INITIAL METACENTRIC HEIGHT OF SMALL SEAGOING
SHIPS AND THE INACCURACY AND UNRELIABILITY OF
CALCULATED CURVES OF RIGHTING LEVERS
Introduction
Since the last war thc interest in the behaviour of ships at sea is increasing in many countries. The subject is complicated, for the movements of a ship in rough water are the result of six simple motions,
viz. 3 rotations (rolling, pitching, yawing) and 3
translations (heaving, surging, drifting). Often,
these 6 motions occur simultaneously but in many
cases some of them are of minor magnitude or
negligible. Anyhow the subject is approached by investigating each component separately, in three
ways:
by collecting and analyzing service
perform-ance data;
by carrying out model experiments;
by a purely mathematical treatment of the
phenomena.This paper is limited to the rolling of actual
ships, and takes into consideration practical
questions of transverse stability.
What are the aims and purposes of collecting and analyzing service performance data?
The answer is that we want a true picture - in
numbers and frequencies; in averages, maxima and minima - of the behaviour of many ships, on many trade routes, under varying weather conditions. The analysis of these statistics will disclose the dominant
factors, dominant either in magnitude or in
fre-quency (or both).
No doubt one of these dominant factors is the
rolling of the ship. What is meant by "safe rolling", "easy rolling" or "a stiff ship"? The answer to these questions requires not only technical and nautical data, but also a personal appreciation of the vessel's behaviour by her master and deck officers.
In the author's opinion, the interest of to-day's
deck officers of the merchant navies in the subject under consideration can be raised, and they will be
willing - at least occasionally - to make and
re-cord the observations asked for in
a specimen questionnaire such as shown in Fig. 1, and give their comment on the ship's behaviour.It seems to be wise to start on such a limited
scale. The scientist would like to ask for much more and more exact information. But in this stage our aim is to make deck officers research-minded by asking their cooperation in the collecting of data
by
Prof. Ir. J. W. BONEBAKKER
with which they are professionally well acquainted; the practical value of the results of the analysis of these data should be obvious to them.
Initial ;uetacc'ntric height
Rolling as an aspect of stability is of particular
importance for smaller ships intended for short crossings and coastal trading. One yardstick is
proposed for judging both rolling and stability; the initial metacentric height. For every ship and for every displacement there is a lower limit to GM,
imposed by considerations of safety, and an upper limit imposed by considerations of rolling behaviour. In some cases the latter is smaller than the former. An example of a normal and of an exceptional case are given below.
The upper limit
Every analysis of statistical material should al-ways begin by checking the data, that is, to decide whether they can be accepted as correct, or must be rejected as erroneous. This has been done for the data of questionnaires as per Fig. i by calculating the radius of inertia from the wellknown formula:
0.554 i
= \/
GM (British units) or(metric units)
where:
t the oscillating period (P to SB),
i radius of inertia
GM initial metacentric height,
t and GM being known from data records.
For merchants ships, i varies approximately from 0.35 B to 0.38 B. Observations resulting in i values
smaller than (say)
0.33 B or larger than (say)
0.40 B should be rejected.
The next step is to prepare frequency diagrams of: displacement-ranges;
GM values for each range of displacement. The first will disclose the displacements with the highest frequencies, and these are the most
impor-tant from the owners point of view.
In the GM frequency diagrams, the appreciations of the ship's rolling behaviour are noted, and the
upper limits for GM become apparent.
4 QUESTIONNAIRE i. Ship data: Name: Owners:
L (b.p.) X B (mid) X D (mid):
Superstructures:Load draught (extreme): Displacement at load draught:
2. Voyage:
From: date
To: date
3. Initial GM:
GM
How was GM computed: from inclining experiment? from mean rolling period?
e. otherwise? how?
Ship's trim at this GM: draught aft:
draught forward:
displacement:
Are GM and displacement corresponding to:
mean draught? actual trim?
4. Ship's behaviour:
Rolling period (P-SB) mean: seconds
maximum: minimum:
Amplitude mean: degrees
maximum: minimum: Number of observations:
5. Personal appreciation of rolling behaviour:
easy:
(rne)
good:
(= g)
rather stiff: ( rs)
too stiff:
(= ts)
Height above keel of the observer's position:
'Was the rolling at any time considered unsafe? If so, please state rolling period and amplitude, and any other relevant facts and considerations.
The lower limit
The fixing of the lower limits for GM requires
the judging of the stability curves for the range
of displacements under consideration, and for GM values ranging from "pius zero" to the upper limit-ing values or more.
Rahola [1] has formulated criteria for judging
stability curves of sinai! vessels in particular; these criteria are based on the analysis of more than 30 disasters, where stability played an important part. Some 20 of these vessels had a gross tonnage less than 1,100 RT. His criteria are concerned with:
= the capsizing angle of heel, fixing the range of the curve of statical levers; = the angle of heel where the lever has
its maximum value;
GZ for q = 20 - 30 - 40 degrees;
(see fig. 6)çt = the angle of heel which should never
occur under service conditions;
d dynamical stability at ço. His criteria are:
minimum 60° Pm 35°
GZforç=20°
5'/2"=l4cm
GZ for = 30°and4ø° 8" 20cm ç°r smaller than 400 çrÇtr smaller than the angle of heel where
non-watertight openings immerse
d at Pr
minimum 3.15" 8 cm.o
8
o
2
45 55E 65 75E <85e <95
Fig. 2a o
4
2o
o u o J C t' a. t, tIt will be noted that Ráhola does not propose
explicitly any minimum for initial GM. Neverthe-less his criteria imply a minimum GM for each in-dividual type and size of vessel.
20 IB o u 6 C t' o a-t' 2x9
I200<l300 ton,
34 observations. GM n cm.40e <50e <60e <70e <80e <90e <100e
Fig. 2b Sxg e. I300e.< 1400 ton. 57 observations. 3cc 9 9 rs GM in cm.
40e <50e <60e <70e <80e -'90e <lOO
6
o >
t
Two Examples
Open Shelterdeck Coaster
In Fig. 2a-2d, frequency-diagrams of GM-values are shown for 3 ranges of displacements, and for all displacements for which data were available. Per-sonal appreciations of the vessel's rolling behaviour
are noted along each line.
Generally speaking, the maximum GM-value for easy rolling is about 26 inches or 65 cm.
Fig. 2e shows the high frequency of loaded dis-placements ranging from 1300-1400 tons (57 trips out of 143 or 40 per cent.). From Fig. 2c it will be
26 24 22 20 IB 16 14 12 IO Q B 6 4 2 Fig. 2d
seen, that 21 loaded trips out of a total of 57 (37 per cent.) were made with GM-values ranging
from 20-24 inches or 5 0-60 cm. The vessel's curve
of statical levers is adequate even with a GM of
16 inches or 40 cm.
The ship's weather deck hatchways are closed by patent rolling steel hatch covers. She has proved herself to possess excellent seakeeping qualities. Raised Quarterdeck Coaster
In Figs. 3a-3e, frequency-diagrams of GM-values are shown for 4 ranges of displacements, and for all displacements for which data were available.
8
sonal appreciations of the vessel's rolling behaviour are noted along each line. Generally speaking, the maximum GM-value for easy rolling is about 32 inches or 80 cm.
Unfortunately, at a load displacement of aboat 1800 tons, an initial GM of 32 inches entails an in-adequate curve of statical levers, because:
its range is only 53 degrees;
the lever reaches its maximum at an angle of heel of only 22 degrees, the corresponding dy-namical lever being only 1.85 inches or 4.7 cm. An adequate curve of statical levers can only be
obtained by increasing the initial GM to ± 47
inches or 120 cm.
At smaller displacements an adequate curve of statical levers
can be obtained with
a smallerinitial GM:
Fig. 3f shows the high frequency (26 per cent.) of displacements ranging from 1700-1800 tons. It will be understood that the vessel in question can hardly be considered as a successful design. Fully loaded she will be either too stiff, or not quite safe in rough weather. 100 a " 1300 ton 42 observatiOns 2a t n ts GM in cm. Is
The need for reliable and accurate stability curves Fig. 4 shows the experimental curve (see Appen-dix A) of statical levers of a 150 ft. coaster having a watertight poop, and two calculated curves. The latter were prepared with the utmost care by two
different members of the staff, fully acquainted with the job. In both cases Fellow's integrator
method was used. The differences between the three
f
8
4
2
45$ "55$ "65a 75 "85e "95e "IOSa «I 5a»125 Fig. 3c 300 " 1500 ton. 25 observations 2* g rs ts ts ts GM in cm. 500m "1700 ton. 29 observations Is G M In cm. 2v , ts 700 1800 ton. 39 observations
Al
A GM in cm. displacement, tons minimum GMH--F 1800 1700
± 47"
± 39"
=
=
120 cm 99 cm ± 1500± 27" = 68 cm
45e "55$ "6S <75e "85$ 'n95 '105$ "115' I25 45$ "55$ "65e "75 "85a "ÇSa "lOSa "1154 "125
Fig. 3a Fig. 3d o t, o t) V 8 6 4
f2
45$ «55 "65 "75$ 85a 'vOS$ I05$
12
curves are truly amazing. This can be gleaned also from table 1.
TABLE i
model without hatchways
all changes being by the head
Rahola's minimum for the dynamical lever at 40 degrees is 8 cm; the actual dynamical levers are
13.2 (curve II), 11 (curve I) and 10.1 cm (experi-mental curve).
It should be mentioned here that when deter-mining the experimental values of the statical
levers, the model was free to change its trim during heeling. The changes of trim arc shown in table 2.
TABLE 2
model with hatchways
At first sight an initial GM of 26.5 inches or 67 cm in conjunction with stability curve II might be considered adequate. But the experimental curve
shows conclusively that the ship is not safe: the statical lever at 30 degrees is about 25 % below
Rahola's minimum, and 8 % at 40 degrees. These
deficiencies are not compensated by the statical
lever at 20 degrees, which is about equal to Rahola's minimum, nor by the excess in dynamical stability
up to 40 degrees.
Anyhow, the calculated curve II gives a grossly exaggerated and misleading picture of the vessel's stability; up to 40 degrees heel its area is 30 % in excess of the area of the experimental curve.
This example shows conclusively that in all those
cases where the stability of a vessel seems to be
doubtful, a reliable and accurate method of com-puting curves of statical levers is simply indispen-sable. The most painstaking calculations are neither reliable nor accurate; Fig. 4 confirms the views of Prohaska [2], Schepers [3] and Steel [4]; several
other striking cases have come to the author's
knowledge.The unreliability and the inaccuracy of calcula-ting methods being beyond doubt, it does not seem
advisable to supersede them by quick approxima-tions, which would lead from bad to worse.
Approximations may be useful in the preliminary stages of a design, provided that they are handled by experts. For several common types of vessels, such as large-freeboard dry cargo ships and tankers,
both of medium and large size, the stability will
always be ample, provided their proportions and
initial GM are within the boundaries of average
practice. But for a great variety of other ships, in
particular those of moderate and small size, an
accurate and reliable curve of righting levers is in-dispensable, if a fair judgment is to be passed on the vessel's stability. Model experiments are the oniy means that meet the case.
A description of the apparatus for measuring the
righting moments of ship models is given in
appendix A.
Inflnence of erections on stability and design As often as not, the poop of a coaster is not an effective watertight erection, and can not
contrib-ute to the vessel's stability. And it would not be safe to treat the cargo hatchways as watertight
erections when they are closed by shifting beams,
wooden hatch covers and tarpaulins. These con-siderations add weight to the conclusion of the
preceding paragraph, i.e.
that the true
(the ex-perimental) curve of statical levers of Fig. 4 is proofthat the
vessel's stability is inadequate. In the design stage this should have revealed the necessityto increase her beam. No doubt this would have
been detrimental to her rolling behaviour.
If, on the other hand, the cargo hatchways are fitted with watertight steel hatch covers, and the
poop designed and fitted out as an effective water-tight erection, the curve of statical levers is greatly improved, as shown in Fig. 5. Such hatchways
con-tribute materially to the area and range of the
curve; it exceeds Rahola's minima. The vessel might now sail safely even with a smaller GM.
Clearly, the inciusion or exclusion of effective superstructures and hatchways has a very important bearing on the stability curve of small vessels; in
the design stage this may be a deciding factor in
selecting their principal dimensions. This proves again that a reliable and accurate method for com-puting curves is indispensable. Only model exper-iments comply with these requirements.
Concluding remark.s
1. Causes of the inaccuracies of methods for
cal-culating stability curves.
At the time when the methods for calculating stability curves now in common use were
con-ceived, their accuracy was taken for granted, angle of heel, degrees 20° 30° 40°
statical lever, curve II, cm 18.2 20.0 27.0
,, ,, , ,, I, erri 13.6 18.4 23.3
experimental curve 14.6 15.1 18.4
Rahola's minimum 14.0 20.0 20.0
14
on the ground that the principles underlying these methods were correct.
In later years it appeared that these curves were always more or less approximate, even when they had been prepared with great care.
Ráhola has emphasized the importance of having accurate standards of minimum stability, especially for smaller vessels. But he seems to be
unaware of the uncertainties inhaerent to all time-honoured calculating methods.
The causes of these inaccuracies may be briefly summarized:
an insufficient number of stations, water-lines and inclinations is used;
cross curves are faired through an insuffi-cient or ill-positioned number of spots;
"k" values are plotted instead of the MS
values (residuary stability) proposed by
Prohaska [2], see Fig. 6;
the scale of the body plan is restricted by the size of the planimeter or integrator to
be used;
errors in the readings of the planimeter or the integrator, even if taken several times for the same area;
the influence of the hull's ends
(cruiserstern, propeller aperture, bossings, raked
stem) is often neglected;
the influence of sheer, camber, hatchways
and watertight erections is neglected or
treated arbitrarily;
changes in trim are not taken into account. 2. Instruments for computing experimental curves
of static levers for ship models.
In the eighties, John Heck devised an apparatus
for ascertaining accurately the position of a
vessel's centre of buoyancy for any displacement, heel or trim, taking into account the influence
of camber, sheer, hatchways and effective superstructures. Its principle can be found in
the Transactions of the Institution of Naval
Architects, 1885.
A mould was made, exactly to scale, of the
ship's hull and effective erections. This mould could be filled with a quantity of water
equiv-alent to the displacement representing the
vessel's condition under consideration.
The empty model was fixed to the apparatus,
which was a sort of balance, and the exact
position of its centre of gravity ascertained ex-perimentally. Then the model was partly filled
with water, and the position of the centre of
gravity of mould and water ascertained over a range of heels. This experiment could be
re-peated with different quantities of water, covering the whole range of displacements from
zero to totally immersed vessel.
Fig. 6
The apparatus described by Heck in his paper
of 1885, made entirely of wood, was of very
primitive construction.
It is rather curious to read that, at the time, the
accuracy of the methods for calculating sta-bility was taken for granted, it being greatly appreciated that the experimental curves did
not differ materially from those calculated! Nowadays we should take the opposite point of
view.
A more elaborate apparatus for model experi-ments on stability is described by Werckmeister
[5], see appendix A. A similar apparatus, of
improved construction, has been acquired by the Technological University of Delf t. Floating
models are tested by registering the heeling
couple at specific angles of heel. The model is free to change its trim, just like the actual ship, and these changes can be recorded too.
3. Priority of smaller types of seagoing ships. It has already been stated that an accurate curve of righting levers is essential for smaller
sea-going ships - such as coasters, trawlers and tugs - in particular when they are of the
low-freeboard type with partial
superstructures. Generally speaking such vessels are employed inshort trades, and it is quite possible to collect
valuable information about their behaviour within a reasonable time.
The necessity of exact information about right-ing levers, and the possibilities to collect data
about seakeeping qualities, are far from
be-come clear that it is no use to fill in simple
questionnaires (Fig. 1), when the amplitudes of roll are less than, say, 5 to 7 degrees (10 to 14 degrees from P to SB). But on the South Amer-ica-U.K./Continent route the chances of meet-ing heavy weather in European waters should
not be neglected. Every opportunity should
then be taken to collect information about the behaviour of such a vessel in bad weather, and
to compare this information with an exact
curve of righting levers for the loading con-dition in which she is sailing. Such data, cover-ing a wide range of ships, should be helpful to the design of more seakindly ships.
4. Future possibilities.
It might be asked whether instruments might become available - within 5 or 25 years - just as foolproof as the camera of a fighter or bom-berplane, that would record automatically, to a time base, the characteristics of all the ship's
motions, and' at commercial prices. Nobody
can give a positive answer to this question. But even if it were considered probable that, in the
Fis. 7
course of the next years, such instruments will
be developed, the merchant navy should not wait for them, but start now collecting data
about seakeeping qualities,
in the rough and
simple ways advocated in this report. Close cooperation with the scientific staff of the ex-perimental tanks is essential. The answers to the questionnaires should be available to them, and
they should advise about the drafting of
ad-ditional questions or questionnaires, as soon as
other phenomena are to be tackled, such as
heaving, pitching, course keeping, shipping of
green seas.
Acknon ledgement
The author wishes to express his indebtedness to
Mr. P. A. van Katwijk, student, for his excellent
assistance in preparing this report. APPENDIX A
THE MOMENT INDICATOR
16
Fig. 8
The moment indicator is, in principle, a balance for moments (see Fig. 8).
The torque required for balancing a model at a certain angle of inclination, is:
p ng sin = p (mg + mn) sin cç
where:
p the displacement;
in the true metacentre; and
n = the false metacentre.
The moment indicator is unloaded in neutral
equilibrium with respect to the axis A. By means of
the weights Q and q the apparatus exerts on the
model a torque which is equal and opposite to the
stability torque of the model. By balancing the
moments for increasing angles, the statical stability of the model can be determined readily.
The metacentric height mg of the model can be determined in two ways:
1. The more accurate method, by carrying out a stability test, during which the angle of in-clination is measured, with a high degree of
accuracy, by means of an optical method.
2.
If the values (mg ± mii)
as measured areplotted against (/ as base, fairing of this curve
to j = O will yield the value of mg, since, if
O, in and n will coincide.
The righting arms of statical stability for the
ship will follow from:
(MG + MN) sin ç = (MG ± a mn) sin q
where:a = the model scale; and
MG the metacentric height of the ship.
The advantages of this method of measurement over a calculation are the following:
The model is free to trim, so that alterations in trim during heeling can be automatically taken into account.
The influence of forecastle, bridge, poop and any other parts of the superstructure that may have to be included in the measurement can be taken into account in a simple manner.
When a model is once available, different con-ditions can be examined in a short time. Measurement is done with a high degree of
ac-curacy; a calibration of the apparatus with a
rectangular model, whose stability is easy to calculate, showed that the maximum error of
the stability torque measured is less than i per
cent. References
Ra'hola, J.: "The Judging of the Stability of Ships and the Determination of the Minimum Amount of Stability". Helsinki, 1939.
Probas/ca, C. W.: "Residuary Stability". Trans. I.N.A., 1947. Schepers, J. A.: "Stabiliteitsgegevens". Schip en Werf, 1956,
blz. 166.
SIed, H. E.: 'The Practical Approach to Stability of Ships".
Trans. I.N.A., 1956.
. Wìerckmeister: "Stabilitätsuntersuchungen mit dem Model eines
