Ship Hydrostatics and Stability 2010 Part 6 potx

Similarly, we should determine the heeling moments and then compare them with the righting moment.. In 1933, Pierrottet wrote in a publication of the testbasin in Rome that the stability

in Chapter 2, shows that the longitudinal position of the centre of buoyancychanges if the heel angle is large It happens so because at large heel anglesthe waterplane area ceases to be symmetric about the centreline If the centre

of buoyancy moves along the ship, while the position of the centre of gravity

is constant, the trim changes too Therefore, cross-curves calculated at constanttrim may not represent actual stability condition Jakic (1980) has shown thattrim can greatly influence the values of cross-curves and, therefore, that influenceshould be taken into account The stability regulations, BV 1033, of the GermanNavy require, indeed, the calculation of the cross-curves at the trim induced byheel Modern computer programmes for Naval Architecture include this option

As we shall show in Chapter 9, waves perpendicular or oblique to the shipvelocity influence the values of cross-curves and can cause a very dangerouseffect called parametric resonance This effect too must be taken into accountand modern computer programmes can calculate cross-curves on waves Thestability regulations of the German Navy take into account the variation of therighting arm in waves (see Arndt, 1965; Arndt, Brandl, and Vogt, 1982)

5.5 Summary

In this chapter, we dealt with the righting moment at large angles of heel, MR =

A(7Z The quantity GZ, called righting arm, is the length of the perpendicular

drawn from the centre of gravity, G, to the line of action of the buoyancy force Weassume that the ship heels at constant displacement This is the desired situation

in which the ship neither loses loads nor takes water aboard Then, the factor A

is constant and the variation of the righting moment with heel is described by

the variation of the righting arm GZ The value of the righting arm is calculated

from

~GZ = 4 - ~KG sin <j>

where £&, called value of stability cross-curve, is the distance from the reference

point K to the line of action of the buoyancy force, KG, the distance of the centre

of gravity from the same point K, and 0, the heel angle It is recommended to take the point K as the lowest hull point The values of the stability cross-curves,

Ik, are usually represented as functions of the displacement volume, with the

heel angle as parameter

One can read in this plot the values corresponding to a given displacementvolume, calculate with them the righting arm and plot its values against the heelangle This plot is called curve of statical stability and it is used to appreciate thestability of the ship, at a given displacement and height of the centre of gravity

To check the correctness of the righting-arm curve, it is recommended to drawthe tangent in the origin To do this, one should draw a vertical line at the angle

of 1 rad and measure on the vertical a length equal to the metacentric height,

GM The tangent is the line that connects the origin of coordinates to the point

found as in the previous sentence

The trim changes as the ship heels That effect should be taken into accountwhen calculating cross-curves of stability Another influence to be taken intoaccount is that of waves

Table 5.2 summarizes the main terms related to stability at large angles ofheel As in Chapter 1, we note by 'Fr' the French term, by 'G' the German term,and by T the Italian term Old symbols used once in those languages are givenbetween parentheses

Table 5.2 Terms related to stability at large angles of heel

English term Symbol Computer Translations

notation (old European symbol) Centre of buoyancy

G Pantocarenenwert bezogen auf K

KG ZKG Fr distance du centre de gravite a la

ligne d'eau zero,

G z-Koordinate des Massenschwerpunktes,

I distanza verticale del centro di gravita

Ship Lido 9 - Cross-curves of stability

Plot in one figure the righting-arm curves and the tangents in origin of the Ship

Lido 9, for V = 50.5 m3 and KG-values 1.8, 2.0, 2.4 and 2.6m Comment theinfluence of the centre-of-gravity height

Exercise 5.2

Draw the curve of statical stability of the Ship Lido 9 for a displacement in sea water A — 35.3 t and a height of the centre of gravity KG = 2.1 m Use data

in Tables 4.2 and 5.1

Simple models of stability

6.1 Introduction

In Chapter 5 we learnt how to calculate and how to plot the righting arm in thecurve of statical stability It may be surprising that for a very long period themetacentric height and the curve of righting arms were considered sufficient forappreciating the ship stability We do not proceed so in other engineering fields

As pointed out by Wendel (1965), one first finds out the resistance to ship advanceand only afterwards dimensions the engine Also, we first calculate the load on

a beam and only afterwards we dimension it Similarly, we should determine

the heeling moments and then compare them with the righting moment It was

only at the beginning of the twentieth century that Middendorf proposed such a

procedure for large sailing ships His book, Bemastung und Takelung der Schiffe,

was first published in Berlin, in 1903, and it contained the first proposal for aship-stability criterion In 1933, Pierrottet wrote in a publication of the testbasin in Rome that the stability of a ship must be assessed by comparing theheeling moments with the righting moment He detailed his proposal in 1935,

in a meeting of INA, but had no immediate followers Thus, in 1939 Raholapublished in Helsinki his doctoral thesis; it was based on extensive statisticsand a very profound analysis of the qualities of stable and unstable vessels.Rahola proposed then a stability criterion that considered only the metacentricheight and the curve of the righting arm The Naval-Architectural communityappreciated Rahola's work and his proposal was used, indeed, as a stabilitystandard and stood at the basis of stability regulations issued later by nationaland international authorities

It was only after the Second World War that the issue of comparing heelingand righting arms was brought up again German researchers used then a very

appropriate term: Lever arm balance (Hebelarm Bilanz) Eventually, newer

sta-bility regulations made compulsory the comparison of lever arms and we show

in this chapter how to do it

Heeling moments can be caused by wind, by the centrifugal force developed

in turning, by transverse displacements of masses, by towing or by the lateral pulldeveloped in cables that connect two vessels during the transfer of loads at sea

In Chapter 5 we have shown that, when the ship heels at constant displacement,

it is sufficient to consider the righting arm as an indicator of stability Then, to

assess the ship stability it is necessary to compare the righting arm with a heeling

arm According to the DIN-ISO standard, we note the heeling arm by the letter

I and indicate the nature of the righting arm by a subscript To obtain a generic heeling arm, £ g , corresponding to a heeling moment, Mg, we divide that moment

by the ship weight

(6.1)

where A is the displacement mass and g, the acceleration due to gravity In older

practice it has been usual to measure the displacement in unit of force Then,instead of Eq (6.1) one had to use

Much attention should be paid to the system of units used in calculation Fromnow on we constantly use the displacement mass in calculations At this point itmay seem that we defined the heeling arm as above just to be able to compare therighting arm with a quantity having the same physical dimensions (and units!)

In Section 6.7, we prove that this definition is mathematically justified

In Figure 6.1, we superimposed the curve of a generic heeling arm, £g, over

the curve of the righting arm, GZ For almost all positive heeling angles shown

in the plot the righting arm is positive We define the righting arm as positive

if when the ship is heeled to starboard, the righting moment tends to return it

towards port In the same figure the heeling arm is also positive, meaning thatthe corresponding heeling moment tends to incline the ship towards starboard.What happens if the ship heels in the other direction, i.e with the port side down?Let us extend the curve of statical stability by including negative heel angles,

as in Figure 6.2 The righting arms corresponding to negative heel angles arenegative For a ship heeled towards port, the righting moment tends, indeed,

to return the vessel towards starboard, therefore it has another sign than in theregion of positive heel angles The heeling moment, however, tends in general toheel the ship in the same direction as when the starboard is down and, therefore,

it is positive Summarizing, the righting-arm curve is symmetric about the origin,while the heeling-arm curves are symmetrical about the lever-arm axis

In this chapter we present simplified models of various heeling arms, modelsthat allow reasonably fast calculations Approximate as they may be, those mod-els stand at the basis of regulations that specify the stability requirements forvarious categories of ships In most cases, practice has shown that ships comply-ing with the regulations were safe The requirements themselves are explained

in Chapters 8 and 10 By the end of this chapter, we briefly explain why thesimplifying assumptions are necessary in Naval-Architectural practice

We can appreciate the stability of a vessel by comparing the righting armwith the heeling arm as long as the heeling moment is applied gradually andinertia forces and moments can be neglected When the heeling moment appearssuddenly, as caused, for example, by a gust of wind, one has to compare the

heeling energy with the work done by the righting moment Such situationsare discussed in the section on dynamical stability In continuation we showhow moving loads, solid or liquid, endanger the ship stability, and we developformulae for calculating the reduction of stability Other situations in whichthe stability is endangered are those of grounding or positioning in dock Weshow how to predict the moment in which those situations may become critical.This chapter also discusses the situations in which a ship sails with a negativemetacentric height.

6.2 Angles of statical equilibrium

Figure 6.1 shows the curve of a heeling arm, £ g , superimposed on the curve

of the righting arm, GZ In general, those curves intersect at two points; they

are noted here as </>sti and 0st2- Both points correspond to positions of staticalequilibrium because at both points the righting arm and the heeling arm are equal,and, therefore, the righting moment and the heeling moment are also equal Onlythe first point corresponds to a position of stable equilibrium, while the secondpoint corresponds to a situation of unstable equilibrium In this section, we give

an intuitive proof of this statement; for a rigorous proof, see Section 6.7.Let us first consider the equilibrium in the first static angle, 0sti, and assumethat some perturbation causes the ship to heel further to starboard by a small

angle, 5$ When the perturbation ceases at the angle 0stl + 8<j>, the righting arm

is larger than the heeling arm, returning thus the ship towards its initial position,

at the angle 0sti Conversely, if the perturbation causes the ship to heel towardsport, to an angle </>sti — 5$, when the perturbation ceases the righting arm is

smaller than the heeling arm, so that the ship returns towards the initial position,

0sti This situation corresponds to the definition of stable equilibrium given inSection 2.4

Let us see now what happens at the second angle of equilibrium, </>st2 If someperturbation causes the ship to incline further to starboard, the heeling arm will belarger than the righting arm and the ship will capsize If the perturbation inclinesthe ship towards port, after its disappearance the righting arm will be larger thanthe heeling arm and the ship will incline towards port regaining equilibrium

at the first static angle, </>sti We conclude that the second static angle, 0st2,corresponds to a position of unstable equilibrium

6.3 The wind heeling arm

We use Figure 6.3 to develop a simple model of the heeling moment caused by

a beam wind, i.e a wind perpendicular to the centreline plane In this situationthe wind heeling arm is maximal In the simplest possible assumption the windgenerates a force, Fy, that acts in the centroid of the lateral projection of the

Figure 6.3 Wind heeling arm

above-water ship surface, and has a magnitude equal to

FV = pyAy

where py is the wind pressure and Ay is the area of the above-mentioned

pro-jection of the ship surface Let us call Ay sail area.

Under the influence of the force Fy the ship tends to drift, a motion opposed

by the water with a force, R, equal in magnitude to Fy To simplify calculations

we assume that R acts at half-draught, T/2 The two forces, Fy and R, form a

torque that inclines the ship until the heeling moment equals the righting moment

The value of the heeling moment in the upright condition is pyAy(hy + T/2), where hy is the height of the sail-area centroid above W^L® The heeling arm

in upright condition is

,ft. Pv A v (h v + T/2)

How does the heeling arm change with the heeling angle? In the case of a 'flat'

ship, i.e for B = 0, the area exposed to the wind varies proportionally to cos </>.

In Figure 6.3, we show that for a flat ship the forces Fy and R would act in the centreline plane, both horizontally, i.e parallel to the inclined waterline W^L^.

Then, the lever arm of the torque would be proportional to cos 0 Summing up, the wind heeling arm equals

' hv H cos 6 = —r —- cos

2; ^A

(6.2)This is the equation proposed by Middendorf and that prescribed by the stability regulations of the US Navy; it can be found in more than one textbook on Naval Architecture where it is recommended for all vessels The reader may feel some doubts about the strong assumptions accepted above In fact, other regulatory bodies than the US Navy adopted wind-heeling-arm curves that do not behave like cos 2 </> The respective equations are described in Chapters 8 and 10 Our

own critique of the above model, and a justification of some of its underlying assumptions, are presented in Section 6.12.

The wind pressure, pv» is related to the wind speed, Vw, by

where c w is an aerodynamic resistance coefficient and p is the air density The

coefficient c w depends on the form and configuration of the sail area An average value for c w is 1.2 Wegner (1965) quotes a research that yielded 1.00 < c w < 1.36, and two Japanese researchers, Kinohita and Okada, who measured c w

values ranging between 0.95 and 1 24 Equation (6.3) shows that the wind heeling arm is proportional to the square of the wind speed In this section, we considered the wind speed as constant over all the sail area This assumption is acceptable for a fast estimation of the wind heeling arm However, we may know from our own experience that wind speed increases with height above the water surface Some stability regulations recognize this phenomenon and we show in Chapters 8 and 10 how to take it into account Calculations with variable wind speed, i.e.

considering the wind gradient, yield lower, more realistic heeling arms for small

vessels whose sail area lies mainly in the low- wind-speed region It may be worth mentioning that engineers take into account the wind gradient in the design of tall buildings and tall cranes.

6.4 Heeling arm in turning

When a ship turns with a linear speed V, in a circle of radius RTC, & centrifugal force, FTC > develops; it acts in the centre of gravity, G, at a height KG above

the baseline From mechanics we know that

F2

Under the influence of the force FTC the ship tends to drift, a motion opposed by the water with a reaction R To simplify calculations, we assume again that the

water reaction acts at half-draught, i.e at a height T/2 above the baseline The

two forces, FTC and R, form a torque whose lever arm in upright condition is (KG - T/2) For a heeling, flat ship this lever arm is proportional to cos </>.

Dividing by the displacement force, we obtain the heeling lever of the

centri-fugal force in turning circle:

speed in turning, V, are not known in the first stages of ship design If results

of basin tests on a ship model, or of sea trials of the ship, or of a sister ship, areavailable, they should be substituted in Eq (6.4) The stability regulations of theGerman Navy, BV 1033, provide formulae for approximations to be used in theearly design stages of naval ships (see Chapter 10) A discussion of this subjectcan be found in Wegner (1965) This author uses a non-dimensional factor

Quoting Handbuch der Werften, Vol VII, Wegner shows that for 95% of 80 cargo

ships the values of CD ranged between 0.19 and 0.25 For a few trawlers thevalues ranged between 0.30 and 0.35

6.5 Other heeling arms

A dangerous situation can arise if many passengers crowd on one side of theship There are two cases when passengers can do this: when attracted by abeautiful seascape or when scared by some dangerous event In the latter case,passengers can also be tempted to go to upper decks The resulting heeling armcan be calculated from

TIT)

ip — — (y cos (f) -f z sin </>) (6.7) where n is the number of passengers, p, the average person mass, y, the horizontal coordinate of the centre of gravity of the crowd and z, the vertical translation

of said centre The second term between parentheses accounts for the virtual

metacentric-height reduction Wegner (1965) recommends to assume that up

to seven passengers can crowd on a square metre, that the average mass of apassenger plus some luggage is 80 kg, and that the height of a passenger's centre

of gravity above deck is l.lm Smaller values are prescribed by the regulationsdescribed in Chapters 8 and 10 Wegner recommends to include in the deckarea all areas that can be occupied by panicking people, e.g tables, benchesand skylights Other heeling moments can occur when a tug tows a barge Thebarge can drift and then the tension in the towing cable can be decomposed intotwo components, one parallel to the tug centreline and the other perpendicular

to the first The latter component can cause capsizing of the tug The process

is very fast and there may be no survivors To avoid this situation tugs must beprovided with quick-release mechanisms that free instantly the towing cable.Lateral forces also appear when fishing vessels tow nets or when two vessels areconnected by cables during replenishment-at-sea operations Special provisionsare made in stability regulations for the situations mentioned above; they arepresented in Chapters 8 and 10 Icing is a phenomenon known to ship crewssailing in very cold zones The accumulation of ice has a double destabilizingeffect: it raises the centre of gravity and it increases the sail area The importance

of ice formation should not be underestimated For example, Arndt (1960a) citescases in which blocks of ice 1 m thick developed on a poop deck, or walls of 60 cm

of ice formed on the front surface of a bridge Therefore, stability regulationstake into account the effect of ice

6.6 Dynamical stability

Until now we assumed that the heeling moments are applied gradually and that

inertial moments can be neglected Shortly, we studied statical stability Heeling

moments, however, can appear, or increase suddenly For example, wind speed isusually not constant, but fluctuates Occasionally, sudden bursts of high intensity

can occur; they are called gusts As another example, loosing a weight on one

side of a ship can cause a sudden heeling moment that sends down the other

side In the latter cases we are interested in dynamical stability It is no more

sufficient to compare righting with heeling arms; we must compare the energy

of the heeling moment with the work done by the opposing righting moment Itcan be easily shown that the energy of the heeling moment is proportional to thearea under the heeling-arm curve, and the work done by the righting moment

is proportional to the area under the righting-arm curve To prove this, let us

remember that the work done by a force, F, which produces a motion from x\

to #2 is equal to

If the path of the force F is an arc of circle of radius r, the length of the arc that subtends an angle d</> is dx = r d(j> Substituting into Eq (6.8), we get

W

r(f>2 f4>

where M is a moment.

A ship subjected to a sudden heeling moment Mh, applied when the roll angle

is 0i, will reach for an instant an angle fa up to which the energy of the heeling

moment equals the work done by the righting moment, so that

or

GZdfi

(6.10)

(6.11)

This condition is fulfilled in Figure 6.4 where the area under the heeling-arm

curve is A 2 + AS, and the area under the righting-arm curve is A\ + A% As A 3 is

common to both areas, the condition is reduced to A\ = A% Moseley is quoted

Curve of statical stability, Lido 9, V = 50.5 m3, KG = 2.6 m

•GZ • • • • • • ; Area under GZ

0 10 20 30 40 50 60 70 80 90

Figure 6.4 Dynamical stability

for having proposed the calculation of dynamical stability as early as 1850 Ittook several marine disasters and many years until the idea was accepted by theNaval-Architectural community.

In Figure 6.4, we marked with </>dyn the maximum angle reached by the shipafter being subjected to a gust of wind An elegant way to find this angle is tocalculate the areas under the curves as functions of the heel angle, </>, plot theresulting curves and find their points of intersection The algorithm for calculat-ing the integrals with variable upper limit is described in Section 3.4

In Figure 6.4, we assumed that the gust of wind appeared when the ship was

in an upright condition, i.e </>i = 0 As shown in Figure 6.5, the situation is less

severe if fa > 0, and more dangerous if fa < 0 In both graphs the maximum

dynamical angle is found by plotting the curve