. For
the each time step, one just has to calculate the wind
fluctuations according to (2) and (3), and wind moment
Mw according to (4) and (5).
From the solution of equation (1) - the time history
of vessel rolling in the beam storm, one can find the
mean value
ϕ
, the standard deviation s
ϕ
, or some other
statistically defined value connected to the vessel rolling.
Then, the probability that the vessel would reach some
given angle of heel
φ
in some given period ts is (see e.g.
[1], [6], [7])
exp exp
exp
2
c
2
c
1
P1 N
2s
1
N
2s
ϕ
ϕ
φϕ
φϕ



−


=− − −









−

≈−





where Nc is the number of cycles in the analyzed period,
found as /
cs
NtT
= , where T is the mean (zero
crossing) period of rolling. Some authors (see [6]), do
not call P the probability, but, because of numerous
approximations used in the calculations, the index of
stability.
The applied method enables development of new,
probabilistic safety criteria for the vessels sailing in beam
storms. Such criteria would have the form: The
probability that the vessel unsecured openings would be
flooded in some given period should be less than... One
of the basic aims of the present investigation is, actually,
to give its contribution to the development and
recognition of such future safety criterions.
5. SAMPLE VESSELS
To examine the correlation of the present inland
stability rules and the probabilistic approach, two typical
European inland container vessels of different size were
used: the large vessel 110 x 11.4 m, and the small vessel
80 x 9 m. The large vessel carries TEU containers in 4
lines abreast and 13 rows in longitudinal direction. The
small vessel has 3 TEU container lines abreast, and 8
TEU container rows longitudinally. The number of
container layers varies from 3 to 5 for large vessel, and
3 to 4 for the small one.

6 Journal of Ship Technology Vol. 2, No. 2, July 2006
The maximal number of container layers is
restricted by the air draught of the vessel, which is limited
by the height of the bridges over the waterway, and by
the stability considerations. Generally, vessels with
maximal number of container layers (5 layers for large
and 4 layers for small vessels) would have problems in
achieving minimal MG required by stability rules.
However, as shown in [8], this difficulty could be
overcome by a careful vertical load distribution.
The vessels are of open type, with no hatch covers.
Both have the minimal freeboard of 0.6 m, required by
UNECE Rules and narrow gangways, close to the
minimal value of 0.6 m. Both vessels have draughts,
when fully loaded, of 2.6 m. The height of hatch
coamings starts from the minimal value required by the
Rules (0.4 m), but is systematically enlarged in some of
the numerical tests. The main vessel particularities and
the levers of so-called additional stability
sinhhMG
ϕ
′=− ⋅
(where h is total stability lever), are presented in Fig. 2.
These curves are given for unrealistically high hatch
coamings of 2 m. However, when applied to smaller
coaming heights, only a part of the curves is accounted,
up to the appropriate flooding angle. From the given h’
curves, for any supposed value of MG, the corresponding
stability moment directly follows :
() ( )
sin
st
Mgh ghMG
∆ϕ∆ ϕ
′
=⋅ = + ⋅
For each of the sample vessels, for different number
of container layers, the minimal values of metacentric
heights according to UNECE Rules have been estimated.
Actually, one can distinguish number of different minimal
metacentric heights that follow from the different
requirements: Inland Weather Criterion, Static Wind-Heel
Criterion, Additional Criteria (Methods A, B). In addition,
the Rules explicitly prescribe minimal metacentric height
of 1 m for non-fixed containers, and 0.5 m (Method A,
only) for fixed containers. The largest of these values -
the actual minimal metacentric heights of sample vessels
prescribed by UNECE Rules, are presented in Table 1.
The Table also gives the critical criterion for each of the
loading conditions.
Large Minimal MG (m)
vessel
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
Method A Method A Method B Method B
1.15 1.42 1.15 1.37
5 (Weather (Add. (Weather (Add.
Criterion) Criterion) Criterion) Criterion)
0.65 1.05 0.65
4 (Weather (Add. (Weather 1
Criterion) Criterion) Criterion)
0.275
3 0.5 1 (Weather 1
Criterion)
Small Minimal MG (m)
vessel
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
Method A Method A Method B Method B
0.549 0.549
4 (Weather 1 (Weather 1
Criterion) Criterion)
0.5 0.26
3 1 (Weather 1
Criterion)
Table 1. Minimal metacentric heights of sample
vessels according to UNECE Rules
It should be noted that Methods A and B, generally,
give similar values of minimal metacentric height, in
spite the fact that the permissible angle of heel in the
two methods (in case of fixed containers) is very
different. The exceptions are the cases where Method B
gives MGmin < 0.5 m. Namely, only the Method A
explicitly defines minimal metacentric height of 0.5 m
in the case of fixed containers onboard. It seems
necessary that the same requirement should be added
to the Method B. We propose and appeal for such an
improvement of the UNECE Stability Rules. If accepted,
Fig. 2. Additional stability levers of sample vessels

Vol. 2, No. 2, July 2006 Journal of Ship Technology 7
the minimal metacentric heights of the two methods
would differ only insignificantly.
6. THE PROBABILITY CURVES
As explained, the probabilistic tools enable us to
calculate the probability that the cargo hold of sample
vessels would be flooded, under the action of beam
gusting wind. In such calculations (for all the cases), the
flooding angle is taken as the angle at which the water
reaches the top of hatch coaming, and the average wind
speed is the one prescribed by the Static Wind-Heel
Criterion of UNECE Rules for zone 2, /v18ms
=.
It was shown in [1], that the most suitable function
to analyze in the context of container vessel safety is the
probability that the open container hold would be flooded
vs. metacentric height. We refer to such P - MG functions
as the probability curves. However, before the numerical
results of these curves are presented and analyzed, it is
necessary to define and emphasize some other important
items.
6.1 Probability in Safety Analysis
When discussing the probability that the vessel
would heel to some prescribed angle, one has to be
aware that the obtained value is extremely sensitive to
small differences in metacentric height and in the stability
curve. In some cases, even the changes in these properties
of the order of numerical errors involved, may
significantly influence the probability. So, if we aim to
use the probability as vessel safety criterion, we should
learn to understand it not as an exact number, but discuss
it in terms of its order of magnitude.
6.2 Storm Duration and Acceptable Probability
When calculating the probability of flooding, one
has to assess the appropriate period in which the vessel
is exposed to wind. This choice of storm duration is
closely related to the assessment of acceptable (permitted)
probability of flooding, as one of the most delicate tasks
in the following analysis.
As we had very little indication which value of
acceptable probability to select for inland vessel analysis,
we exploited some experience with seagoing ships. It
was shown in paper [1] that the probability of capsize in
two hours of seagoing ships with minimal safety
according to IMO Weather Criterion largely scatters
(from 10–2 to 10–5) depending on the ship type, her
dimensions, loading, bilge keels, etc. However, after a
careful analysis presented in paper [1], an acceptable
probability of capsize of O(10–3) in two hour, in beam
storm with fluctuating wind and waves, was proposed*.
Now, we go a step further, and adopt the same
probability of flooding, that is Pa = O(10–3) in two hours,
as the acceptable value for inland vessels also. If the
probability that the angle of flooding is reached in two
hours is greater than Pa , the inland container vessel
would be less safe than a seagoing ship in appropriate
beam storm conditions. It would be, therefore, considered
unsafe in the present investigation.
It should be recognized that the above choice of
acceptable probability of flooding is somewhat arbitrary.
For instance, Mc Taggart et al [9] proposed (by a very
different approach) values of order 10–4 as acceptable
probability of capsize for seagoing ships. The question
is, obviously, still open, and some new contributions to
the matter urgently needed.
However, the present choice could be justified by
the fact that the changes of acceptable probability (in
reasonable limits) would not principally alter the
conclusions, but influence some numerical details only.
In addition, the results of the analysis are put in a
graphical form in which any different choice of
acceptable probability could be easily analyzed.
6.3 Probabilistic Criterion
So, we can define now the probabilistic safety
criterion to be applied to inland container vessels, and
compared to the UNECE stability rules:
The probability that the container hold would be flooded
in 2 hours, in beam gusting wind of mean speed
18 m/s, should be of O(10-3).
The concrete values used: 2 hours of storm
duration, /v18ms
=, Pa = O(10-3) are, as mentioned,
somewhat arbitraryand still open for discussion.
6.4 Influence of Roll Damping.
The estimation of roll damping is, as known, one
of the most delicate tasks in any roll motion calculations.
* the order symbol “big oh” used here has the meaning: of order
10–3, or less

8 Journal of Ship Technology Vol. 2, No. 2, July 2006
On one hand, the results very much depend on the
damping coefficients used, and on the other (without
proper experimental data) one has only approximate,
rough means to assess these coefficients.
In paper [1], nonlinear damping following from
the seagoing Weather Criterion, was applied. That was
relatively rough, but simple method, relying basically
on the so-called Bertin’s damping coefficient. It seemed
correct for the seagoing ships, but the extension of the
method to the inland vessels may be doubtful.
In the present approach, the more advanced,
nonlinear Ikeda’s method was applied (see e.g. [9], [10]).
This semi-empirical method takes account of vessel speed
and details of her form. Although approximate, the
method is believed to be a reliable engineering tool for
roll damping prediction, and is incorporated in most of
the present seakeeping software codes.
In spite the fact that numerical tests showed a good
correlation between Ikeda’s and Weather Criterion
method, and did not give any unreasonable result, there
are still some doubts concerning the damping coefficients
used for the full forms of inland container vessels. The
authors can only regret for not having some experimental
results for the roll damping of the analyzed forms.
6.5 Influence of Terrain Roughness.
In applying the probabilistic approach, one has to
accept so-called terrain roughness coefficient
κ
. This
coefficient depends on the terrain category, and its
approximate values are given in Table 2 (see e.g. [5]).
In case of seagoing ships, the appropriate value of
roughness coefficient is obvious:
κ
≈ 0.0024. In the case
of inland vessels, the roughness is higher than in the
open sea, and changes as the waterway passes through
the different surroundings. It is, however, hard to asses
the proper value for inland vessel safety analysis,
especially as its impact on the final result turned to be
very significant.
It was decided, therefore, to use two different
values, which are believed to comply with the extreme
conditions on inland waterways. The value
κ
= 0.003,
appropriate for open smooth environment as the minimal
value, and
κ
= 0.015, appropriate for suburban areas, as
the maximal value.
6.6 Numerical Experiments
Now, after the previous remarks have been fully
understood, we can present and analyze the probability
curves found by the numerical experiments.
The experiments were carried out to find the
probability of flooding the open container holds of two
sample vessels in different loading conditions and
different environments. In these tests, the following
particulars were varied:
– Metacentric height, in wide range of technically
realistic values;
– Hatch coaming height, from the minimal value of
0.4 m prescribed by UNECE Rules, up to 1.6 m;
– Number of container layers, 3-5 for the large vessel,
and 3-4 for the small vessel;
– Terrain roughness:
κ
= 0.003 for smooth
environment, and
κ
= 0.015 for rough environment.
In all the experiments, the minimal freeboard of
0.6 m according to UNECE Rules, was assumed. Also,
in all roll damping calculations, the vessel speed of 16
km/h was accepted.
The obtained probability curves, for different hatch
coaming heights, are presented in Figures 3-12. In all of
these figures, in addition to the probability curves, the
range of unacceptable probability P > O(10-3) is marked.
Also, the minimal metacentric heights for fixed and
non-fixed containers (Methods A and B), according
to Table 1, are given. The rightmost probability curve
in all the diagrams presents the case of minimal
Table-2 : Coefficient of terrain roughness
Terrain category Range of
κκ
κκ
κ
Recommended
value
Open sea. 0.0019 - 0.0028 0.0024
Smooth flat country
Open country 0.0033 - 0.0075 0.0047
Sparsely built up
urban areas. 0.0075 - 0.030 0.013
Wooded areas
Densely built-up 0.017 - 0.025 0.022
urban areas
Centres of very 0.014 - 0.022 0.022
large cities

Vol. 2, No. 2, July 2006 Journal of Ship Technology 9
Fig. 8. Large vessel, 3 cont. layers, rough terrain
Fig. 6. Large vessel, 4 cont. layers, rough terrain
Fig. 7. Large vessel, 3 cont. layers, smooth terrain
Fig. 5. Large vessel, 4 cont. layers, smooth terrain
Fig. 3. Large vessel, 5 cont. layers, smooth terrain
Fig. 4. Large vessel, 5 cont. layers, rough terrain

10 Journal of Ship Technology Vol. 2, No. 2, July 2006
Fig. 12. Small vessel, 3 cont. layers, rough terrain
Fig. 11. Small vessel, 3 cont. layers, smooth terrain
Fig. 9. Small vessel, 4 cont. layers, smooth terrain
Fig. 10. Small vessel, 4 cont. layers, rough terrain
hatch coaming height according to UNECE Rules,
hc=0.4 m.
6.7 Discussion of the results
An overall view of the diagrams shows that there
are cases in which the minimal requirement of the Rules
(minimal MG and minimal hatch coaming height) does
not provide the required safety according to the
probability approach. These results are summarized in
Table 3. It can be seen that the discrepancies between
the two approaches are not as extreme as found in paper
[1]. It is, mainly, because UNECE Rules are somewhat
stricter than the previously applied rules due to different
definition of permissible angle of heel. Still, the
deficiency of the Rules from the probabilistic point of
view (especially for fixed containers, large vessel, rough
terrain) seems obvious.
To provide the required safety, that is to reduce the
probability of flooding to desired value of O(10–3) in
two hours, either metacentric height, or hatch coaming
height should be increased above the value prescribed
by the Rules. The necessary increase of hatch coaming
height, for the minimal MG, is presented in Table 4.
The necessary increase of metacentric height, for the
minimal hc, is given in Table 5.
A detailed analysis of the results given in the tables
and diagrams shows the following.
– Both vessels, if fulfilling the minimal stability
requirements of the Rules, are considerably safer

Vol. 2, No. 2, July 2006 Journal of Ship Technology 11
if intended to carry non-fixed containers, because
of the explicit demand for MGmin > 1 m.
– The increase of terrain roughness, at the same mean
wind speed, significantly decreases the safety of
the vessels. In other words, the vessel fulfilling the
minimal stability requirement of the Rules, would
be less safe when passing the urban (rough) areas,
than in the open country.
– Small vessel, fulfilling the minimal requirements
of the Rules, satisfies most of the probabilistic
safety requirements, also. The only exception is
the case of fixed containers in rough environment.
In that case, either the hatch coaming height, or
the metacentric height have to be increased above
the minimal requirements of the rules.
– Large vessel is, generally, less safe than the small
vessel. If such a vessel has the minimal
requirements of the rules (minimal MG, minimal
hc), it would be safe enough in the case of non-
fixed containers and smooth environment, only.
The results indicate that the proper way to improve
the Rules is to increase the demands for minimal hatch
Large vessel, minimal MG according to Table 1
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
5h
c > Cannot be hc > 1m
0.55 m satisfied
4h
c > Cannot be hc >
0.55 m satisfied 0.9m
3 hc >
1.2 m
Table 4. Increase of hc to provide the necessary
probabilistic safety
Small vessel, minimal MG according to Table 1
Layers Fixed Non-fixed Fixed Non-fixed
cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
4h
c>1.5 m
3h
c>0.6 m
Large vessel, minimal hc = 0.4 m
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
5 MG > MG > MG >
1.25 m 1.9 m 1.9 m
4 MG > MG > MG >
0.8 m 1.3 m 1.3 m
3 MG >
0.9 m
Table 5. Increase of MG to provide the necessary
probabilistic safety
Small vessel, minimal hc = 0.4 m
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
4 MG >
1 m
3 MG >
0.6 m
Table 3. Probabilistic safety of vessels satisfying the
minimal requirements of UNECE Rules
(✓ satisfied, X not satisfied)
Large vessel, with minimal MG (according to Tab. 1)
and minimal hc = 0.4 m
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
5X✓XX
4X✓XX
3✓*✓X✓
Small vessel, with minimal MG (according to Tab.
1) and minimal hc = 0.4 m
No. Fixed Non-fixed Fixed Non-fixed
layers cont. cont. cont. cont.
smooth smooth rough rough
terrain terrain terrain terrain
4X✓
3✓*✓X✓
* If MGmin > 0.5 m, in Method B, as recommended.

12 Journal of Ship Technology Vol. 2, No. 2, July 2006
coaming height. This would also be very much in
accordance with the other requirements that hatch
coamings have to fulfill. Namely, in addition to UNECE
Rules, which should provide the necessary safety against
flooding, the hatch coaming participates in vessel
strength, providing the minimal sectional modulus
prescribed by the classification societies. The hatch
coaming also protects the crew on the narrow gangway
from falling inside. This last requirement (usually) sets
the rail height of, at least, 1.1 m. So, the typical solution
found on the open-hold container vessels is to satisfy all
the three demands by watertight hatch coamings, which
are 1.1 - 1.3 m high.
Such a situation seems to explain why the stability
and hold-flooding of the vessels are not jeopardized as
much as expected, in spite of the deficiency of the Rules.
There are a number of other interesting items that
a careful reader could deduce from the presented
diagrams. For instance, there is a strange decrease of the
probability in case of small metacentric heights, (see
Fig. 7 and Fig 11). The explanation of this, and some
other effects is, however, left for some other occasion.
7. CONCLUSIONS
In the present investigation, nonlinear rolling of
inland container vessels due to gusting beam wind was
calculated, and the probability that their open container
hold would be flooded was found. In accordance with
the analysis for seagoing ships given in paper [1], the
probability of Pa = O(10-3) in two hours was accepted
as the safety criterion. The new probabilistic criterion
for vessel safety was defined, compared to the recent
version of UNECE Stability Rules, and the correlation
of the two approaches thoroughly analyzed.
In previous paper [1], a similar probability approach
was applied and compared to inland stability rules [2].
Although the analysis was only brief (the investigation
was concerned with some other aspects of ship stability)
it showed an excessive disagreement between the two
approaches. The vessel satisfying the stability rules was
found to flood and capsized in numerical test for less
than ten minutes; she was extremely unsafe from the
probabilistic point. The UNECE Stability Rules used in
the present investigation were found considerably more
strict than the rules [2], mainly due to the different
definition of permissible angle of heel. This angle,
according to UNECE Rules, is the angle of deck
submergence, while the rules [2] (similar to number of
other rules, e.g. [4]) consider, as permissible, the angle
of flooding. The UNECE Stability Rules therefore
showed much better correlation with the probabilistic
approach. Still, these rules did not provide the necessary
safety against flooding in all the cases analyzed.
With all the risks of generalizing the results
obtained from the analysis of just two sample vessels,
the presented analysis proves that the container holds of
the vessels with minimal safety against flooding
according to UNECE Rules (the vessels having the
minimal values of metacentric height and hatch coaming
height) could be flooded and capsized in beam storms
prescribed by the Rules. Such result shows not only
the vulnerability of inland container vessels to severe
beam wind, but also indicates some insufficiency of the
applied stability standards.
To reduce the probability of flooding to the
acceptable value of O(10-3) in two hours, either
metacentric height, or the hatch coaming height should
be significantly increased above the value prescribed by
the Rules.
The analysis of the sample vessels shows that if
the hatch coaming height remains minimal (0.4 m), the
metacentric height in some cases has to be increased to
some technically unrealistic values. For instance, for large
sample vessel with 5 layers of containers onboard, in
rough surroundings, MG should be over 1.9 m. Such
high MG could hardly be achieved by suitable vertical
load distribution, so the only way to increase the safety
would be to remove the top layer of containers.
On the other hand, the probability of flooding could
be considerably reduced, if the hatch coaming height is
increased to the values which are (because of the other
requirements) already in practice. For instance, in the
case of usual hatch coaming heights of 1.1 - 1.3 m and
the minimal values of MG according to stability rules
for non-fixed containers, the probability of flooding of
both sample vessels, regardless of the terrain roughness,
would be of the required order.
Therefore, a suitable way to improve UNECE
inland stability rules is to increase the minimal
requirement for hatch coaming height. The best choice
would be, perhaps, to adjust that requirement to the crew-

Vol. 2, No. 2, July 2006 Journal of Ship Technology 13
safety standard of 1.1 m. To repeat, the hatch coamings
of 1.1 - 1.3 m are already in practice because of the
structural, and the crew-safety requirements. Thus, the
more stringent demand of UNECE Rules would only
formalize the already used technical solutions.
If the hatch coaming height is increased as
proposed, the sample vessels carrying non-fixed
containers would become safe enough in the beam
storms. However, in case the vessels carry fixed
containers, the increase of hatch coaming heights would
not always be sufficient. According to the present
analysis, their minimal MG should also be increased to
approximately 1.3 m.
The reader should remember the dilemma of paper
[1]: why there are no accidents of inland container vessels
in spite of anticipated high probability that the container
holds would be flooded? Ths dilemma actually triggered
the present investigation. The answer is in typical hatch
coaming heights used on inland container vessels.
Namely, most of the vessels have hatch coamings higher
then prescribed by the Rules, because of the other
(structural, crew-safety) requirements. Therefore, they
are, at least in case of non-fixed containers, safe enough
according to the present probabilistic approach, also.
The present detailed analysis proved the
shortcoming of the inland stability rules detected briefly
in the paper [1], indicated the possible way to improve
the rules, and gave a probable reason why there are no
accidents of inland container vessels, in spite of the
deficiency of the rules. However, some questions remain
open for further investigation.
The investigation indicated terrain roughness as a
very relevant parameter for the inland vessel analysis.
Namely, in contrast to seagoing ships, the roughness of
waterway surroundings (open country, sparsely or densely
built urban areas, etc) significantly influences the vessel
safety. In the present analysis, the two extreme
alternatives (smooth and rough terrain) were investigated,
both for the same mean wind speed prescribed by the
stability rules. It is not fully clear, however, if it is correct
to assume the mean wind speed independent of the terrain
roughness? Should the mean wind speed be reduced (and
how) as the terrain roughness is increased?
In one other sense, the present analysis is, also,
incomplete. It used the model of motion with a single
degree of freedom - the rolling only. The more complex
model of motion, taking account of vessel drift due to
steady wind, and her sway due to wind gusts, would,
undoubtedly, give more realistic results. Some
preliminary results indicate that the rolling of vessels
obtained by such model would be somewhat reduced,
compared to the present analysis. That would present
the inland stability rules in somewhat better light and
(perhaps) moderate some of the criticism given in this
paper.
As stated earlier, the present paper is a part of
much broader research trend, aiming to substitute the
present stability rules by the new approach based on
seakeeping and probabilistic analysis. Such an approach
(we believe) has the potential to predict and rank the
vessel safety against capsizing much more accurately.
However, before the new criteria of the form presented
in paragraph 6.3 are developed and substitution of
present stability rules proposed, there is a long way ahead.
It includes numerous systematic studies on large
variety of ships and clarification of many open problems,
some of which (acceptable safety, number of degrees of
freedom, terrain roughness) are already mentioned in
the previous text. We can only hope that the presented
investigation contributes towards that general aim.
8. ACKNOWLEDGMENTS
The paper is a part of long-term project
“Development of New Generation of Inland Cargo
Vessels” executed by Department of Naval Architecture,
Faculty of Mechanical Engineering, University of
Belgrade and Institute of Technical Sciences of Serbian
Academy of Science and Art. The project is partly
financed by Serbian Ministry of Science and Ecology,
Contract No. TR-6317A.
9. REFERENCES
1. Hofman, M. and Backalov, I., “Weather Criterion
for Seagoing and Inland Vessels - Some New
Proposals” Proceedings of International Conference
on Marine Research and Transportation,
ICMRT’05, University of Naples “Federico II”,
2005, pp. 53-62.
2. Yugoslav Register of Shipping, “Classification
Rules for Inland Waterways”, Part 4, Belgrade (In
Serbian).
3. United Nations Economic Commission for Europe

14 Journal of Ship Technology Vol. 2, No. 2, July 2006
(UNECE), Working Party on Inland Water
Transport, “Amendment of the Recommendations
on Technical Requirements for Inland Navigation
Vessels”, March 2006.
4. European Agreement Concerning the International
Carriage of Dangerous Goods by Inland Waterways
(ADN), Part 9.
5. Lungu, D. and Rackwitz, R., Joint Committee on
Structural Safety Probabilistic Model Code, Section
2.13, Wind, Part 2, Loads, 1996.
6. Bulian, G., Francescutto, A., “A Simplified Modular
Approach for the Prediction of the Roll Motion
due to the Combined Action of Wind and Waves”
Proceedings. Inst. Mech. Eng. Vol. 218, Part M,
Journal of Eng. for the Maritime Environment 2004,
pp. 189-212.
7. Vassalos, D., Jasionowski, A., Cichowicz, J.,
“Weather Criterion - Questions and Answers”,
Proc. 8th STAB, pp. 695-705, Madrid, 2003.
8. Hofman, M., “Inland Container Vessel: Optimal
Characteristics for a Specified Waterway”,
Proceedings of International Conference on Coastal
Ships and Inland Waterways, RINA, London, 2006.
9. Himeno, Y., “Prediction of Ship Roll Damping -
State of the Art” Report No. 239, College of
Engineering, University of Michigan, 1981.
10. Chakrabarti, S., “Empirical Calculation of Roll
Damping for Ships and Barges”, Ocean
Engineering No. 28, 2001, pp. 915-932.
Milan Hofman is professor at Department of Naval Architecture, Faculty of Mechanical Engineering, University of
Belgrade, Serbia. He teaches Theory of Ships (buoyancy and stability), Seakeeping, Ship Dynamics and various other
subjects of shipbuilding to undergraduate and postgraduate students. With more than thirty years of teaching and
research experience, he has educated numerous generations of Naval Architects. His main research interests are
Marine Hydrodynamics (especially wave resistance in shallow and restricted waters), Stability, Seakeeping, etc.
Ivan Maksic is a young researcher at the Institute of Technical Sciences, Serbian Academy of Science and Art,
Belgrade. He is a candidate for Ph.D. studies at Department of Naval Architecture, Faculty of Mechanical Engineering,
University of Belgrade, and this is his first paper.
Igor Backalov is a young teaching assistant and Ph.D. student at the Department of Naval Architecture, Faculty of
Mechanical Engineering, University of Belgrade. He lectures on Theory of Ships (buoyancy and stability), Seakeeping,
Ship Systems and Ship Design. His main research interests are Ship Stability and Seakeeping.
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