The influence of trim on the directional stability of a ro-ro ship in shallon water

THE INFLUENCE OF TRIM ON THE DIRECTIONAL STABILITY OF A RO-RO SHIP IN SHALLOW WATER

by

W. Beukeiman Report No. 854-E May 1990

Presented (in Dutch) at a Symposium about the "Safety of Ro-Ro ships" _{held at TU Delft, The Netherlands on} March 23, 1990 and organised _{by Directoraat-Generaal} Scheepvaart en Maritieme Zaken (.DGSM), _Nederlandse Ver-eniging van Technici _{op Scheepvaartgebied} (NVTS),Stich-ting Coordinatie Maritiem Onderzoek _(CMO), Scheepsbouw-kundig Gezelschap "William Froude".

CONTENTS

abstract 1. Motive

l..i Simulation carried out by BMT 1.2 Previous research

I.,3 Proposed research

2 Nodeldata and testcondjtjons

3. :'Rêsults, of .lexperiments

4.:Discussjon' of reauits

4.1 Restricted watercond'j!tjons

4.2 Hydrostatic- and dynamic _derivatives 4.3 Stabiiity roots

4.3.1 Criteria for stability roots 15 _{Conclusions and recommendations}

Nomenclature

References Figures Tables

abstract

On account of the disaster with the 'Herald _{of Free Enterprise'} the Shiphydrontechanics Laboratory _{of the Delft University of}

Technology carried out an investigation into _{the directional}

stability of a Ro - Ro ship as function of waterdepth, trim-condition and forward speed. This study was commissioned by the

'Directoraat - Generaal Scheepvaart _{en Maritieme zaken (DGSM)'} of the Ministry of Public Works in The _Netherlands.

Forced oscillations were carried out by means of a Planar Motion Mechanism (PNM) with the model of a Ro-Ro ship at four different

waterdepth' s.

Except the evenkeel position, four trimconditions, _{including one} for trim by stern were also taken into _{consideration.}

The mode], was equipped with _{one rudder and three propellers.} After measurement of the static drift _{coefficients the}

hydrodyna-mic derivatives

_{were determined for both the sway- and}

yaw motion. The different coefficients _{together with the stability} roots are presented in tabular form for three low frequencies of oscillation related to each condition considered. For all these conditions it was concluded that _{the vessel possessed negative} directional stability characteristics.

1. Motive

On 6 March 1987 the roll on/roll off passenger and freight ferry 'Herald of Free Enterprise' capsized 23 minutes after departure from the

inner

_{harbour of Zeebrugge. During the last moments} before capsize the ship turned rapidly to starboard against the

rudder. She capsized 900 to _{port and came to rest in shallow}

water some 930 metres from the

channel

_centreline.

After passing the "Zand 1" buoy the _{quartermaster reported that} he was having difficulty with the handling of the ship and that she was not answering the helm. _{Very soon afterwards capsizing} started. Also in the turn to starboard which followed, the ship did not answer the helm to port.

. .l.1.. Simulation carried out by B

.. The hydrodynamjc:

'Br.itsh Maritime . Technology.:. Ltd'for.whjchuse; has:'been made

of, computer. modelling,. modeL- tests _{. on a}

sister ship,the. 'Prideof FreeEnterprise' to check anumber of

suppositions [.1] . _{The research had a twofold purpose: at first}

-purpose . was to investigate the .:water ingress _{through the opened}

bow doors and secondly the dynamic _{aspects of the capsize which} followed. Because. there _{was some doubt about the exact draught} and the ship's trim at the time of. the. accident, model tests were performed to determine the water ingress on the main vehicle deck for a series of draughts and trim _conditions.

Computer simulations suggested that the likely speed of the ship as it approached ,the..accjdent site _{was somewhere between 15-and} 18 knots. Estimates of squat and trim based on calculations and model tests, _{see Figure 1 [1], confirm that at high speed the} ship shows significant sinkage _{at the.bow..more than at the stern} which is augmentedby shallow _{water effects.}

For a speed of more. than 16 knots _{.. the bow. sinkage is predicted to}

be of the orderof _{one.metre whichfor.a..static freeboard at the} bow of 2 metre means _{a. very substantia]....:reductjon,. of .the}

free-board.

.The-. 'Herald, of .Free

:Enterprise'..'..possessed..f:adequate....power to achieve speeds between .16 and _{.l8knotsin'theapproachchanne1 at} a. waterdepth of 16.5. metre. _{The Froude-number related to the} waterdepth is'.for.awaterdepth..of _{16.5metre and a ship speedof} 18 knots: _{- 0.73, so that significant shallow water} influen-ces may be expected for such a situation.

Moreover the height of the bow wave is also important in reducing the effective freeboard substantially. _{See Figure 2 from (1).} Combination of the static bow trim, _{which at departure from the} harbour amounts about 0.80 _{m, the sinkage and trim by}

ted to waterdepth and ship speed was achieved as shown in Figure 3 from [1].

Modeltests showed an ingress of substantial quantities of water on the vehicle deck at that critical speed, _{after which the model} maintained position at _{a heeling angle of about 300 to 40°. At} the end of the turn the starboard propeller was above the water-level and the equivalent of _{1600 ton water on the vehicle deck} was present.

Very important is the following _{conclusion of BMT [1]:}

No problems in steering were _{encountered with the self propelled} model before water entered the vehicle deck. As speed increased substantial quantities of water entered the vehicle deck once a 'critical speed' had been reached. The ship heeled to port after

water ingress which heeling had _{been enlarged by centrifugal}

forces during the uncontrollable turn to starboard. The combined

effect of heeling by water ingress

_{on deck and the dynamic}

effects during the sheer to starboard _{caused a rapid. capsizing:} 51 seconds at full scale.

The certification that no steering problems could be established

with the modeltests, at least _{not before water ingress on the}

vehicle deck, does not _{seem to fit with the warning of the} helms-man of the 'Herald of Free Enterprise'.

It is not known whether the test _{with the self propelled model}

was carried out at the 'self propulsion point of ship'. If no correction was applied for the relatively larger friction coeffi-cient of the model, for instance by means of an air-propeller _or a small added propeller under the model, _{then a more significant} influence of the model-rudder upon the directional stability _may

be expected on account of the _{relatively higher load of the}

model-propellers.

In other words: the model of _{the 'Herald of Free Enterprise'}

might have had a substantially higher directional stability _than the ship itself. So one might _{also suppose that as a result of} initial directional instability _{the ship made an uncontrollable} sheer to starboard in such a way that due to centrifugal forces _a heeling angle to port side arose, allowing asymmetrically water ingress in large quantities.

1.2 Previous research

In the past research has been performed _{by Gerritsina [2] into the} directional stability of a Todd 60 series model. For this _case hydrodynamic derivatives were determined as a function of draught and forward speed by means of _{PMM tests.}

These experiments were carried _{out in deep water for two} condi-tions namely without rudder and propeller and with rudder _and ro-tating propeller. In the last case a positive directional

stabil-ity could generally be _{established for the ship type considered.} It also appears that the influence of forward speed on the

tional stability is small compared to that of the draught varia-tion in terms of the load condivaria-tion _{and the trim.}

Without rudder and propeller directional _{instability was always} established except for the condition of trim by stern.

The influence of trim on the hydrodynamic derivatives has also been investigated by moue et al _{as reported in [3] and (4].}

From these studies it appears that just as from Gerritsma's study [2] the lateral drift force increases for trim _{by the stern.}

Also research carried out by Beukelman _{[5] and (6] shows that for} trim by the stern the drift force indeed increases but the dis-tribution over the ships's length is more symmetrical fore and aft resulting in a reduction of the drift _{force moment.}

The influence of waterdepth on the hydrodynamic derivatives for

manoeuvring has been investigated by Hirano _{et al} [7] and by

Beukelman and Gerritsma (8].

These investigations found an increase of the damping coefficient (drift force coefficient) Y' and the added mass coefficient with decrease of the waterdepth. _{Measurements and strip theory}

calculations taking into account the influence _{of restricted} waterdepth agree reasonably except for the _{damping coefficient}

Yv' in the aft part of the ship.

Bishop and Price published a theoretical study entitled 'On the dangers of trim by the bow' (9]. _{They put forward there that 'the} vessels loss by hydrostatic instability was preceded by loss of control. She (Herald of Free Enterprise) _{became directionally} un-stable at high speed in shallow water _{while trimmed by the bow'.} With the aid of linear equations of motion for the ship motion in

the horizontal plane the authors _{put forward as condition for}

positive stability of the yawing, swaying _{and rolling system:}

pgVGM [YvNr + (mu _{- Yr)Nv] > 0 (1)} The first _{factor should be positive because the metacentric} height GM > 0.

The term between the brackets is the _{well known criterium of} sta-bility for the coupled yaw and sway motion for which the separate stability derivatives ', _{Nr, '1r' Nv only depend on the} underwa-ter hull form and consequently _{also on trim and sinkage; mn is the} mass and U the forward speed of the ship.

The stability criterion:

(YvNr + (mU - Yr)Nv] > 0 ₍₂₎

may be written as function of the trim TAPP - TFPP

mUc(y) = Yv(i)Nr(i) _{+ Y.(-y) (4)} L

_N(y)

J

in which Uc _{= critical speed with respect to stability.}

After some suppositions about the stability derivatives and their dependence on - analysis of condition (4) leads Bishop and Price

to the conclusion that trim by stern increases the critical speed above which instability appears.

1.3 Proposed research

The supposition mentioned under 1.1 and _{discussed in the} 'Com-missie Zeegaande Ro-Ro schepen' was the motive for research into the influence of waterdepth, trim _{and speed on the directional} stability of the 'Herald of Free Enterprise' _{on .modelscale.}

Generally it seemed desirable to _{carry out qualitative research} into the influence of trim, speed and waterdepth on the directio-nal stability of a Ro-Ro passengership.

In this case forced oscillation tests

_{by means of which the}

linear stability derivatives may be determined were selected.

Harmonic motion tests with the shipmodel _{were carried out for}

both sway and yaw motions.

These tests were performed with rotating propellers in such a way that the thrust agreed with the self propulsion point of ship in order to avoid scale influence in this case. The stability of the system is easy to quantify based _{on the measured stability} deri-vatives. For instance by means of the criterion of stability or

by requirement that the real

_{part of the roots}

_{a should be}

negative. See (8).

A complication at these modeltests is _{caused by the restricted}

width of the towing tank while _{at shallow water the critical}

speed is also determined by the breadth/waterdepth ratio of the waterway.

The speeds to be investigated have been chosen in such a way that the critical speed _{as result of bottom- and wal].influence was not} achieved so that also the _{water].eve]. fall beside the model could} be neglected. See pt. 4.1 and _{the tables lOa and lOb.}

For the largest waterdepth only the even keel and one trim

condi-tion and 3 high speeds have been _{taken into consideration with}

the maximum revolution number to obtain some impression about the tendency of the directional stability _{at these high speeds.}

2. Nodeldata and testconditions

The model is almost equal to the 'Herald of Free Enterprise', but not an exact copy of this ship.

Data of ship and model are presented in table 1. The model _was manufactured to a scale 1:40. The longitudinal radius of inertia was known for each trim condition and amounts about O.25Lp.

For the

testconditjons

_{see table 2.}

Including

even keel four trim conditions were considered of which three were with bow trim and _{one with trim by the stern.}

For all trimconditiong, waterdepths, _{oscillation frequencies and}

static drift angles three forward _{speeds have been considered}

namely Fn = 0.08, 0.10 and 0.12. _{Three hicth speeds corresponding} to ship speeds of 14, 16 and _{18 knots have been tested only for} the highest waterdepth, the even keel condition (nr. _{0) and the} smallest bow trim

(condition

nr. I).

At these high speeds there was a substantial fall of the water level beside the constrained model. _{The resistance for this} con-dition was so high that insufficient thrust at the maximum number of revolutions could be delivered. _{Nevertheless this condition}

has been taken

_{into consideration}

_{as a special addition to}

determine a possible tendency in directional stability at speeds for this ship type when leaving _{the harbour.}

For oscillation use has been made of the Planar Motion Mechanism (PMM) _{which was connected to a lengthened bridge construction} under the towing carriage in such a way that the required oscil-lations at shallow water could _{be realized. See figure 4.}

The model was connected to the legs of the PMM by means of dyna-mometers at two points situated 0.5 _{m fore and aft the centre of} gravity for the trim-condition considered.

These dynamometers, situated in the _{ship longitudinal symmetrical} plane were sensitive for forces _{perpendicular to this plane.}

Before carrying out tests for a new trim condition and waterdepth these dynamometers _{were turned 90 degrees to measure the} resis-tance in order to adjust the required _{number of propeller}

revolu-tions with respect to the _{correction for the frictional}

resis-tance. Two motions were carried out by the PMM,viz. sway and _yaw, for 3 frequencies: _{w = 0.25, 0.50 and 0.75 rad/s.}

The experiments were executed on 4 waterdepths viz. 1.4, 1.6, 2.0 and 2.2 times the models draught in _{even keel position.}

Static drift tests were performed for all trim conditions consid-ered and waterdepths with the _{rudder in zero position and} rota-ting propellers. The adjusted drift angles for these tests _{were ß} = ± 2, 4 and 6 degrees. Except for the largest waterdepth and the three highest model speeds the number of revolutions of the

pro-pellers had been adjusted to _{a value corresponding to the ship} self propulsion point. For this purpose the frictional resistance correction was determined according to the procedure as denoted by the ITTC 1973 friction _{extrapolator:}

0.075 Cv

= (log Re - 2)2

For the adjusted number of propeller revolutions and the fric-tional resistance correction,

02 -B

+ J

B2 - 4AC

al'

-2A -B -

J

B2 - 4AC' 2A } ₍₈₎

-7

3. Results of experiments

The transverse drift force Y and _{the moment N as measured for the} ted in the tables 5-9 in a non-dimensional way as _{tv'' Nv', 1r"} to those for non-dimensional _{forces and moments:}

oscillatory motions and the static drift experiments are presen-etc. This non-dimensional values _{for the derivatives correspond}

Y _Y

y' - en N' =

bu2

Lp2

For a review of the way in which _{each of the measured derivatives} had been made non-dimensional _{see table 4.}

The mass moment of inertia _{with respect to the vertical} z-axis as used in the derivative _{(I - Nr) has been determined for} each condition. _{The structural mass radius of inertia had an} average value = O.25L.

The static drift test showed a strong non-linear behaviour for all conditions.

As an example the driftforce-coeffjcient

A' has been presented

in figure 5 for h/T _{= 2.2 and trimcondiion IV (trim by the}

stern).

For calculation of the directional _{stability roots in table 5}

- 9

the value of the static drift coefficients Yß' and Nß' is based on the lowest drift angle, that _{means ß = ± 2 degrees.}

There are some exceptions, but _{generally these values do not} dif-fer much from the results obtained _{from forced oscillation tests,}

'v' and Nv'. The static and hydrodynamic derivatives _determine the roots _{and 02' for the straight line stability.}

The equation of motion for the _{sway- and yaw motion of a model} with fixed rudder _{are as follows:}

(m - v)'ß' + Yß'ß + Yj'r + (r - m)'r' ₌

Nv'ß' - Nß'ß _{+ (' - Nr)'r' - Nr'r' O} ) (7) The stability roots derived from _{equations (7) are:}

in which:

A = (m - Y)' (I - Nr)' - Yj.'N'

B _{= ß'(1zz - Nr)' Nr'(m} -C = Nß'(Yr - rn)' _{- Yß'Nr'}

The calculated stability roots are presented in the tables 5 - 9,

one set derived f röm the static coefficients Yp' and Nß' together with the remaining hydrodynamic derivatives and one set deter-mined with hydrodynamic derivatives only, which means -Y.' and-Nv' instead of Yp' and Np'.

The differences between these 'static" and 'dynamic' stability

roots,

a'

(stat) and

a'

(dyn) are generally small with some excep-tions. This could be expected due t'o the small differences between _{- YV and Yp' and - N' and Np'.}

As an example

a1'

(dyn) with positive value 'has ,been plotted, in the figures 6 and 7' as fun'ction of' speed(Fn) for each waterdepth, trim-condition and

w = 0.25.

Figure 8 shows the stability root

a1'

_{(dyn) in the same way for} h/T = 2.2 in case of the two trim-conditions considered and three higher speeds.

4. Discussion of results

4.1 Restricted waterconditions

At.firs:t it was' the intention to consider.:higher speeds viz. Fn 0.15,. 0.20 and 0.25,. but on account of the strong, bottom and wall influence of 'the towing tank t ese speeds were not tested.

At these speeds a strong water level fall could be established leading to a substantial trim and sinkage if the model had not been constrained', in such conditions the model power and nthnber of propeller revolutions were too much restricted to apply t'he required friction correction. It was therefore decided to lower

the speeds so that no significant wall and bottom influence

should arise. 'See table lOa and 10b.

Previous investigation on deep water carried out by Gerritsma [2'] showed that t"he speed 'influence on the hydrostatic and dynami,c derivatives and' the stability roots were small.

From the present research, however it appeared that the influence

of restricted water conditions is rather important especially

related to trim.

Further investigations into the influence of the bottom with or

without channel walls partly under water on the trith, sinkage, hydrodynamic derivatives and st-ability roots as function of high' forward speeds are required' for this ship type.

it should be remarked that at hi:,gh speeds (Fn = 0.205, 0.233 and' 0.262 corresponding to ship speeds of V = 14, 16 and 18 knot's) which for h/T = 2.2 only have been taken into account the above

-mentioned

_phenomena

_{as a result of} curred. The adjusted maximum number 1er was insufficient related to the The reason for still taking these presented under pt.2.

To give an Impression of the critica], boundary speeds, sinkage and return speeds calculations have been carried out for ail the

speeds considered in accordance with _{the method presented by}

Bouwmeester in (10] and drafted by Schijf. _{The critical boundary} speed for the condition considered has been determined with the

graphic curves (a _{= 1.1) of Schijf as presented in figure}

9

(figure 36 from (10]) which _{curves have, been determined, :according} to: Fr. = Vcr/jlj = 0.78(1 .As/Ac]225 _'(9) in 'which: .

Ac.':Bóho

. As = surface sectiòn Vcr' .

= ratio of critIcal boundary speed in a channel to

critical speed. on shallow water' _{c =} a = correct.ion coefficient

= average waterdepth (= Ac/B0)

Three conditions are d'istinguished: Um < Vcr

water level.'

- Um2

Z/h . ('a'(,l "As/Ac -Z/h)2-i)/2.

gh

and the 'retun flow u 'with:

u/j'

' Um/j' (1/(i - As/Ac - Z/) - I)

Um = Vcr Then will be:

- '1

Zcr/h = - {1 - Ag/Ac - V02/gh) Fr

critical boundary speed in a channel

bottom and wall

influence

oc-of revolutions oc-of the propel-speed's considered.

'speeds into consideration is

9

('10;)

and

Ucr/j = {-('1 - As/Ac +

jv2/g)

_{) - Vcr/J (13)}

3.'Um>Vcr

The maximum transported quantity of water per time unit is now first 'determIned:

Qmax = VcrAc _(14.)

and the waterlevel fall ,than"amounts:

z Qmax/(Um + Ucr) A As

. (15)

B0

:. The' Calculated_{. val.}

l0a and1Ob for the conditions considered.. _{.. From!! these': it»appears:.:thatf or the} .three lowest speeds, Fn _{= 0.08,. 0.10 and 0.12, the model speed}

remains far below the critical boundary speed, so Um < Vcr The average, water level fall amounts about 0.005 m.

u+Uia

The comparable speed on unrestricted _{water .is.presen.ted by}

For the three highest speeds at hIT _{= 2..2 it ye1ds, tha.t Um >} Vcr. while water level fall varies from .0.065to 0.103 m.

The lowest one of these three _{speeds, Fn = '0.205, corresponds'on} unrestricted water to:

u+Um

Fn = . - 0.2.76

which-means a ship speed of. _{l8.,9':knotg.}

4.2 Hydrostatic- and dynamic derivatives.

From the tables. 5 'to 9 it. appears'. that the 'values 'for the

hydro-static derivatives Yß'and _{NA' generally agree well with} respecti-vely the measured hydrodynámic _derivatives

-'i' and Nv' for the lowest .and considered frequency of' 'oscillation w = 0.25.

An exception should be made for the _{even keel condition (0') for} which. the hydrodynamic derivatives _{are a good deal lower than the} hydrostatIc derivatives..

'The influence of waterdepth _{.on the hydrostatic- and dynamic' force} coefficients may be characterized, as follows:

Yß' and _{v' increase with decreasing h/T}

while the moment-coefficients _{Nfl' and -Ng' experience little} in-fluence of variation in waterdTepth. _{This agrees with what has} been found in [5] and [6].

The influence of forward speec _{on the hydrostatic- and dynamic} derivatives is small for both the drift _{force and moment.}

This only concerns the three lowest _{speeds up to Fn = 0.12. For} the three highest speeds at h/T _{= 2.2 the moment-coefficients Nß'} and _{Nv' appear to decrease with increasing speed.}

The influence of trim _{on the hydrostatic- and dynamic coef f}

i-cients may be described _{as follows:}

- the drift force coefficients Y' and -Y.,' _{are independent} of trim by the bow, but they increase _{for trim by stern.} - the drift force moment coefficients Nß' and _{-Ny' ,however,}

mostly increase with trim by the bow _and.reduce considera-bly for trim by stern.

Similar behaviour of the hydrostatic- _{and dynamic coefficients}

related to trim in deep water was also ascertained in [2] and [3].

4.3 Stability roots

The static- and dynamic stability _{roots are presented in the}

tables 5 to 9 for all conditions _{considered. From these tables it} appears that the lowest oscillation frequency _{generally shows the} lowest absolute value for the stability roots with exception of the three highest speeds at the deepest water. For this condition that relation is not clear and _{sometimes even reversed.}

It also appears from the tables 5 _{- 9 that generally good} agree-ment is shown between the 'static' _{and 'dynamic' stability roots.}

Exceptional is the even keel condition nr.0 which phenomenon was also valid for the hydrodynamic derivatives _{as denoted under pt.}

4.2. As most principal _{characterization it appears from the}

tables 5 - 9 that for all conditions a negative directional sta-bility should established because in no case did both roots _show a negative value. Henceforth for discussion _{of the}

_influence

_of

speed, waterdepth and trim the degree _{of negative directional}

stability or in other words the degree of directional

instabili-ty will be used as a basis for _{comparison. For this reason the}

positive value of the dynamic stability _{root a11(dyn) is taken} into consideration.

It should be remarked that directional _{instability had not been} observed with experiments related to the 'Herald of Free Enter-prise' as presented by Dand in _[1].

il-The dynamic stability root _{o i (dyn) as function of forward speed} is shown in figures 6 and _{7 for each waterdepth related to the} lowest frequency of oscillation w

0.25.

in

_{figure 8 the basis for h/T = 2.2 has been extended up to the}

highest speed considered (Fn _{= 0.262, V = 18 knots).}

From the figures it is clear that directional _{instability reduces} with increasing speed _{up to a certain value (Fn 0.18) after} which the directional instability _{increases with forward speed..} One should, _{however, keep in mind that the}

_conditions

_{at the} higher Speeds were not completely _{realïstic as discussed under} pt. 4.1.

The same tendency may also be _{perceived for both lowest} water-depths. in., case .of some. trim conditions .related to lower speeds

(Fn = 0.08 - 0.12).

.The influence of...waterdepth _{.;on-.the::, directional' instability.}

considered area h/T .,jg _{in', general,; :'smally. (see '.the.':'figu'res 6 and}

7) . In

for speed Fn =

0.12.

Trim.

by .the,,-bow. appears .Yto;...proyide _{little.improvement'jn. the}

.'directional stability for both;:iow.:speeds. _{'in 'r:Bhailow.water-(h/T}

1.4 and 1.6) and for Fn _{= 0.12 ..in .h/T=.:2.O.and2.2.}

-.

In the remaining cases trim by the'.:bow _{causes' a-. deterioration of} the directional stability. .Trimby _{-the stern, condition:iv,.} gene-rally provides a pronounced improvement _{in directional, stability,} especially as result of the strong reduction of the

hydrostatic-and dynamic moments,.

For manoeuvrability preference is given to this condition because the rudder vili react more effectively even if the Ship is still lightly directionally instable.

It should be remarked that .a directionally instable ship may be made stable in a cybernetic way by a correct choice of the

constants of the automatic pilot. _{The question, however, remains} up to whichconditions such.a control still _{may be valid.}

It is questionable if for _{a > 1. 'to. 2. the ship considered 'remains} manoeuvrable. Definite . answer. about this can only be. obtained by

full scale..s'imulation,,manoeuvrjng.eerjmentg.

.5. 'Conclusions and recommendations

From the. .preceding; investigation.. .the.; following.. conclusions and

recommendations may be derived:

It appears that for the considered Ro-Ro ship type direc-tional instability could be established for all investigated vaterdepths, forward speeds and trim _conditions,.

This directional instability decreases if speed increases _up to a certain value (Fn _{0.18) after which probably a strong} growth occurs in directional instability.

The influence of variation of waterdepth in _{the directional} Instability is generally' small for the considered area.

Trim by. the bow shows light improvement _{of the directional} instability in. shallow water, but _{some decrease in deeper} water.

Trim by the stern generally provides a clear rectification of the directional instability mainly as resuit of the decrease of. the hydrostatic- and/or dynamic moment. The rudder will be more. effective.. for. this _condition.

:'6'.Drift1' forces.increa.se' With;déCr.easing,:waterdepth'and;:trj: _by

. the stern.

Trim by the bow and speed show littleinfluence on the drift

force coefficients,.

Drift, moment coefficients _{-.are;rather independent..} of..varIa-tion of waterdepth. and forward . speed,.,

The drift 'moment increases with _{bow trim and}

.reduces with

trim by the Stern.

It. is recommended to carry out further _{investigations in.to} the influence, of restricted _{waterdepth with and without} .channe. walls under 'waker. :dn the trim., _{'.s»inkage, 'hydr.odynainic'}

derivatives and' stability roots. _{as function of high forward} speeds..

Restriction of forward'speed dependent., _{on .'the.."waterdepth is} also strongly recommended.

6. Nomenclature

A _{propeller 'blade area}

Ac area of undisturbed cross channel profile As area of submerged part of the ship's cross'

section

B _{breadth, beam}

B0 _{width of waterievel of undisturbed}

channel

D. _{propeller diameter}

13-X, y, z

U

Fn - _____ Froude number ,/gLpp

ratio of critical boundary speed in _{a channel to} critical shallow water speed

centre of gravity

acceleration due to gravity waterdepth (maximum)

average waterdepth (= Ac/B0) mass moment of inertia

radius of 1uass.]noiuent of. inertia length between:perpendicu1ars overall length

mass of ship model moment of drift force number of revolutions pitch friction correction Reynolds number yaw velocity draught trim (=TVLL - TALL) forward speed

return speed or flow critical boundary speed

transverse forward speed

_component

_{(positive to} starboard)

coordinate-system fixed to model G g h h I k Lpp, L LOA m N n P Ra UL J, r T t U u Vcr V

drift force waterlevel fall

z _{nuber of propeller blades}

a

_{correlation coefficients}

ß drift angle

-y trim coefficient

kinematic. viscosity coefficient

w :- .circuiarfrequency'of;oscjliajon

p _{density of water}

a stability root

-7. References

Dand., i.W.,

Hydrodynainic aspects of the sinking of the ferry "Herald of. Free Enterprise",

Royal Institution of Naval Architects (RINA),

p. 145-165., Aprii 1989. Gerritsma, J.,

Hydrodynamic derivatives as a

function

of draught and speed, Report 477, Ship Hydromechanics _Laboratory, University of Technology Delft,.January _1979.

moue Shosuke., Kyima Katsuro and Moriyama Fuinio, Presumption.'ofchydrodynamjc :derivatives.on 'ship manoeuvring. in trimmed condition.

Transactions of the West-Japan Society _{of Naval Arch} ,

No. 55, 1978.

[4]. _{moue Shosuke,. :Hïranp .Masayoshiand,: KyimaKatsuro,}

'Hydrodynainjc derivativeer: onshipïrmanoeuvrjng,;

mt. Shipbuilding Progress,voi.28, _{No.321,.-May.::l98i.}

(5] Beuke].man, W.,

Longitudinal distribution of drift _{forces for a ship} model, Report 810, Ship Hydromechanics _Laboratory, University of-Techno1ogy. beift, December 1988.

[6] Beukelman, W.,

Cross flow drag on a. segmented model.

Fourth Int. Symposium on Practical Design. of Ships and Mobile Units cPRADS'a9), October _{1989, Varna, Bulgaria}

J. Hirano Masayoshi,., Takashina Junshi, Moriya Shuko and

Nakamuro Yoshja]cj,,

An experimental study. on _{manoeuvring .hydrodynamic}

forces in shallow water,

Transactions

of. theWest-Japan :Society.. of Naval. Architects, No. 69, March 1985.

(8] Beukelman,, W. and J. Gerritsma,

'The longitudinal. distributionof _{..1owfrequency}

hydro-dynamic

derivatives for lateral

:rnotions in

_{shallow water,} 'Report 562-A, Ship Hydromechancg _Laboratory,

University of Technology Delft, _{September 1983.} Bishop, R.E.D. and W.G. Price,

On the dangers of trim by bow,

Evening Meeting, London and Joint Evening Meeting, Glasgow, The Royal

Institution

_{of Naval Architects}

(RINA), 1988. [10] Bouwmeester, J., Binnenscheepvaart en Scheepvaartwegen, Koliegediktaat .f12N, (31 [ [9)1

-o.'

-06

9 )0 lt 12 I) U. IS 16 17

I

I-

- I

/

WAT _{OPT Iii}

165m

IcEY -. ASJD(TFP S8rn Ti,p.5.55)Tpp -OflAUC,4T AT FP

O ASLJCO(FP -6OSm TAP_S.25m)TÄP_DÇUUGHTAT AP

PREDCIEOwATER 'cCPTH'-165rn

- P

DCT(DWAT-ER -OCP1H-t?,.2rn

Figure 1. Predicted and measured squat from model _{tests [1].}

o

-WAlER ccPTH l22rn

Figure 2. Calculated bow váve profiles _(1].

2:0 ¡9 _{18 19}

STATJCt(

ColcIoIed nse ol

WQI«

I( Gloe

4I

nt

bGw I

_'HroId

f Fur Enttrplsr'.

_{L H S. rn lrrot Uway,R H S. Uw}

_thrcry.

14

1:3

.GEEN WATER

OVER BOW SILL.

SHIP SPEED (KNOTS)

MEA DRAUGHT - 56 METRES

+ MODEL MEÂSURMENTS

Figure 3. Critical speed plots (1].

NOARY LIFE

INDICATING

I:TIJ' SPEED

FOR 165 MET RES

WATERr.:DCpTH

NO WATER OVER BOW SILL

/

/.

LINE

FOR i22 METRES

WAiER DEPTH. 1O

20

J'O FREEBOARD AT BOW - 19 2:0 19 18 1 7 16 1 I,

't

i hIT = '2.2' Trim IV O En. 0.08 1 Fu 0.10

fl

Fn 0.12 500 Stati'c measurement's 2000

Figure 5. Drift force coefficient _{A' as function of drift angle} for h/T = 2.2 and trim coidition IV.

-6 0 2 4'

1.25 1.00 0. 75 a'(dyn) ò.50 0.25 o Fn. Fu 0 _{.08 .10 .12,}

1.25 i ..00 0.75 c/1(dyn) o.5o 0.25. o 1.25 i .o0 o. is

a(dyn)

0..50 0.25 O .08 .08 Fn

Figure 7. The dynamic stability root

_{o1(dyn) as function of}

forward speed for h/T

_{= 2.0 and 2.2.}

.12

.12

Figure 9. Diagram for Speéds and väterÏevèi related to shipS 'in 'a channel .(Schijf,.a = 1.1).

Table 1. Main dimensions of ship, model and _propeller.

Table 2. Testconditons.

Ship Modelnr. 280 Length over all _{LOA 132.000 In 3.300 In} Length between perpendiculars _{Lpp 126.100 In 3.152 Ifl}

Beam _{B 22.700 In .567 m}

Draught (even keel) _{T 5.685 In .142 in}

Displacement _{8874 ton 138.660 kg}

LCB forward of Lpp/2 _{-2.870 In -0.720 In} Radius of gyration for yaw

(adjusted for each

condition)

kzz 0.250 Lpp Propeller NSMB - Serïe B-380

Diameter _{D 80 mIn}

Pitch ratio _{P0/D 0.90}

Blade area ratio of propeller _{AE/AO 0.80}

Number of blades z ₃

Number of propellers ₃

Model scale _1:40

Ship Model

Trirnconditjon O (even keel)

0.375 in 0.750 in 1.125 in 0.375 in 5.5 knots 6.8 knots 8.2 knots 14 knots 16 knots 18 knots 0.04 rad/s 0.08 rad/s 0.12 rad/s 1.4 1.6 2.0 2.2 ± 2 degrees ± 4 degrees ± 6 degrees 0.0094 m 0.0188 In 0.0281 in 0.0094 m 0.445 In/s 0.556 rn/s 0.667 rn/s 1.140 rn/s 1.300 in/s 1.460 rn/s 0.25 rad/s 0.50 rad/s 0.75 rad/s 1.4 1.6 2.0 2.2 ± 2 degrees ± 4 degrees ± 6 degrees

I _{(trim by the bow)}

II _{(trim by the bow)} III (trim by the bow) IV (trim by the stern) S.eeds Fn = 0.08 0.10 0.12 Fn = 0.205 for h/T = 0.234 } 2.2 and 0.262 cond. 0 and i only Oscillation-frequencies Waterdepth/draught ratio h/T

Table 3. Frïction correction and number of revolutions.

maximum number of revolutions and insufficient

_thrust.

**

_{nominale value for deep water.}

Table 4. Non-dimerisjona]. derivatives.

Yß'

Nß'

(in

-Nv'

Nv'

('

- Nj)

Nr'

Yß IpL2U2 Nß pL3U2

m-Y,r

pL3 Yv pL2U N pL4 Nv pL3U

-pL5

Nr

pL4U Yi p L4

inYr

pL3 - 27

-speed

_Friction

correction

revolutions

number of

h/T

Fn

Ship

Model

RA in N

1.4

1.6

2.0

2.2

0.080

5.5 kn.

0.455 in/s

0.479

_{6.25 6.45 6.95 7.10}

0.100

6.8 1m.

_{0.556 in/s}

_0.706

_{8.00 8.10 8.55 8.95}

0.120

8.2 kn.

0.667 rn/s

0.969

_{9.80 10.1 10.1 10.5}

0.205

14

kn.

1.140 rn/s

2.476**

-

-

-

26*

0.234

16

kn.

1.300 rn/s

3.120**

-

-

_-

26*

0.262

18

kn.

1.460 rn/s

3.827**

-

-

_-

26*

LOAD-'CONDITION cò O

i

II III iv +Yß' - 2450 2069 2216 L _{2213 2391} -i-r' - 677 780 ₈₁₂ H 855 592' ' ' .25 , 2101 2053 2116 2107' 2053 , (m-Y,) .50 2174 2:181 2283 2168 2136 75 2454 2453 2443 2426 2462

-"V

.2.5 .50 2421 2950 2167 2912 2298 272'7 2271 , 2849 2470 2896 75 5115 4917 5020 4976 5444 25 -66 -74 -57 -91 -121 .50 ' -58 48 13 5'7 - 86 .75 -52 -46 _H -44 -4.7 -.25 680 752 H 852 842 564 Nv' .50 736 875 9100 900 667 75 887 1038 H 1051 1144 _79,5 ¡ .25 255 300 2165 2'48 1910 (I.zzѱ) .50 203 226 221 234 181 .7.5 224 244 246 230 209 25 -247 -199 -209 -160 -167 Nr' 50 -260 250 -298 -310 23b0 .75 -233 -245. -2318 . -279 -225 .25 435 442 470 459 401 .50 229 228 237 258 194 .75 148 149 162 :155 ₁₃₇ ' .25 H 62 , 89 98 244 128 (m-Yr) .50 324 385 3194 _{295 362} .75 .1. 4.70 449 H _{506 421 455} .25 1.2.47 .964 1.178 1.149 1.174 a1' (stat.) .50 .1.615 1.531! 1.8.11 1.727 1.564

75

1.355 1.3.5.9 1.368 1.586 1.346 .25 - .920 - .770 - ,784 - .868 - .922 a2' (stat.) .50 .75 -1.166 -1.177 H-1.060 -1.042 -1.066 -1.128 -1.041 -1.090 -1.211 1.,14i5 .25 1.249 : 1.193 1.134 1.152 a1' (dyn.). .50 1.620 1.523 1.824 1.710 .75 1.331 1.321

1320

1.565 1.324. .2:5 - .910 - .814 . - .804 - .889 - _.960 02' (dyn:.) . .5'O 1.365 1..3'88 1..261 .1.287 1.4'21, Table 5a h/T = 1.4 Coefficients * lO = 0.08

- 29 - LOAD-CONDITION o i ii iii iv - 24,0:8 2134 _L 2278 2225 2366 - 631 749 801

81

600 .25 2216 2191 2143 2093 2:108 (rn-Y.)' .50 2140 2212 2166 2183 2183 .75 2542 2572 2540 2:682 262,5 .25 225.8 _{2346 198:3 .21.70 2270} -f,' .50 29:85 _{2712 2691 2758 '2:847} 75 4936 4785 4737 4920 4507 .25 -.61

-9

L .56 _{-95 -118}

-N'

.50 -65 -51 -6:0 -61 - 8.0 ; .75' -77 -8:1 -72 -92 89 .25 6:80 780 812 8:09 518. .510 797 904 941 967 ₆₉₉ .7.5 1025 1139 1129 1280 925 .25 283 26.7 376 305 24.8 ('Iz.z'Ñ.)' .50 198 23:2 H 222 249 .192 .75 222 241 L 2:35 239 214 .25 -1,68 i34, L -172 - 72 : -154 Nr1 .50 -2:32 -283 -233 : -2182 -239 .75 -258 -26.3 -25.7 -288 -247 .25 54:0 '539 62:6 642 459 .50 2175 _{273 32:6 363} 289 .75 191 222 H _{230 2:16 193} ' .25 - 3.1 134 _Y - 97 _{159 - 76} (mYr) .5:0 2:41 : 206. 301 2:19. 142' .75 439 4312' 4'62 43.1 _"I 43:0 .25 .789 .:889 L .623 .627 .763 a,' (stat..) .5:0 1.486 1.546 1.5:07 ' 1.534 1.500 75 i 438 i 430 1 486 1 541 i 411 .25 - .699

-.600

.659

-.724

'°2' (stat.): .50 .75. -1.083 -1.079 H .930 .970 -1.023 1.04:8 - .928 - .972 - .981 -1.031. .25 .818 .885 .64.6 .633 .741 a1' (dyn.) .50 1.544 : 1.584 1.559 1.579 1.528 75 1 502 1 466 1 504 1 604 1 471 .25 - 6:63 - .760 - .483 . - .643 - .726 02' (dyn ) 50 -1 297 -i 142 -1 167 -1 108 -1 154 .75 -1.982 -1.812' ' -1.8:85 1.8:5'7 -1.775 Table 5b h/T = 1.4 Coefficients * 5 = 0.10

LOAD-CONDITION o L

m

iv

- 249:6 2252 H 230:5 2291 2502. - 691 807 H 8:32 824 625 25 2180 2190 2060 2193 2166 (m-Y)' .50

257

2147 2167 2225 2253 .75 3003 2955 2919 30.17 2822 25 2187 2249 2441 2078 2428 .50 H 2752 2737 3044 ₂₆₀₉ 2919 7:5 _{5462 5357 5113} .49:64 .5641 .2:5 H -: - 96 H -138 -1.08 -112 -N./ .50 - 14 - 77 H - ₈₃ - 65 - 87 75 -149. -149 -140 -162 -172 Nv' .25 .50 667 83i 785, 949 815:6 ' 100'5 . &19: 993 574 748 .75 1352 1352 1364 1507 1092 .25 ' 314 312 372 309 257 .50 206 . 256 237 245 19.1 .75 228 247 235 219' 196. .25 _H - 72 -177 - 64 - 83 Nr' .50 -176 ' -211 -203 -273 -187 .75 ' -259 '-181 -192 -244 -19i8 .25 _{707 70:1 663' 786 578} -Y.' .50 H .75 338 277 ' 370 327 391 319' 386 ' 287 332 264 ' .25 _{59 63} ' -146 : 97 23 (m-Yr) .5:0 223 177' 245 . '107 124 .75 331 423 497 448 3:97 .25 .479 .543 .626 .593 .52:0

a1'

(,stat.)l _.50 ' 1.216 1.183 1.306 1.4:69 1.257 .75 1.366 1.103 1.262 1.461 1.261 .2:5 - .658 - .523 - .534 -, .501 - .688

u2' (tt.)'

.50 -1.040' - .904 - .942 - .830 H .944 .75 - .883 H- .812 - .944 - .898 - .985 .2:5 .496 .53.1 .625 : .621 .503

a1'

.dyn.) .50 1.279 1.216 1.344 1.531 1.2:99 .75 1.5:07 1.16,9 1.355 1.640 1.359 .25 - .566 - .53.1 - .573 - .449 - .688

a2' (dyn.).

.50 -1.15' 1.0!6'i -1.177 - .914 -1.070

Table 5'c,

- 31 - LOAD-CONDITION w O : II III . IV +Yß - 1740 ' 1748 1652 1855 1900 +Nß' - 655 640 _{628 601 539} . ' 25 1914 1939 1834 1990 1865 (m-Y) . . 50 1999 1976 19:67 1985 1922 .75 2161 210:3 2112 2150 2084 .25 .50 1811 2375 1795 2150 2086 2085 1893 2289 197:3, 2484 75 3470 3354 3195 3379 3414 .25 -139 -9.3 -9.1 -77 -62

-N/

.50 - 77 -51 -57 -39 -44 .75 - 4:6 -38 -41 -31 -39 .25 H 347 603 596 ₆₇₅ 547 - Nv' . 50 539 603 547 690 623 75 _{703 186 71.7 838 667} .25 158 263 203 2:37 251 .50' 174 198 194 2:06 189 75 207 224 210 220 204 .25 -105 -201 -154' -149 -116 .50 -274 -275 . -290 -315 -287 75 -258 -275 -282 -300 -275 25 447 i 476 463 508 401 -Yj.' .50 225 22,1 2!5:5 246 219 75 136 141 146 145 129 ' .25 _L 102 19 141 138 13:6 (rn-Yr) .5.0 319 361 359 388 345 75 505 533 547 554 522 .25 1.154 1.047 1.221 1.000 720

o'

Cs.tat..) .50 L 1.9:64 1.786 1.9:13 1.917 1.85.1 .75 L 1.633 1.610 H 1.741 1.723 1.667 .25 - .593 - .627 - .678 - .703 - .827

a2'

(stat.) .50 : - .956 - .990 - .9:30 -1.019 -1.059 .75 -1.053 -1.071 -1.034 -1.081 -1.121 .25 .866 1.023 1.147 1.051 .718

a1'

(dyn.) .50 1.851 1.736 1.826 1.949 1.874. .75 1.566 i 1.594 1.704 1.766 1.664 .25 H- .716 - .657 : .853 - .699 - .856 U2' (,dyn.) .50 -1.217 -1.158 -1.101 -1.2:12 -1.328 .75. -1.761 -1.767

-1689

-1.753 -1.798 Table La h/T = 1.6 Coefficients * Fn 0.08

LOAD-CONDITION H o

i

ii

iii

iv

+Yß' - 1718 1741 1525 1800 20219 +Nß' - 603 667. 628 616 604 .25 2008 1936 1809 2055 1949 .50 1797 1972 1952 1987 1887 .75 2194 2136 2150 21182 2134 .25 1794 1803 1762 1696 1986 .50 2291 « 215.6 2256 2300 2424 75 3087 2853 2886 3018 2945 .25 -128 -97 -115 -78 -82

-N'

5:0 - 80 -'60 - 68 -49 -61 .75 - 64 -52 - 53 -42 -48 .25 413 584 569 652 565 .50 577 652 641 717 634 .75 749 H 791 775 844 671, .25 138 271 2182 ₂₅₈ 287

(IzN±)'

.150 _{170 210 185 205} 1 18:7 .75 : 199 224 207 211 211 .2:5

-79

_{-100 -161 -120:}

_-81

.50 i -'225 -223 -234 -268 -247 .75 2'83 -300 -308 -317 -29.3 529 6106 '538 60 5:65 .50 299 310 . 316 324 2816 75 188 197 198 198 1918 .25 - 8 69 - 23 124 6,9 1 .50 221 218'4 296 302 23.4 .75 431 435 474 :536 438 .25 1.059 .725 .836 .892 .514

a'

(stat.)

.50

1.714 1.500 1.738 1.780 1.689 .75 1.759 1.709 1 1.876 : 1.933' . i.726 .25 - .358 - .520 - .483 .582 - .66,1

a2'(stat,)

.50

- .906 .8,90 - .816 .911 -1.027 .75 - .946 - .973 - .898 -1.03,0 ,i.08:3 .25 .851 .661 .784 .887

499

a1'

(dyn.) .50 1.678 1.455 1.690 1.770 ' 1.68:4 .75 1.772 1.711 1.886 1.960 1.717 1.25 -.472

-.572

-.604

!.556

-.66.1 02' (dyn.) .50 -1.16,3 ' -1.0:58 -i.:iió -1.112 -1.198 Table 6b h/T = 1.6 Coefficients * ]5 _{Fn = 0.10}

- 33 - LOAD-CONDITION o i ii 'iii iv +Yß' - 1796 1717 1691 1881 1908 - 622 677 585 634 580 ' .25 2054 2057 2016 2103 1922 ('m-Y.) .50 2078 1909 2001 2049 1909 .75 2287 2218 2191 2278 2176 .25 1815 1882 1982 1827 : 1827 .50 2252 2225 2208 2241 2101 .75 2793 3023 3008 2:87:4 3002 .25 -153 -113 -112 -87 , _-71

-N'

.50. - 74 - 88 - 69 -52 -67 .75 - 7,5 - 68 - 71 -60 -61 25 494 64.8 625 665 569 .50 587 723 619 718 559 .75 794 863 8.09 .898 L ₁₃₀ .25 286 320 298 313 32:9 .50 .179 2:03 193 184 200 .75 200 2]8 208 211 188 .25 - 74 -Iii -.1.09 - 65 -112 Nr' . .50 -'169 -193 -196 -224 -191 .75 -270 -282 -279 -286 -296 ' .25 612 ' 678 695 757 633 .50 362 341 388'. 346 350 75 248 257 274 272 223 .25 19 : .15 - 9 - H - 20 ('rn-Yr)' .50 75 164. 33 183 374 194 393 , 22.8' 457 143 337 .25 .473 .623 .613 .427 .515 'a.1'(StatJ .50 1.3.58 1.362. 1.43.0 1.669 1.291 .75 1.682 1.6.76 1.696 1.754. L 1.886 .25 . -.449 -.438 -.431. -.388 -.520 a2'(Statj .50 -.749 -.8:00 -.753 -.858 -.855

.75-.8'6:8-.878

-.871 .950 '-.95.4 .25 .412 .591 .606 .451 ' .517 a1'(dyn.) ' .50 1.295 . 1.349 1.414 1.705 1.263 .75 1.728 ' 1.703 1.761 1.860 1.922 .25 - .512 - .504 - .511 - .356 :- .552 o2'(dyn.) .50 - .923 - .997 - .947 - .988 ,- .94.1 .75 -1.243 _-1.378 -1.383 i..33'3 -1.397 Table 6c h/T '= 1.6 Coefficients * 5 Fn 0.12

LOAD-CONDITION o i ii iii iv - 1397 1219 1305 1347 1483 - 50.8 526 599 568 369 .25 ' 1825 1807 1774 1822 1842 (m-Y.)'' .50 1751 1760 1756 1770 1762 .75 1773 1780 i771 1799 1807 .25 1572 1544. 158:3 _{1496 1540} .50 1810 1921 1828 1841 1845 .75 2264 2167 215:1 .2148 2215 .25 -71 -73 -65 -39 99 Nv' .50' -17 -19 - '7 - 4 -58 75 -11 - 14 _{- 4} 2 -36 .25 ₄₄₈ 502 524 512 367 .5:0 456 504 5:07 5:17 377 75 534 557 557 606 464 ' .25 H 223 : 232 . 252 2:1,6 171 (IzzNt) .50 175 171 192 :189 158 .75 194 ' 20,3 199 209 192, .25 ' -102 -103 -14.7 -108 -119 .5.0 .75 H -2:93 308 3015, -330 -'291 -.3,54 -334 -358 -297 -294 .25 452 482 589 5:02 ' 423 .5:0 .75 230 133 217 , 136 .248 132 243 143 224 131 ¡ 25 ' 156 172 17.0 139 116

(mr)

.50 4:07 403 437 410 372 .75 541 541 ' 560 538 545 i' (stat.)' .25 .50 .825 ' 2.146 .866 , 2.26.9 ' 1.109 ' 20,99 i .9:95 2.2:96 .953 2.132 .75 1.997 '2.039 2.25.8 _{2.156 1.793} .25 . - .612 - .54.9 - .545 - .546 - .645 021 (stat.) .50 - . 925 - 843 - .902 - . 886 - .930 .75 -1022 927 . -1.0.00 - .972 -1.010 .25 .756 .801 .995 .917 .947 oi (dyn.) .50 2.065 2.19:5 . 1.965 2.210 2.124. .75 1.957 1.996 2.182 2.125 1.826 .25 ,: - .694 - .6.74 - .666 ' - .614 - .668 02' (dyn.) .50 -1.117 .I17'8 . -1.131. . -1.115 -1.112 Table 7a. h/T = 2.0 Coefficients. * o5 Fn = O. 0:8'

- 35 - LOAD-CONDITION o i Ii iii iv +Yß' - 1347 1366 1287 1330 1366 - 518 529 547 5.10 479 25 1846 2002 1878 1846 1792 (m-Y) ' .50 1728 1737 1773 1758 1802 .75 1792 1792 1788 1806 1821 .25 l523 1469 1418 1424 1675 .50 1904 1969 1891 1835 i830 .75 2104 2056 2016 2140 2130 .25 -61 33 54 -49 -146 -Ni' .50 -38 -33 -11 . -22 . - 54 .75 17 -15 - 9 - 7 - 38 .25 499 556 535 ' 554 452 50 496 546 552 565 41.0 .75 525 545 545 618 486 .25 278 .290 254 254 162

(I-N)'

.50 168 188 188' 187 160 .75 1.84. 191 199 203 18:6 .25 -.114 -14.0 - 96 -107 -124 .50 -247 -254 -258 -291 -218 .75 -339 -34.1 . -354 -366

-10

.25 .50 54Ó, 287 562 31.6 5:69 313 578 314 545 295 .75 .179 192 132 198 185 ' .25 34 39 . 126. 67 116 (m-Yr) .50 250 2:86 3.02 282 , 279 .75 460 :, 488 560 503 470 .25 ' .671 .764 .790 .818 1.179 01' (stat.) .50 1.925 1.825 . 1.911 2.082 ' .1.78,3 .75 2.260 2.165 2.2:1,0 2.2.81 2,028 25 - 464 - 463 - 480 - 444 - 516 02' (stat.) .50 - .797 - .8.10 - .773 - .784 - .795 .7'5- .924 : .93:9 - .966 - .921 .9:25 .25 .640 .769 .760 .790. 1.117 o' (dyn..) .50 1.858 1.786 1.854 2.030 1.681 .75 2.216 2.129 2.159 2.256 1.988 2" (dyn..) .25 , .50. ' .75 ' - .542 -1.062 -1.295 - .493 -1.091 -1.276 - .528 -1.048 -1.322 - .483 -.1.018 -1.31S - .630 -1.004, -1.293 Table 7b h/T = 2.0 Coefficients * iø = 0.10

LOAD-CONDITION I o

i

II III IV - 1284 1317 1342 13.53 1463 - 544 545 571 660 523 .25 1889 1889 1861 1904 19Db (m-.Y...)' .50 1784 1.780 1790 1795 1792 .75 1831 1821 1810 1832 1860 .25. 1535 1273 1450 1516 1374 Yv' .50 1724 1758 1614 1786 1780 .75 1980 1940 1981 2003 ', 2:073 H .25 -'77 -60 -73 -82 . -104. L .50 -36 -31 -2.0 -'34 - 63 L .75 -1,7 -18 -14 20 - 41 .25 558 532 562 642 448 .50 477 517 496 610 437 L . 75 522 538 5.61 633 50:9 .25 : 406 308 310 249.

-(Izz) ¡

.50 . . 185 19:5 192 2.09 178 .75' ' 18:0 185 192 201 174 Y .25 -1.02 - 3.3 - 40. 6 - 62: .50 -218 -199 2'15 223 1176 L .75 -311 -318 -335 -341 . : 295: .2:5 712 933 759 776 658 .50 349 367 381 290 363 .75 240 243 25,1 256 237 L ¡ .25 ' 24 42 61 179 110 ' (m-Yr) .50 ' 127 . 186 187 ' 20S 181 .75 3.86 418 424 L '43iO 39.2 .2.5 .578 .305 . .454 ,464 . .609

o'(stat)

.5.0 1.61.4 1.493 ' 1.652 1.529 : 1.412 .75 2.211 2.207 2.258

2258

2.111

25.3.63

_{-.264 -.310} ' -.362

-.450'

'°2' (stat.) .50 - .631 - .679 - .68.1 ' - .746 - .729 .75 .826 -.858 .873, _.8:84 ' .25 .554 .305 .425 .4.04 . .556 (dyn.) H _{.50 1.517 1.421 1.549 '1.451 1.306} .75 2.139 2.15.2 . 2.197 2.180 2.059 25 - 446 - 257 - 345 - 401 - 447 a2'(dyn..) . .50 - .84.0 -.876. .80:8 ' .94.0 -.879 Table 7c

hIT = 2.0

Coefficients * Fn = 0.12

37 - LOAD-CONDITION o i ii iii iv - 1369 1338 1351 1214 1172 +Nß' - 608 622 568 561 512 ' .25 1763 1746 1682 1770 1863 (m-Y) .50 3.701 1717 1715 1665 174.4 .75 1732 1730 1712 1696 1738. .25 1466 1424 1407 1510 1714. .50 1870 1.745 1891 1647 1926 .75 2160 2133 2088 :2108 2325 25 -67 -57 . -83 -31 -12 .50 -27 -'16 -24 -15 1 .75 -20

-4

' -1,4 9 -16 .25 H 484 472 428 539 498 Nv' .50 460 463 469 449 444 75 501 531 505 536 424 .25 .230 247 225 217 284 .50 187 205 184 191 18,9 .75 ' 209 211 207 '2:10 202 .2:5 144 - 40 -164

-47

-110 .5:0 -286 -29:6 -331 -329 -306 .75 -3.07 -318. -352 -345 -314 .2:5 .414 437 46:8 450 491 .50 222 238 236 236 232 .75 127 133 138 135 129 .2:5 116 - 86 101 142 124 (m-Yr)' .5:0 4.03 39,9 392 417 349 .75 534 H _{517 523} .25 . 1.025 3.689 1.137 1.170 .724 a' (stat.) .5.0 . 2.052 1.973 2.291 2.2:51 2.090 .75 1.920 1.9'7OE 2.126 2.092 1.955 .25 - .603 -1.664 - .588 - .554 - .496 02' (stat.) .50 - .957 - .918 - .910 - .875 H .781 .75 ' -1.064 1.034 -1.020 - .970 .916 .25 .922 .127 1.011 1.107 .649 a' (dyn.) .50 1.884 _H 1.80:6 2.164 '2.110 1.974 .75 1.791 ,, 1.850 2.0,33 2.0'09 1.822 02' (dyn.) 25 ' .5:0 - . 669 -1.180' '- .282 -1.091 - . 649 -1.165 - .678 1.075 - . 21, -.1.145 .75 -1.425

-1404

-1.379 -1.421 -1.475 Table Sa h/T = 2.2 Coefficients * Fn = '0.C8

LOAD-CONDITION w 0

I

II

III

IV +Yß' - 1353

1321

1302 1213 1205 - 541 601 54:6 554 546 '

.25

1795

1770

1707

1727

1694 i (m-Y,,,) 50 1729 1715 1662

1667

1777

.75

1750

1761

1727

1719 1742

.25

1375

1366 1364 1333 1408

.50

1764

1740

1822 1779

i777

.75

2029 1988 1993 :19:92 1976 H

.25

-80

-77

_97

-61

-68

-Ñ'

.50

-36

-40

-57

-33

- 2

.75

-26

-16

-20

-1.0

-18

.25

44,7 514 447 506 453

.50

459

477

462 494 428

.75

498

515 492 522 460

.25

292 342 271 2S6 318

(izzN)'

.50

189 206 183 193 197

.75

20,1 205 199 20,6 19:9 .25.

-112

-173

-131

-143

-156

Nr'

.50

-223

-23:9 -244.

-258

-229

.75

-334

-322

-365

-373

-331.

.25

516 477 553 574 552

.50

284

296 _L 311

313

312 75 178 184 18:8 191 183

.25

16 - 77 0 29

- 25

.50

281

260 322 316 269 75 463 485

470

461

448 25 612 659 766 837 733

a1'

(stat.)

5i0

1.635

1.629

L 1.829

1.868

1.684.

75

_2.067

_2.055

_2.266

_2.262

_2.10:0

.25

-.463

-.430

-.431

-.428

-.413

a21(stat.)

.50

-.824

-.793

-.838

-.787

-.716

.75

-.954

-.958

-.929

-879

-.875

.25

. 560 628 . 694 78.6 . . 663

a1'

(dyn.)

.50 H 1;.527

1.491

1.699

1.752

_1.52:0

75

_1.991

1.939

_2.179

_2.182

_1.982

25 - 509 - 488 -

498

-

494

- 545

a2'

(dyn ) 50

-1 015

- 992

-1 085

-1 057

- 979

75

_-1.28:0

_-1.26.1

_H-1.267

_-i.266

_-1.240

Table 8b

h/T = 2. 2

Coefficients * lO5 Fn = 0.10

- 39 - LOAD-CONDITION w O

I

II

III

iv

+Yß1 - 1333 1335 _{1301 1192 1220} +Np ' -: 5:59 608 547 53.6 504 ' .25 1831 1841 1690 1873 1733 (m-Y) .50 1722 1770 1694 1721 1738 .75 1775 1768 1735 1737 1747 25 1328 1373 1434 1470 1267

5Ø

1668 1633 15109 1559 1669 -. 75 1965 1920 1890 ' 1881 1940 .25 -'85 -96 -124 -56 -67 i .50 -64 -42 - 63 -31 -41. .75 -39 -28

-33

-18 -23 .25 488 511. 498 580 478 .50 430 464 402 4.68 428 .75 487 510 480 512 472 ' .25

32

338 314 380 352 i . 5i0 192 227 _20.0 175 ₁₉₃ 191 208' _L 187 201

189,

25 -118 - 98 - 75 - 72 - 90 I .50 1.71 i40 199 212 173 75 -289 -312 -338 -354 -314 .25 597 640 L 699 720 651 .50 349 366' 3181 338 ' 348' 7i5 ', 229 249 236 245 236. ' .25 34 23 62 20 23 (rn-Yr) .50 156 225 181 221 151 75 390 379 383 395 355 2'5 582 553 543 405 ₄₇₃ 1a11(stat.) .50 1.274 . 1.090 L 1.44.0 1.744 1.30.5 .75 1.954 1.979 H 2.25:1

2220

2.082. ..215.,448 _-.383 .392 _{-.3,17 -.392} o21(stat.) .50 -.681 _-.712 -.6.85

.683

-.636 .75 _L -.887 -.872 H -86:8 -.807 -.804 .25 .544 .494 H .487 .390 .453

a11(dyn..)

.5.0 1.139 .940. H 1.287 1.633 1.197 .75 1.850 1.856 2.154 2.14.8 ,2..Ó0'4. .47.0 - .43.3 - .460

- .400 -

.423 a2'

(.dy.)'

.50' - .852 '- .850. »- .799 - .852 - .851 .75 -1.177 -1.139 -1.150 -1.143 -1.154 Tabie 8c h/T = 2.2 Coefficients t o5 = 0.12

LOAD-CONDITION O I II III IV +Yß' - 1281 1277 +Nß' - 52'3 _L 599 .25 1988 2034 (m-Y..)" .50 1926 1987 .75 2579 2664 .2:5 130:8 1285 .50 1548 H 1474 .75 2774 _H 2841 .25 -165 H -219 .50 -105 -13.0 75 -'219 -280 .25' 48.8 598 Nv' .50 515 . 575 75 758 861 ' 604 544

(1zzr)

50 326 3158 75 183 150 .25 -89 -93 .50 -46 H -2:6 5 -42 25 1294 1406 5:0 .587 :: 686 75 H 562 571 .25' ₅₃ H 6 '(rn-Yr)' .5& L 11 36 .75 .243 415 .25 .343 : .43.6

'cf

(stat.) .5:0 H .29.0 .262 75 . 4.45 877 .25 : -.292 ' - .199 a2' (stat..') .50 ,' - .324 : - .261 7.5 - . 456 - . 616 .25 .321 . .434 i' ('dyn.) .50

261

' .223 .7:5 .425: : 979 25 - . 3.14 : - .201 ''2" (dyn.') ': .50 - .427 . ' , 3:3 Table 9a hIT = 2 ..2 Coeff'iciènts * _{Fn = .205}

- 41 - LOAD-CONDITION o i ii III IV -I-Yß' - 1238 1276 _L - 493 578 .25 2177 2229 (m-Y)' .50 2008 1957 .75 865 981 .25 .50 1326 2066 1352, 1963 H .75 2703 2868 25 -412 -449 .50 -340 33:7 H .75

-64

-112 H .25 424 507 Nv' .50 282 300 .75 -102 - 44 H .25 31 - 27 ('IzzNt) 510 179 19:0 .75. 271 274 25 -173 -180 .510 _{-154 i68} 75 -364 -381 .25 1951 H 2139 H .50 797 839 .75 259 304 .25 54 110 (m'-Yr)' .50 3:61 L 293 .75 410 348 .25 _{1.636 L 1.968}

o(:stat.)

.50 1.099 1.207 1.782 1.7915 25 - 169 - 166 °2'

(.stat.)H

.50 - .532 - .486 .75 -1.460 -1.264 25 1. 502 1. 819 o1' (dyn.) .50 .850 .886 75 1 . 2,115 I,. 2:6.9 .25 H- .193 _L- .183 02' (.dy.) .50 75 H .784 -3 . 0:89 , ' - .72:0 -2 . 8:03 Table 9b h/T = 2.2 Coefficients * _{Fn = 234:}

LOAD-CONDITION o i

ii

III

IV - 1498 1545 - 466 53.3

.25

1165 120:5

.50

1153 1190

.75

1025

1014

H

.25

1437 1494 Yv1

.50

1612 1670 75 1475 1542.

.25

-290

-354

-N,'

.50

- 14,6

-i 60

.75

-75

-87

.2:5 394 472

Nv'

.50

360 423

.75

312 3.7:0

.25

-123

-211

(IzzÑ),"

50 .2:15 2.11

.75

213 213

.25

- 73

-157

HNr' .5'O

.75

-

-132

98

-loi,

-176

12:13 1323 -

'i"

.50 688 ' 769

.75

449 . H

480

.25

- 65 , L

-131

(m-Yr) '

.50'

' 51 - .29 .7:'5 ' H 25

4184

6050

ai'

(stat.) .50

.789

L

.952

75 888

1 215

.25

-9.064

Hr .133

a2'

(stat.)'

.50

_L-

.448

_-

.395

.75 H- .771

H .792

.25

3.740

5.633

a1'

(dyn.) 50 .6.65 . 790

.75

.774

1.0.5.7. .2:5

- .102

- .143

2'

(.dyn. ) i .50 - . 603 . - .. 529

.75

- .912

- .935

Table

9c

h/T = 2... 2

Coefficients *

Fn = .263

i Table _{lOa. Critical- and returnspeedg, fall of vaterlevel.} - 43-Um h/T 1.4 h/T = 1.6 = 2.0 _{h/P = 2.2} h _{0.199 0.227 0.284} 0.312. = * _0.8049 0.9256 1.167 1.285 0.190 0.218 0.275 0.303 1.365 1.462 _{L642 1.724} 0.094 0.082 _{0.065 0.058}

V/J

0.58 0.60 0.63 0.65

U/JÎì 0.326 Um/./ _{O.3O4 Un/j= 0.271 um/J= 0.258}

= O 08 0 445 _Um Z/h 0.021 Um <.V Z/h = 0.016 Um <

V.

Z/h = 0.011 U < Vcr. Z/h = 0.0086

U,/J = 0.042 'u/j' = 0.033. u/j.'= 0.022 uj _{= 0.018} Z = 0.004 m Z = .0.003 m Z = 0.003 m Z = 0.003 m u = 0.057 ,/s u = 0.048

Ws

u = 0.036 s u= 0.031 's

U+Um

____ - 0.09 .U«

-

0.09 = 0.09

'Um

-

0.09

'Um//i 0.407Un/,/ 0.380'UinJJ 0.338T.Ljj _0.323 = 0.10 0.556 _{Um <'V Um <}

Um <

V.

Um < Vc.

Z/h = 0.038 Z/h = 0.028 Z/h = 0.018 Z/h = 0.014

U/,/ _{0.062TJ/' 0o047U/j= 0.03]. u/j=}

0.025:

Z = 0.007 m Z = 0.006 m Z = 0.005 m Z = 0.004 m u = 0.085

Ws

u = 0.069 /s u = 0.051 /s 'u' = 0.031 _nVs

-

0.12

-

0.11 - 0.11 'SUm

-

0.11 Um/j 0.489U/,/ 0.456 _{Um/d 0.406 U/,/ 0.387} Fn = O 12 . 0.067 . Um

<V.

Z/h = 0.067 ' Um < Z/h = 0.046 Um < V Z/h =0.028 _Z/hUm < Vc1.= 0.023 'Uj = 0.094 'uj 0.067 U/= 0.042 u/ _{= 0.034} Z = 0.013 m Z = .0.010 m. _{Z = 0.008 m Z .= 0.007 m.} u 0.128 /s u = 0.098 _{/s u = 0.069}

_Ws

_{u = 0.059} MVS l+Um

-0.14 _{- 0.14 - b..13} 'U-fUm

-0.13

Table lOb. Critical- and returnspeeds, fall of waterlevel. 1VT=2.2 h 0.312 1.285 0.303 1.724 As/Ac 0.058 0.65 Um/jh'= 0.661 = 0.205 1.140 _{Um > Vcr}

Z/h

0.173

u/j= 0.230

Z = 0.065 m = 0.397Ws Ucr+Um - 0 276

UJ= 0.754

= 0.234 1.300 _{Um >} V Zr./h = 0.180

Ur/J

0.230 Z 0.085 m = 0.397ny's Ucr+Um - 0 305 Um/ji' 0.847 Fn = 0.262 1.460 _{Um >} Z/h = 0.180 0.230

Z=0.103m

0.397nVs Ucr+Urn - 0.334