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The Radii of Gyration of Merchant Ships
Predicting the motion response of a ship at sea requires,
among other things, values for the respective radii of
gyration with reference to the three principal co-ordinate axes through the centre of gravity of the ship. Mr Peach
began his presentation by pointing out that, until the
development of modern computers, direct calculations for radii of gyration were considered to be too time consuming in relation to the value of their application. Instead, various empirical formulae were developed and have been used formany years. In fact, as late as 197.7 Mr Peach discovered,
from an approach to the US Maritime Administration
(MarAd), that no directly calculated values were available. He suggested that a computer program should be developedfor this purpose and MarAd agreed, adding two
require-ments. The program was to be produced as an entity i.e. not
trying to add radii of gyration calculations to any existing
program for weights and centre of gravity. They also
wanted the program to produce values for weight per foot of ship length as a longitudinal distribution.Apart from the use of the program to perform direct
calculations, Mr Peach thought that it was useful to
consider the results in comparison with some existingempirical formulae. First he presented the simplest type
K =C1 B
where K is radius of gyration for roll B is breadth of ship
C1 is a coefficient dependent upon ship type. Traditionally, values have been quoted forC between 0.40
and 0.44. References i and 2 give corresponding values in
the form of C = l.108C1. He then considered the radii of gyration for a thin-walled rectangular tube, which has a
radius of gyration
K = (B+D)/jT
or
C1 = (1 + B/D)/AJ12
This form led him to suggest that perhaps (B + D) may be a better parameter than the frequently used (B2 + D2)+
Another equation given by K ato3 is
(K/B)2=Cl2=F[CbCU+ l.IOCU(l-Cb)
(HIT -2.20) + (H/B)2 J where
= Block coefficient
= Deck area coefficient = (Deck Area)/LB = Equivalent hull depth = D + A/Lp
= Profile projected areas of erections and deck-houses
= Draft
= Beam
= Constant for the ship type
= 0.125 for passenger, passenger and cargo, and
cargo ships = 0.1.33 for tankers = 0.177 for whalers
Finally, Mr Peach quoted from his own report4 to MarAd:
K = 0.30 (B2 + D2)+
based on the results of the full-length calculations by
computer. TECHNISCHE UNIVERSITEIT Laboratorium vooc Scheepshydromechanlcs Archief Mekelweg 2,2628 CD D&ft TeL: 015- 786873 - Fax: 015- 781836R W PEACH, MSE, PE, CEng. Member Na va/Architect and Marine Engineer, USA A K BROOK, BSc, MSc, CEng, British Maritime Technology Ltd
Report on a presentation and discussion 1 6th February 1987
115
So far all the equations refer to radius of gyration for
roll. The computer program was also used to obtain values of radii of gyration associated with pitch and yaw, and Mr Peach went on to describe the computations in some detail.
Three ships, each of a characteristically different type,
were used in the investigation: a Roll On/Roll Off (Ro/Ro),
a Lighter Aboard Ship (LASH), and an LNG carrier.
Particulars of the ships are given in Table
iand the
calculated radii of gyration are given in Table 2. Mr Peach
Table I-Ship characteristics
* Not including cantilevers at stern At side
La = Length overall
= Length between perpendiculars
Table 2-Radii of gyration
explained that the calculation used relevant information for
every individual piece of structure, outfit and equipment,
involving between 20000 and 40000 items. The results are
converted into coefficient form associated with various parametric terms, and presented in Table 3. For
compari-son, values of Kr/B estimated from Kato's equation are
included in Table 3. These are considerably higher than the computed values and can only be correlated by using much
SHIP TYPE Ro/Ro LASH LNG
Builder Avondale Bath Avondale
Design C7-5-95A C8-5-8lb LG9-5-107a
La, Feet 684.00 772.00 931.50 Metres 208.48 235.31 283.92
L,
Feet 640.00 724.00 887.00 Metres 195.07 220.68 270.36 B, Feet 102.00 100.00 140.50 Metres 31.09 30.48 42.82 D. Feet 69.50 60.00 94.00 Metres 21.18 18.29 28.65 Displ. Tons 15805 14874 30298 Tonnes 16059 15113 30784 LCG Feet 382.20 400.94 477.05 Metres 116.49 122.21 145.40 VCG Feet 38.57 36.59 48.68 Metres 11.76 11.15 14.84 TCG Feet -0.16 0.14 0.18 Metres -0.049 0.043 0.055 KM Feet 52.20 58.30 118.24 Metres 15.91 17.77 36.04 GM Feet 13.63 21.71 69.56 Metres 4.154 6.617 21.20RollPeriod, Seconds
T=2
7TKr10.89 8.81 6.70 /g GM
Feet Metres Feet Metres Feet Metres RO/RO 36.2732 11.06 187.8276 57.25 187.3549 57.11
LASH 37.0488 11.29 205.0905 62.5 I 204.9517 62.47
LNG 50.4408 15.37 239.2502 72.92 239.9572 73.14
SHIP Kr (Roll) K (Pitch) K (Yaw)
Cb C H A T B F
Table 3-Radii of gyration coefficients
L = Length between perpendiculars
La = Length overall
smaller values of the coefficient F, compared with those
recommended by Kato.
From the same table it can also be seen that the
calculated values of radii of gyration Kp and Ky, for pitch and yaw, are significantly higher than the frequently used K
=0.25L.
Mr Peach concluded by emphasising that the computed values did not take into account the effect of entrained
water in the context of added virtual mass.
A complementary presentation was then given by Mr Keith Brook of British Maritime Technology. He started
by identifying himself as a user of this type of data as
distinct from an analyst such as Mr Peach. His experiencehas been particularly in the context of using estimated
values of radii of gyration in computer calculations predic-ting ship motion response and his comments were largely in relation to rolling. Mr Brook had tried Kato's equation but was more confident using that proposed by Bureau Ventas for radius of gyration in rollK/B = 0.289-/ i + 4 (KG/B)2
He expressed the view that it should be remembered that Kato's equation was derived some years ago and it was probably unfair to expect it to relate to modern vessels of
special types; the limitations of empirical formulae are not
always appreciated. In comparison with the range of K/B from 0.40 to 0.44 given in References I and 2, Mr Brook mentioned that values used in the UK were usually in the
range 0.35 to 0.40 for a loaded cargo ship. The significance of this was raised in the subsequent discussion.
Describing a recent sensitivity study with which he had
been involved, Mr. Brook presented information which Table 4-Effect of error in roll radius of gyration on roll
response
116
showed that, although a 10% variation in radius of gyration caused a significant change in the natural period of roll, the
roll amplitude did not have a direct correlation. This was
based upon results from calculations giving RMS roll
amplitude in various sea states as shown in Table 4.
Following this, he suggested that highly accurate and
detailed calculations were not essential, although logicallythe best possible estimate should be used. He was more
concerned about roll damping coefficients and considered
these to be a more serious problem than radii of gyration
because damping coefficients are difficult to estimate from available data. In his opinion, more experimental work was necessary in this respect. Accurate radii of gyration can be calculated if required, but the same is not true for damping coefficients at the present time.
THE DISCUSSION
Dr I L Buxton, opening the discussion, referred to the
long history of estimation of radii of gyration and reiterated
Mr Brook's warning about the misuse of empirical data
especially, for example, Kato's equation based upon
information from the 1950s. He considered it timely to update such information because of the increasing
aware-ness of the need to predict ship motions. This is very
important, for example, in the way that pitching affects ship performance. He referred to a paper by Swaan and Rijker5which demonstrated that speed loss at sea could be a
consequence of particular longitudinal weight distributions.Dr Buxton asked Mr Peach whether loaded conditions had
been examined and was informed that they had not been included in the study. The calculations were for 'empty' ships. This aspect was referred to by other contributors
during the discussion and Mr O M Clemmetsen pointed out that typically the cargo weight was about two thirds of the total loaded ship weight.
Mr J Whatmore, doing post-graduate work on midship section scantlings by computer at Newcastle University,
indicated that he expected the work to progress towards the
inclusion of radius of gyration. He asked Mr Peach if his
computation had indicated whether the results were sensi-tive to particular items, i.e. could some items be treated in less detail than others? Mr Peach indicated that all items had been treated in full detail. However, Mr W Hills reworded
Mr Whatmore's query and asked if some degree of
'lumping' was not possible, i.e. using blocks or groups ofitems, without significant loss of accuracy while achieving a
worthwhile reduction in data preparation. In reply, Mr Peach simply reiterated that no lumping of data had been
considered.
Mr D Brown referred to the importance of good values of
roll radius of gyration for warships and fleet auxiliaries
because MOD (Navy) ask for design estimates of the
motions of specific locations such as helicopter decks. Dr
Buxton added a comment in a similar context, referring to
the importance of all three radii of gyration in the current
debate on the 'short fat' versus 'long thin' warship hull.
It was then pointed out by Professor J B Caidwell that though roll amplitude may not be very sensitive to accuracy of radius of gyration, rolling accelerations were likely to be
more relevant because they are dependent upon radius of gyration squared. This could be important in relation to acceleration forces on such items as containers stowed on
deck.
Mr J Davison suggested to Mr Peach that while (B+ D) may be a relevant parameter for the radius of gyration ofa
hollow rectangular shell, it might be more appropriate to use (B2 + D2) for loaded ships by analogy with the radius of gyration of a solid rectangle. Referring to added virtual
mass, he pointed out that it was not just radius of gyration
which was affected, but also the total moving mass. it
VESSEL K/B T RMS ROLL (DEGS) SIG.WAVEHEIGHT(m) (SECS) 3 5 8 OFFSHORE SUPPLY 0.36 0.40 0.44 6.8 7.5 8.2 5.5 5.1 4.6 8.3 7.9 7.4 10.7 10.6 10.3 FISHERY PROTECTION 0.36 0.40 0.44 9.4 10.5 11.5 4.8 3.7 2.7 8.6 7.3 5.8 13.0 12.0 10.6 0.36 12.7 1.1 2.2 4.5 CONTAINER 0.40 14.1 0.8 1.7 3.5 0.44 15.5 0.6 1.3 2.7
SHIP TYPE Ro/Ro LASH LNG
Kr/B 0.3556 0.3705 0.3590 K/(B2 + 0.2939 0.3177 0.2979 F 0.125 0.4278 0.3400 0.5831 Kato's 0.133 0.4413 0.3508 0.6014 K /B 0.17 7 0.5090 0.4046 0.6938 Required F 0.0483 0.0707 0.0336 K p/La 0.2746 0.2657 0.2568 KI(La2 + D2)+ 0.2732 0.2649 0.2555 0.2935 0.2833 0.2697 K /(L2 + D2)+ 0.29 18 0.2823 0.2682 Ky/La 0.2739 0.2655 0.2576 K/(L + B2)+ 0.2709 0.2633 0.2547 0.2827 0.2831 0.2705 Ky/(Lp2 + B2)+ 0.2891 0.2804 0.2672
i
h50
would seem preferable, he suggested, to treat added virtual mass separately as in vibration calculations. Both Mr Peach
and Mr Brook agreed that there was scope for more
experimental work in this respect.The President, Dr Mime, asked if any data had been collected from the actual behaviour of ships at sea. Mr Brook replied that consideration had been given by the
offshore industry to continuous monitoring of the
metacen-tric height of a vessel by measuring a vessel's natural roll period, i.e. using an estimated radius of gyration in the equation T = 2 r KA/g GM. However, the difficulties of
determining the vessel's natural period accurately mean this
method is problematic. Mr Whatmore asked if Mr Peach was able to provide, for publication, information on the
weight per foot of length distribution which had been
produced as a supplement to the calculations for radii of
gyration. Mr Peach readily agreed and Fig. 1 is included for this purpose. TONS PER FXT - 220 - 200 - 180 - 160 - 140 -120 - 100 -80 -60 20 0
Dr T Svensen, in proposing a vote of thanks to Mr Peach
and Mr Brook, mentioned that he would treat radii of
gyration with much greater respect in future, and summed up the discussion by referring to the general consensus thatthe presentations had re-emphasised the need for further research on this important topic, especially in relation to increasing interest in the prediction of slamming, deck
wetness and similar seakeeping problems. REFERENCES
Principles of Naval Architecture, SNAME, 1941 Principles of Naval Architecture, SNAME., 1967
KATO, H. On the Approximate Calculation of Ship's Rolling Period, JSNAJ, Vol 89, 1956
PEACH. R. W. ENGINEERING ASSOCIATES,FinaiReport on Study of
Ship Radii of Gyration for US Department of Commerce, Maritime Administration, April 1979
SWAAN, W. A. andRIJKER. H.Speed Loss at Sea as a Function of
Longitudinal Weight Distribution, Trans. NECIES, 25 Jan 1963
CONVENTIONAL TRAPEZOIDAL WEIGHT DISTRIBUTION CURVE
117
Fig. 1Distribution of weight per foot of length for LNG (LG9S-107a) 300
0O 350
