Ship Hydrostatics and Stability 2010 Part 6 potx

Statical stability at large angles of heel 117 in Chapter 2, shows that the longitudinal position of the centre of buoyancy changes if the heel angle is large. It happens so because at large heel angles the 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 constant trim may not represent actual stability condition. Jakic (1980) has shown that trim can greatly influence the values of cross-curves and, therefore, that influence should be taken into account. The stability regulations, BV 1033, of the German Navy require, indeed, the calculation of the cross-curves at the trim induced by heel. Modern computer programmes for Naval Architecture include this option. As we shall show in Chapter 9, waves perpendicular or oblique to the ship velocity influence the values of cross-curves and can cause a very dangerous effect called parametric resonance. This effect too must be taken into account and modern computer programmes can calculate cross-curves on waves. The stability regulations of the German Navy take into account the variation of the righting 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. We assume 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 displacement volume, calculate with them the righting arm and plot its values against the heel angle. This plot is called curve of statical stability and it is used to appreciate the stability 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 draw the tangent in the origin. To do this, one should draw a vertical line at the angle 118 Ship Hydrostatics and Stability 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 account when calculating cross-curves of stability. Another influence to be taken into account is that of waves. Table 5.2 summarizes the main terms related to stability at large angles of heel. 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 given between 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 Centre of gravity Curve of statical stability Heel angle (positive starboard down) Keel point - reference point on BL Projected centre of gravity Righting lever Value of stability cross-curve z-coordinate of centre of gravity B Fr centre de carene (C), G Verdrangungsschwepunkt (F), I centre di carena G Fr centre de gravite, G Massenschwerpunkt, I centro di gravita Fr courbe de stabilite, G Stabilitatskurve, I curva di stabilita 4> HELANG Fr angle de bande, angle de gite, G Krangungswinkel, I angolo di inclinazione traversale, sbandamento K F point le plus has de la carene, G Kielpunkt, I intersezione della linea base con la sezione maestra Z G Projizierte Massenschwerpunkt ~GZ GZ F bras de levier (GK), G Aufrichtenden Hebelarm, I braccio radrizzante Ik LK Fr pantocarenes, bras de levier du couple de redressement, 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 Statical stability at large angles of heel 119 Ship Lido 9 - Cross-curves of stability 20 V(m 3 ) Figure 5.5 Three-dimensional cross-curves of stability of Ship Lido 9 5.6 Example Figure 5.5 is a three-dimensional representation of the cross-curves of the Ship Lido 9. 5.7 Exercises Exercise 5.1 Plot in one figure the righting-arm curves and the tangents in origin of the Ship Lido 9, for V = 50.5 m 3 and KG-values 1.8, 2.0, 2.4 and 2.6m. Comment the influence 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. 6 Simple models of stability 6.1 Introduction In Chapter 5 we learnt how to calculate and how to plot the righting arm in the curve of statical stability. It may be surprising that for a very long period the metacentric height and the curve of righting arms were considered sufficient for appreciating 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 advance and 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 a ship-stability criterion. In 1933, Pierrottet wrote in a publication of the test basin in Rome that the stability of a ship must be assessed by comparing the heeling moments with the righting moment. He detailed his proposal in 1935, in a meeting of INA, but had no immediate followers. Thus, in 1939 Rahola published in Helsinki his doctoral thesis; it was based on extensive statistics and a very profound analysis of the qualities of stable and unstable vessels. Rahola proposed then a stability criterion that considered only the metacentric height and the curve of the righting arm. The Naval-Architectural community appreciated Rahola's work and his proposal was used, indeed, as a stability standard and stood at the basis of stability regulations issued later by national and international authorities. It was only after the Second World War that the issue of comparing heeling and 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 pull developed 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 122 Ship Hydrostatics and Stability 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, M g , 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. From now on we constantly use the displacement mass in calculations. At this point it may seem that we defined the heeling arm as above just to be able to compare the righting 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 2.5 1,5 b I CD 0.5 0.5 Curve of statical stability, Lido 9, V = 50.5 m 3 , KG = 2.6 m 10 20 30 40 50 60 Heel angle (°) 70 80 90 Figure 6.1 Angles of static equilibrium Simple models of stability 123 towards port. In the same figure the heeling arm is also positive, meaning that the 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 are negative. For a ship heeled towards port, the righting moment tends, indeed, to return the vessel towards starboard, therefore it has another sign than in the region of positive heel angles. The heeling moment, however, tends in general to heel 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, models that allow reasonably fast calculations. Approximate as they may be, those mod- els stand at the basis of regulations that specify the stability requirements for various 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 the simplifying assumptions are necessary in Naval-Architectural practice. We can appreciate the stability of a vessel by comparing the righting arm with the heeling arm as long as the heeling moment is applied gradually and inertia forces and moments can be neglected. When the heeling moment appears suddenly, as caused, for example, by a gust of wind, one has to compare the 2.5 2 1.5 0.6 £ O 0 I CD -0.5 -1,5 -2 -2.5 -GM : -100 -80 -60 -40 -20 0 20 40 60 80 100 Heel angle (°) Figure 6.2 Curve of statical stability extended for heeling towards both ship sides 124 Ship Hydrostatics and Stability heeling energy with the work done by the righting moment. Such situations are discussed in the section on dynamical stability. In continuation we show how moving loads, solid or liquid, endanger the ship stability, and we develop formulae for calculating the reduction of stability. Other situations in which the stability is endangered are those of grounding or positioning in dock. We show 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 negative metacentric 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 </> st i and 0 st 2- Both points correspond to positions of statical equilibrium 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. Only the first point corresponds to a position of stable equilibrium, while the second point 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, 0 st i, and assume that some perturbation causes the ship to heel further to starboard by a small angle, 5$. When the perturbation ceases at the angle 0 stl + 8<j>, the righting arm is larger than the heeling arm, returning thus the ship towards its initial position, at the angle 0 st i. Conversely, if the perturbation causes the ship to heel towards port, to an angle </> st i — 5$, when the perturbation ceases the righting arm is smaller than the heeling arm, so that the ship returns towards the initial position, 0 st i. This situation corresponds to the definition of stable equilibrium given in Section 2.4. Let us see now what happens at the second angle of equilibrium, </> st 2. If some perturbation causes the ship to incline further to starboard, the heeling arm will be larger than the righting arm and the ship will capsize. If the perturbation inclines the ship towards port, after its disappearance the righting arm will be larger than the heeling arm and the ship will incline towards port regaining equilibrium at the first static angle, </> st i. We conclude that the second static angle, 0 st 2, 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 situation the wind heeling arm is maximal. In the simplest possible assumption the wind generates a force, Fy, that acts in the centroid of the lateral projection of the Simple models of stability 125 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^. 126 Ship Hydrostatics and Stability Then, the lever arm of the torque would be proportional to cos 0. Summing up, the wind heeling arm equals PvAv cos 6 (' T\ p v Av(hy -f T/2) 9 ' 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 Pv = CwP^w ( 6 - 3 ) 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 F 2 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 Simple models of stability 127 water reaction acts at half-draught, i.e. at a height T/2 above the baseline. The two forces, FTC an d 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: T (6.4) 9 #TC V 2 The speed V to be used in Eq. (6.4) is the speed in turning, smaller than the speed achieved when sailing on a straight line path. The turning radius, RTC> an d the 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, are available, they should be substituted in Eq. (6.4). The stability regulations of the German Navy, BV 1033, provide formulae for approximations to be used in the early design stages of naval ships (see Chapter 10). A discussion of this subject can be found in Wegner (1965). This author uses a non-dimensional factor (6.5) where Vb is the ship speed in turning and VQ, the speed on a straight line path. Substituting into Eq. (6.4) yields - / _ T\ (KG }cos(f> (6.6) gL pp \ 2 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 the values 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 the ship. There are two cases when passengers can do this: when attracted by a beautiful 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 arm can 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 [...]... dynamic angle are 0 = 0, < 0 (6- 20) Substituting the first part of Eq (6. 20) in Eq (6. 16) , we obtain _ GZd(f>= — d0 (6. 21) Equation (6. 21) represents the condition of equality of the areas under the righting and the heeling arms The second part of Eq (6. 20) when applied to Eq (6. 12) yields the condition ~GZ > ^ (6. 22) Simple models of stability 133 Curve of statical stability, Lido 9, V = 50.5 m3... Curve of statical stability, Lido 9, V = 50.5 m3, KG = 2 .6 m •GZ • • • • • • ; Area under GZ 0 10 20 30 Figure 6. 4 Dynamical stability 40 50 60 70 80 90 130 Ship Hydrostatics and Stability for having proposed the calculation of dynamical stability as early as 1850 It took several marine disasters and many years until the idea was accepted by the Naval-Architectural community In Figure 6. 4, we marked... 60 80 100 a5 £ o o o -0.5 Work difference GZ -40 0 -20 60 100 Heel angle (°) Figure 6. 7 Two limiting cases of instability Figure 6. 7 shows two limiting cases In the upper plot the first part of condition (6. 19) is fulfilled, while the second is not Therefore, in this case there is no angle of stable statical equilibrium and the ship is lost In the lower Figure 6. 7 the areas under the righting-arm and. .. Chapter 9) 6. 9 Loads that adversely affect stability 6. 9.1 Loads displaced transversely In Figure 6. 8, we consider that a mass m, belonging to the ship displacement A, is moved transversely a distance d A heeling moment appears and its value, for any heeling angle 0 is dm cos As a result, the ship centre of gravity G moves to a new position, GI, the distance GGi being equal to (6. 30) 1 36 Ship Hydrostatics. .. endangering thus the ship stability A partially filled tank is known as a slack tank 138 Ship Hydrostatics and Stability M Zeff Figure 6. 10 Effective metacentric height Figure 6 1 l(a) shows a tank containing a liquid whose surface is free to move within a large range of heeling angles without touching the tank top or bottom Let us consider that the liquid volume behaves like a ship hull and consider the... (6. 14) 132 Ship Hydrostatics and Stability and integrate between an initial angle, 0o, and a final angle, f, /*0f J r4>f _ . -0.5 -1,5 -2 -2.5 -GM : -100 -80 -60 -40 -20 0 20 40 60 80 100 Heel angle (°) Figure 6. 2 Curve of statical stability extended for heeling towards both ship sides 124 Ship Hydrostatics and Stability heeling. 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 130 Ship Hydrostatics and Stability for. (6. 24) Substituting the above expression into Eq. (6. 23) and rearranging yields 9GM -T7T dt 2 (6. 25) 134 Ship Hydrostatics and Stability With the notation (6. 26) the solution of this equation is of the