Intro to Naval Architecture 3 2010 Part 3 docx

FLOTATION AND STABILITY 51Figure 4,16 Fluid free surface Effect ofUquidfree surfaces A ship in service will usually have tanks which are partially filled withliquids.. Differentconsidera

Similarly the vertical moment of volume shift is:

From the figure it will be seen that:

This is called the wall-sided formula It is often reasonably accurate for

full forms up to angles as large as 10° It will not apply if the deck edge

is immersed or the bilge emerges It can be regarded as a refinement of

the simple expression GZ = GM sin <p.

Influence on stability of a freely hanging weight

Consider a weight w suspended freely from a point h above its centroid.

When the ship heels slowly the weight moves transversely and takes up

a new position, again vertically below the suspension point As far as theship is concerned the weight seems to be located at the suspensionpoint Compared to the situation with the weight fixed, the ship's

centre of gravity will be effectively reduced by GGi where:

This can be regarded as a loss of metacentric height of GGj

Weights free to move in this way should be avoided but this is notalways possible For instance, when a weight is being lifted by ashipboard crane, as soon as the weight is lifted clear of the deck orquayside its effect on stability is as though it were at the crane head.The result is a rise in G which, if the weight is sufficiently large, couldcause a stability problem This is important to the design of heavy liftships

FLOTATION AND STABILITY 51

Figure 4,16 Fluid free surface

Effect ofUquidfree surfaces

A ship in service will usually have tanks which are partially filled withliquids These may be the fuel and water tanks the ship is using or may

be tanks carrying liquid cargoes When such a ship is inclined slowlythrough a small angle to the vertical the liquid surface will move so as

to remain horizontal In this discussion a quasi-static condition isconsidered so that slopping of the liquid is avoided Differentconsiderations would apply to the dynamic conditions of a ship rolling.For small angles, and assuming the liquid surface does not intersect thetop or bottom of the tank, the volume of the wedge that moves is:11);2 <p dx, integrated over the length, I, of the tank.

Assuming the wedges can be treated as triangles, the moment oftransfer of volume is:

where I\ is the second moment of area of the liquid, or free, surface.

The moment of mass moved =pff»/1, where pf is the density of the liquid

in the tank The centre of gravity of the ship will move because of thisshift of mass to a position Gj and:

where p is the density of the water in which the ship is floating and V

is the volume of displacement

The effect on the transverse movement of the centre of gravity Is to

reduce GZby the amount GGi as in Figure 4.16(b) That is, there is an effective reduction in stability Since GZ= GMsin (p for small angles, the

influence of the shift of G to Gj is equivalent to raising G to G2 on the

centre line so that GGj = GGg tan <p and the righting moment is given

by:

Thus the effect of the movement of the liquid due to its free surface, is

equivalent to a rise of GG^ of the centre of gravity, the 'loss' of GM

being:

Free surface effect GGg = pf /i/pV

Another way of looking at this is to draw an analogy with the loss ofstability due to the suspended weight The water in the tank with a freesurface behaves in such a way that its weight force acts through some

point above the centre of the tank and height I\/v above the centroid

of the fluid in the tank, where v is the volume of fluid In effect the tank

has its own 'rnetacentre' through which its fluid weight acts The fluid

weight is p f v and the centre of gravity of the ship will be effectively

raised through GG^ where:

This loss is the same whatever the height of the tank in the ship or itstransverse position If the loss is sufficiently large, the metacentricheight becomes negative and the ship heels over and may even capsize

It is important that the free surfaces of tanks should be kept to aminimum One way of reducing them is to subdivide wide tanks intotwo or more narrow ones In Figure 4.17 a double bottom tank is shownwith a central division fitted

figure 4.17 Tank subdivision

FLOTATION AND STABILITY 53

If the breadth of the tank is originally B, the width of each of the two

tanks, created by the central division, is J5/2 Assuming the tanks have

a constant section, and have a length, 4 the second moment of area

without division is IB 3 /12 With centre division the sum of the second moments of area of the two tanks is (//12) (B/2) 3 X 2 = 1&/48

That is, the introduction of a centre division has reduced the freesurface effect to a quarter of its original value Using two bulkheads todivide the tank into three equal width sections reduces the free surface

to a ninth of its original value Thus subdivision is seen to be veryeffective and it is common practice to subdivide the double bottom ofships The main tanks of ships carrying liquid cargoes must be designedtaking free surface effects into account and their breadths are reduced

by providing centreline or wing bulkheads

Free surface effects should be avoided where possible and whereunavoidable must be taken into account in the design The operatorsmust be aware of their significance and arrange to use the tanks in waysintended by the designer

The inclining experiment

As the position of the centre of gravity is so important for initial stability

it is necessary to establish it accurately It is determined initially bycalculation by considering all weights making up the ship - steel, outfit,fittings, machinery and systems - and assessing their individual centres

of gravity From these data can be calculated the displacement andcentre of gravity of the light ship For particular conditions of loadingthe weights of all items to be carried must then be added at theirappropriate centres of gravity to give the new displacement and centre

of gravity It is difficult to account for all items accurately in suchcalculations and it is for this reason that the lightship weight and centre

of gravity are measured experimentally

The experiment is called the inclining experiment and involves causing

the ship to heel to small angles by moving known weights knowndistances tranversely across the deck and observing the angles ofinclination The draughts at which the ship floats are noted togetherwith the water density Ideally the experiment is conducted when theship is complete but this is not generally possible There will usually be

a number of items both to go on and to come off the ship (e.g staging,tools etc.) The weights and centres of gravity of these must be assessedand the condition of the ship as inclined corrected

A typical set up is shown in Figure 4.18 Two sets of weights, each of

w, are placed on each side of the ship at about amidships, the port and starboard sets being h apart Set 1 is moved a distance h to a position

Figure 4.18 Inclining experiment

alongside sets 3 and 4 G moves to GI as the ship inclines to a smallangle and B moves to Bj It follows that:

<p can be obtained in a number of ways The commonest is to use two

long pendulums, one forward and one aft, suspended from the deck

into the holds If d and / are the shift and length of a pendulum respectively, tan <p = d/L

To improve the accuracy of the experiment, several shifts of weightare used Thus, after set 1 has been moved, a typical sequence would be

to move successively set 2, replace set 2 in original position followed byset 1 The sequence is repeated for sets 3 and 4 At each stage the angle

of heel is noted and the results plotted to give a mean angle for unitapplied moment When the metacentric height has been obtained, theheight of the centre of gravity is determined by subtracting GM fromthe value of ,KM given by the hydrostatics for the mean draught at which

the ship was floating This KG must be corrected for the weights to go

on and come off The longitudinal position of B, and hence G, can befound using the recorded draughts

To obtain accurate results a number of precautions have to beobserved First the experiment should be conducted in calm water withlittle wind Inside a dock is good as this eliminates the effects of tidesand currents The ship must be floating freely when records are taken

so any mooring lines must be slack and the brow must be lifted clear Allweights must be secure and tanks must be empty or pressed full to avoid

FLOTATION AND STABILITY 55

free surface effects If the ship does not return to its original positionwhen the inclining weights are restored it is an indication that a weighthas moved in the ship, or that fluid has moved from one tank toanother, possibly through a leaking valve The number of people onboard must be kept to a minimum, and those present must go todefined positions when readings are taken The pendulum bobs aredamped by immersion in a trough of water

The draughts must be measured accurately at stem and stern, andmust be read at amidships if the ship is suspected of hogging orsagging The density of water is taken by hydrometer at several positionsaround the ship and at several depths to give a good average figure Ifthe ship should have a large trim at the time of inclining it might not

be adequate to use the hydrostatics to give the displacement and thelongitudinal and vertical positions of B In this case detailed calcula-tions should be carried out to find these quantities for the incliningwaterline

The Merchant Shipping Acts require every new passenger ship to

be inclined upon completion and the elements of its stabilitydetermined,

Stability when docking or grounding

When a ship is partially supported by the ground, or dock blocks, itsstability will be different from that when floating freely The example of

a ship docking is used here The principles are the same in each casealthough when grounding the point of contact may not be on thecentreline and the ship will heel as well as change trim

Figure 4,19 Docking

Usually a ship has a small trim by the stern as it enters dock and asthe water is pumped out it first sits on the blocks at the after end As thewater level drops further the trim reduces until the keel touches theblocks over its entire length It is then that the force on the sternframe,

or after cut-up, will be greatest This is usually the point of most criticalstability as at that point it becomes possible to set side shores in place

to support the ship

Suppose the force at the time of touching along the length is w, and that it acts a distance ~x aft of the centre of flotation Then, if t is the

change of trim since entering dock:

Should the expression inside the brackets become negative the ship will

be unstable and may tip over

Example 4.2

Just before entering drydock a ship of 5000 tonnes mass floats atdraughts of 2.7m forward and 4.2m aft The length betweenperpendiculars is 150m and the water has a density of 1025kg/

m3 Assuming the blocks are horizontal and the hydrostatic datagiven are constant over the variation in draught involved, find theforce on the heel of the sternframe, which is at the afterperpendicular, when the ship is just about to settle on thedockblocks, and the metacentric height at that instant

The value of w can be found using the value of MCT read from the

hydrostatics This MCT value should be that appropriate to the actualwaterline at the instant concerned and the density of water As the

mean draught will itself be dependent upon w an approximate value

can be found using the mean draught on entering dock followed by a

second calculation when this value of w has been used to calculate a

new mean draught Referring now to Figure 4.19, the righting momentacting on the ship, assuming a very small heel, is:

FLOTATION AND STABILITY

Hydrostatic data: KG= 8.5 m, KM= 9.3 m, MCT 1 m = 105 MNm,

LCF = 2.7m aft of amidships

Solution

Trim lost when touching down

Distance from heel of sternframe to LCF

Moment applied to ship when touching down

Trimming moment lost by

ship when touching down

Hence, thrust on keel, w

Loss of GMwhen touching down

Metacentric height when touching down

LAUNCHING

The launch is an occasion in the ship's life when the buoyancy, stability,and strength, must be studied with care If the ship has been built in adry dock the 'launch' is like an undocking except that the ship is onlypartially complete and the weights built in must be carefully assessed toestablish the displacement and centre of gravity position Large shipsare quite often nowadays built in docks but in the more general casethe ship is launched down inclined ways and one end, usually the stern,enters the water first The analysis may be complicated by the launchingways being curved in the longitudinal direction to increase the rate ofbuoyancy build up in the later stages

An assessment must be made of the weight and centre of gravityposition at the time of launch The procedure then adopted is to move

a profile of the ship progressively down a profile of the launch ways,taking account of the launching cradle This cradle is speciallystrengthened at the forward end as it is about this point, the so-called

fore poppet, that the ship eventually pivots At that point the force on the

fore poppet is very large and the stability can be critical As the ship

Figure 4.20 Launching

enters the water the waterline at various distances down the ways can benoted on the profile From the Bonjean curves the immersed sectionalareas can be read off and the buoyancy and its longitudinal centrecomputed The ship will continue in this fashion until the moment ofweight about the fore poppet equals that of the moment of buoyancyabout the same position

The data are usually presented as a series of curves, the launching curves, as in Figure 4.21.

The curves plotted are the weight which will be constant; thebuoyancy which increases as the ship travels down the ways; themoment of weight about the fore poppet which is also effectivelyconstant; the moment of buoyancy about the fore poppet; the moment

of weight about the after end of the ways; and the moment of buoyancyabout the after end of the ways

Figure 4.21 Launching curves

FLOTATION AND STABILITY 59The maximum force on the fore poppet will be the differencebetween the weight and the buoyancy at the moment the ship pivotsabout the fore poppet which occurs when the moment of buoyancyequals the moment of weight about the fore poppet The ship becomesfully waterborne when the buoyancy equals the weight To ensure theship does not tip about the after end of the ways, the moment ofbuoyancy about that point must always be greater than the moment ofweight about it If the ship does not become waterborne before the forepoppet reaches the after end of the ways it will drop at that point This

is to be avoided if possible If it cannot be avoided there must besufficient depth of water to allow the ship to drop freely allowing for thedynamic 'overshoot' The stability at the point of pivoting can becalculated in a similar way to that adopted for docking There will be ahigh hogging bending moment acting on the hull girder which must beassessed The forces acting are also needed to ensure the launchingstructures are adequately strong

The ship builds up considerable momentum as it slides down theways This must be dissipated before the ship conies to rest in the water.Typically chains and other energy absorbing devices are brought intoaction during the latter stages of travel Tugs are on hand to manoeuvrethe ship once afloat in what are usually very restricted waters

STABILITY AT LARGE ANGLES OF INCLINATION

Atwood's formula

So far only a ship's initial stability has been considered That is for small

inclinations from the vertical When the angle of inclination is greaterthan, say, 4 or 5 degrees, the point, M, at which the vertical through theinclined centre of buoyancy meets the centreline of the ship, can nolonger be regarded as a fixed point Metacentric height is 110 longer a

suitable measure of stability and the value of the righting arm, GZ, is

used instead

Assume the ship is in equilibrium under the action of its weight andbuoyancy with W0Lo and WjLj the waterlines when upright and when

inclined through <p respectively These two waterlines will cut off the

same volume of buoyancy but will not, in general, intersect on thecentreline but at some point S

A volume represented by WgSWj has emerged and an equal volume,

represented by LoSLj has been immersed Let this volume be u Using

the notation in Figure 4.22, the horizontal shift of the centre ofbuoyancy, is given by:

This expression for GZ is often called Atwood 's formula.

Figure 4.22 Atwood's formula

Curves of statical stability

By evaluating v and h e hj for a range of angles of inclination it is possible

to plot a curve of GZ against <p A typical example is Figure 4.23 GZ

increases from zero when upright to reach a maximum at A and thendecreases becoming zero again at some point B The ship will capsize ifthe applied moment is such that its lever is greater than the value of GZ

at A It becomes unstable once the point B has been passed OB is known as the range of stability The curve of GZ against (p is termed the

GZ curve or curve of statical stability

Because ships are not wall-sided, it is not easy to determine theposition of S and so find the volume and centroid positions of theemerged and immersed wedges One method is illustrated in Figure4.24, The ship is first inclined about a fore and aft axis through O onthe centreline This leads to unequal volumes of emerged and

Angle of inclination 0

Figure 4,23 Curve of statical stability

FLOTATION AND STABILITY 61

Figure 4.24

immersed wedges which must be compensated for by a bodily rise orsinkage In the case illustrated the ship rises Using subscripts e and ifor the emerged and immersed wedges respectively, the geometry ofFigure 4.24 gives:

For very small angles GZ still equates to GMq), so the slope of the GZ curve at the origin equals the metacentric height That is GM = dGZ/

d<p at (f = 0 It is useful in drawing a GZ curve to erect an ordinate

at (f) = I rad, equal to the metacentric height, and joining the top of

this ordinate to the origin to give the slope of the GZ curve at theorigin

The wall-sided formula, derived earlier, can be regarded as a specialcase of Atwood's formula For the wall-sided ship:

If the ship has a positive GM it will be in equilibrium when GZ is zero,that is:

This equation is satisfied by two values of <p The first is sin </> = 0, or if* = 0 This is the case with the ship upright as is to be expected The

second value is given by:

With both GMand B^M positive there is no solution to this meaning

that the upright position is the only one of equilibrium This also

applies to the case of zero GM, it being noted that in the upright

position the ship has stable, not neutral, equilibrium due to the term inWhen, however, the ship has a negative GM there are two possible

solutions for <p in addition to that of zero, which in this case would be

a position of unstable equilibrium These other solutions are at <f> either side of the upright <p being given by:

The ship would show no preference for one side or the other Such an

angle is known as an angle of loll The ship does not necessarily capsize although if (p is large enough the vessel may take water on board through side openings The GZ curve for a ship lolling is shown in

Figure 4.25

If the ship has a negative GM of 0.08 m, associated with a B$M of 5 m,

<p, which can be positive or negative, is:

This shows that small negative GMcan lead to significant loll angles Aship with a negative GM will loll first to one side and then the other inresponse to wave action When this happens the master shouldinvestigate why the stability is so poor

Figure 4,25 Angle of loll

FLOTATION AND STABILITY 63

Metacentric height in the lolled condition

Continuing with the wall-side assumption, if <pi is the angle of loll, the

value of GMfor small inclinations about the loll position, will be given

by the slope of the GZ curve at that point Now:

Unless <pi is large, the metacentric height in the lolled position will

be effectively numerically twice that in the upright position although ofopposite sign

Cross curves of stability

Cross curves of stability are drawn to overcome the difficulty in defining

waterlines of equal displacement at various angles of heel

Figure 4.26

Figure 4.26 shows a ship inclined to some angle <p Note that S is not

the same as in Figure 4.24 By calculating, for a range of waterlines, thedisplacement and perpendicular distances, SZ, of the centroids of thesevolumes of displacement from the line W through S, curves such as

those in Figure 4.27 can be drawn These curves are known as cross

Figure 4,27 Cross curves of stability

curves of stability and depend only upon the geometry of the ship and

not upon its loading They therefore apply to all conditions in whichthe ship may operate

Deriving curves of statical stability from the cross curves

For any desired displacement of the ship, the values of SZ can be read

from the cross curves Knowing the position of G for the desired

loading enables SZ to be corrected to GZ by adding or subtracting

SG sin <p, when G is below or above S respectively.

Features of the statical stability curve

There are a number of features of the GZ curve which are useful indescribing a ship's stability It has already been shown that the slope of

the curve at the origin is a measure of the initial stability GM The

maximum ordinate of the curve multiplied by the displacement equalsthe largest steady heeling moment the ship can sustain withoutcapsizing Its value and the angle at which it occurs are both important

The value at which GZ becomes zero, or 'disappears', is the largest

angle from which a ship will return once any disturbing moment is

removed This angle is called the angle of vanishing stability The range

of angle over which GZ is positive is termed the range of stability.