SHIP BUOYANCY AND STABILITY

Transcription

1 SHIP BUOYANCY AND STABILITY Lecture 04 Ship stability-z curve 09/11/2017 Ship Buoyancy and Stability 1

2 Literature J. Matusiak: Laivan kelluvuus ja vakavuus Biran A. B., Ship Hydrostatics and Stability, 2003 J. Matusiak: Short Introduction to Ship Theory (Part 1) Rawson, K. J J. Basic Ship Theory Volume Buoyancy and Stability of a Ship, Finnish text book A Ship Stability book in English. Shorten version of the Finnish textbook, in English Check the library Available on KNOVEL as e-book

3 BEFORE THIS LECTURE Ship initial stability Small angle assumptions Metacentric height Transversal stability VS Longitudinal stability Now, you should be able to: Define the concept of ship stability and how to use it Define the metecentric height and why the metacenter has to be above the center of gravity Describe the metacenter evolute Motivate why generally transversal stability is more important than the longitudinal one. Calculate how the trim changes by moving/loading weights longitudinally 09/11/2017 Ship Buoyancy and Stability 3

4 Introduction Following lecturers Ship equilibrium and introduction to hydrostatics Ship initial Stability The stability curve (Z curve) Cross curve of stability Z curve according to the ship design Heeling Moments Preparation for the laboratory test Dynamic stability Second generation of intact stability criteria Ship Damage Stability Stability special topics Ship Buoyancy and Stability 4

5 Ship stability at large (or finite) angles M ulk It is usually that a ship experiences heeling angles over than In that case the initial stability approach does not work properly, as the centers of buoyancy do not belong to a circular arc anymore. Ship under wind and wave external loads will have high heeling angle, for which the initial stability approach is no more valid. The centers of buoyancy move on a curve that is not circular and do not belong to the YZ plan. The shape of this curve depends on the hull shape. From the equilibrium of the momentum of the forces W-, let s evaluate the righting moment arising at high or finite angle. M ulk + M st = 0 f WL 0 WL f B 0 K W M f M 0 h N f Z B f D

6 Transverse stability at large heel angles M st = Z = h = N sin = B 0 N sin B 0 sin. Prometacenter or false metacenter lim 0 N M 0 The B-curve is not any more a circular arc. Neither metacenter remains at the same position. For a normal ship forms (vertical sides), large inclination is followed by a shift of metacenter ship s plane off the symmetry plane and upwards. Only ships with circular sides will continue to have the B-curve as a precise circular arc

7 We defined the metacentric radius B 0 M 0 for the initial equilibrium condition with no trim and no heel as the radius of the circumference, with center in M 0, on which laid the center of buoyancy for small heeling angle. For finite heeling angles, the center of buoyancy usually do not belong to the YZ plan and do not laid on the circumference of the metacentric method. Is possible to evaluate for each heeled condition the relative metacenter (on the actual buoyancy line, not the z axix) and metacentric radius. Let s define metacentric evolute the locus of the metacenters at different heeling angles. Evolute in geometry is defined as the locus of the centers of curvature of another curve. Metacenter evolute For the metacenter evolute properties, the tangent to this curve in a generic point M is the buoyancy direction. On this direction the quantity B M define the current center of buoyancy B. The intersection of the buoyancy direction with the vertical axis, for large angles identifies the false prometacenter N Only for small angle prometacenter N and metacenter M coincides with the initial metacenter M 0 WL 0 WL WL' g 1 M' M M 0 F B 0 B 0 M 0 = I T B B' g 2

8 Z curve Ship Buoyancy and Stability 8

9 Stability restoring moments Z KN K sin Z curve written in this way highlighs the dependance on a hull geometry term and on the vertical center of gravity term. KN is the value of the stability cross curve and depends on the ship shape only. N The results regarding the restoring moments, obtained throught graphic way, could be obtained solving the moments equation (see next page). It could be notice that (see also slide #10): KN y B cos z sin B 09/11/2017 Ship Buoyancy and Stability 9

10 Cross curves of stability The general formula to express the righting arm without approximation is: Z N sin ( B0 N B0)sin KN Ksin WL 0 WL B 0 W M ulk M M 0 h L N Z B Shape stability- weight stability vakavuusmomenttivarsi painovakavuusvarsi B 0 jäännösvakavuus M 0 N Z B The shape stability term, KN, depends only on the ship geometry, and it is defined as the distance of the buoyancy line from the origin of the reference system K. KN(, *,) takes the name of cross curves of stability K N muotovakavuusvarsi KN KN sin y B cos z B sin 09/11/2017 Ship Buoyancy and Stability 10

11 KN sin tai B 0 N sin [m] Cross curves of stability [m 3 ] = 10 0 KN(, *,) The cross curves calculation is based on the ship hydrostatics. Fixing the trim, *, the curves are function of the heeling angle and of the displacement i.e. the volume iven a ship with a known weight distribution (W, LC, TC, VC) find the equilibrium condition ( 1, *, * ). The ship, normally, presents zero heel equilibrium condition. With the cross curves of stability for the trim *, enter with the volume 1 and read all the value of KN for the different heeling angle. Then, knowing the VC (i.e. K) value, it is possible to evaluate the Z curve following the general formula. Z( ) KN( ) Ksin 09/11/2017 Ship Buoyancy and Stability 11

12 Cross curves of stability Usually several cross curves of stability diagrams are provided per different trims, including no trim. The calculation of the Z curve, for trim different by those provided, the interpolation of the KN values for the closest trims is necessary. It should be noticed that the KN curves take into account the longitudinal equilibrium for each heeling angle for a finite heel, the ship LCB will change, changing the trim of the vessel. The approximate methods are affected by a lot of errors. Before the computer age, approximate methods were the fastest way to create a stability diagram. Anyway it took even a day. Such approximations led to errors in evaluated ship stability, with critical consequences. Computer aided calculation is the only way to calculate the Z curve, avoiding significant errors. 09/11/2016 Ship Buoyancy and Stability 12

13 SOME CHARACTERISTICS AND FEATURES OF THE Z CURVE 09/11/2016 Ship Buoyancy and Stability 13

14 Static stability lever curve lim N 0 M 0 Z N sin ( B N B0 0 ) sin dz d d B 0 N d sin B0 N cos B0 cos dz d 0 B 0 M 0 B 0 M 0 ( r a) A 1rad=57.3 dz d 0 tan AB 1rad h Z M 0 M 0 AB B at small heel angles initial stability model applies Ship Buoyancy and Stability 14

15 Heel angle due to a static load Using the Z curve (named also h curve) it is possible to find the ship heeling angle due to external loads M ulk At the equilibrium M ulk + M st = 0 Z h h [m] h max C A B l ulk dh/df > 0 dh/df = 0 dh/df < 0 M ulk l ulk M st = Z l ulk = Z f A f C f B f K kallistuskulma f [aste] Two equilibrium conditions: only A is a stable equilibrium. dm st /df < 0 dh/d > 0 B is an unstable condition dh/d < 0 The maximum statical external moment sustained by the ship is: h max (see point C) 09/11/2017 Ship Buoyancy and Stability 15

16 h [m] h max Stability range dh/df = 0 C A B l ulk dh/df > 0 dh/df < f A f C f B f K kallistuskulma f [aste] The ship can experience istantaneous angles over C without capsizing (transient stages-dynamic stability) For all < K the righting arm is positive i.e. h > 0. That means that the ship for every < K will come back into the initial position as soon as the external perturbation is removed. The range between the initial position (typically =0) and = K, takes the name of stability range. Over K the ship will capsize. K takes the name of capsizing angle or angle of vanishing stability 09/11/2017 Ship Buoyancy and Stability 16

17 Freeboard effects on stability Z y B cos z B sin Ksin B * f tan 2 * 2 f arctan B Freeboard is the distance between the draft and the main watertight deck (weather deck). A lower freeboard means that the ship will immerse the deck for smaller angles. With the deck underwater the I T reduces and the ship will have soon a lower metacenter. As consequence, the ship will put underwater the hatch and other openings for smaller angles. 09/11/2017 Ship Buoyancy and Stability 17

18 Freeboard effects on stability 2 f 2 f 1 f 2 >f 1 Let s compare two ships: Ship1 and Ship2 have the same displacement, the same breadth, the same center of gravity and f 2 >f 1. The two ships will have the same metacentric radius and the same metacentric height M r a BM B The initial stability will be the same for the two ships i.e. the righting arm curve will have the same tangent in the origin. The Ship2, instead, will put the deck underwater for higher angles, being more stable than Ship1. IT BM r 09/11/2017 Ship Buoyancy and Stability 18

19 Freeboard effects on stability 3 Ship1 and Ship2 have the same displacement, the same breadth and f 2 >f 1 Increasing the freeboard, means to increase the depth of the ship to the main deck. clearly the center of gravity of the Ship2 will have the vertical coordinate higher than Ship1: a 2 > a 1 The metacentric radii remain the same but the metacentric height will change: r-a 2 < r- a 1 The initial stability for a ship with a higher freeboard is lower that the same ship with a lower freeboard. The Ship2 will be more stable than Ship1 for the main part of the stability range. 09/11/2017 Ship Buoyancy and Stability 19

20 Breadth effects on stability c B LBT K 0. 7D T1 T 0 1 r0 Ship0 and Ship1 have the same displacement, i.e. volume, the same block coefficient and the same freeboard. The first ship is larger that the initial: B 1 > B 0 The ships will have the same a, i.e. height of the center of gravity over the center of buoyancy; The larger ship will have an higher metacentric radius; The initial stability of the larger ship is higher: r 1 -a > r 0 -a The larger ship immerges the deck for smaller angles. Anyway the larger ship, having also a smaller K, presents higher righting arms. r I I * arctan 2 f B 09/11/2016 Ship Buoyancy and Stability 20

21 K effects on stability r-a 3 < r-a 2 < r- a 1 The vertical center of gravity coordinate has a strong impact on stability Higher VCs mean reduction in stability At the same time r-a is reduced too. 09/11/2016 Ship Buoyancy and Stability 21

22 Angle of loll h [m] f 1 f f [aste] epävakavuuden alue The timber carrier assumed for the example, presents an high center of gravity with a consequent negative M. The ship has a negative initial stability: as could be noticed by the diagram, it will have two possible equilibrium positions 1 and 2, even if y =0. This example well shows the loll phenomenon. At high angles the ship presents a good stability For small angles the gravitational effects are greater than the shape effects, leading to negative rigthing arms Z KN Ksin 09/11/2017 Ship Buoyancy and Stability 22

23 Sailing boat stability The range of stability of same sailing boat could extend almost from < < Positive effect of the weight stability Z y B cos ( z B z ) sin 09/11/2017 Ship Buoyancy and Stability 23

24 Submarine example alusta kallistava ulkoinen momentti a) vakaa b) epävakaa B W z B z' B z' z B W voimapari tasapainottaa alusta voimapari kallistaa alusta lisää Sinusoidal shape of the Z. Only positive weight effects. Stability range up to 180 Z ( z ) B z sin 09/11/2017 Ship Buoyancy and Stability 24

25 The effect of moving loads on static stability lever curve Cargo and liquid movements have a great influence on ship stability rain shift, liquid sloshing, and ore cargo and small container movement on the ship side have been the causes of several accidents regarding ship loss of stability. The engineer has to take into account all these aspects in the design process, trying to provide appropriate solutions in order to avoid cargo shifts In situation in which is not possible to avoid the problem, the engineer has to introduce in the stability calculation, all the effects arising from the cargo and liquid movement. For example, is not possible to avoid that a fluid moves inside a compartment; it is impossible to avoid that the passengers on a cruise vessel crowd on the ship side looking at the harbour. Let s focus in the following on the cargo movement effects on the ship static stability. 09/11/2016 Ship Buoyancy and Stability 25

26 Load displaced Q D D K K Q K D Q K D K K ) ( ) ( ) ( ) ( ) ( ) ( ) ( Moving generically a weight Q from the point D1 to a point D2, the center of gravity will pass from the initial position 1 to the final position 2 according to the following formula: The displacement remains the same, but the ship equilibrium floating plan will change. ),, ( ( 1) 2 z y x D D Q z z z Q y y y Q x x x /11/2016 Ship Buoyancy and Stability 26

27 In the previus lecture ( lecture 3) it was studied the longitudinal equilibrium and stability regarding weights moved in the longitudinal direction (i.e. along the x-axis). Now we want to study the transversal stability problems by moving weights transversally and vertically! 09/11/2016 Ship Buoyancy and Stability 27

28 Loads displaced on the side Moving a load transversally, from the position 0 to the position 1 a destabilizing effect will arise due to the following moment: cos ' cos s y Ql M lq y y y ' 1 2 Q z z Q l y y Q x x /11/2016 Ship Buoyancy and Stability 28

29 Loads displaced on the side effects on stability y' cos [m] 2 1 h [aste] 90 h' -2-3 h '= h - cos y' The ship righting arm is reduced and h' max < h max. The effect of a transversely shifted ship s CO is to decrease the vessel s capability to handle the external heeling moment and to cause a static heeling of 0. 0 is the angle of heel or list. 09/11/2016 Ship Buoyancy and Stability 29

30 Loads displaced vertically Q l z z Q y y Q x x Moving a weight vertically, from the position 0 to the position 1 a destabilizing effect will arise due to the following moment: sin ' sin s z Ql M lq z z z ' /11/2016 Ship Buoyancy and Stability 30

31 Load displaced vertically effects on stability If the quantity Ql is high enough, then the center of gravity could go above the metacenter. In that case there is an initial instability. The ship is not stable in the initial position and will find a new equilibrium position with non-zero heel. The ship stability is reduced. M x ' Z Qlsin In particular even the initial stability i.e. the metacentric height is reduced by: ' M M ' M Ql The ship with y =0, M<0 can have 0 0 of stable equilibrium: LOLL ANLE An angle of loll can be corrected only by lowering the centre of gravity, not by moving loads transversely, or by filling ballast tanks on the higher side 31 0

32 Effect of free Liquids on transverse stability The effect of the liquid in tanks not completely full tends to reduce ship stability The reason is that the liquid tends to move on the inclined side of the ship, changing its center of gravity That reduces the righting arm of the ship If the liquid in the tank is frozen, or the tank is fully loaded, no freesurface effect arise, as the fluid does not change shape i.e. center of gravity In analyzing the free liquid effects, the main assumption is that the liquid moves slowly and its freesurface is parallel to the floating plane. We are assuming that no sloshing is involved in the phenomenon (sloshing: irregular movement of fluid inside the tank, responsible of dynamic effects). 09/11/2017 Ship Buoyancy and Stability 32

33 Free surface effects at small inclination M 0 1 ' m Assuming small inclination dφ the center of the liquid mass will move on a circular arc according to the metacenter assumptions, as for the ship: b b B M FS = Qd = (vρ t g)(bmsinφ) M st = (ΔM 0 Qbm)sinφ Ship metacentric height corrected by free surface effects M corr = r (a + Q Δ bm) = BM 0 (B + Q Δ bm) The ship stability reduction due to free surface effects, has the same behavior of the stability reduction due to vertical shift of weight. According to this, the stability reduction due to the free surface effects is treated as a VIRTUAL INCREASE OF THE VERTICAL CENTER OF RAVITY 09/11/2017 Ship Buoyancy and Stability 33

34 Free surface effects at small inclination bm = i x v M FS = Qd = γ t i x sinφ There is no direct effect of tanks capacity (volume of fluid) on stability. Taking into account all the tanks characterized by free surfaces, the apparent rise of VC is given by: 1 = 1 n Δ γ i i xi i If the tank is of rectangular shape, then the moments of inertia with respect to the longitudinal axis is: i x = b3 l 12 There is a pronounced effect of tank s breadth: the larger is the tank, the more will be the freesurface effects due to the moving liquid. 09/11/2017 Ship Buoyancy and Stability 34

35 Free surface moment at Large inclinations The small angle assumptions are not valid anymore. It is necessary to evaluate the effective position of the center of the fluid mass, for each percentage of loading of the tank at each inclinations, to correct the righting arm. For large inclination angle, for rectangular tanks, can be demonstrated that the heeling moment to a percentage of loading Cr is equal to the heeling moment due to the complementary percentage of loading of the tank Cr =1-Cr. So it is sufficient to work till Cr 0.5 Cr =1-Cr t Cr h t b t Approximate formulas are given by the rules, as for example: 09/11/2017 Ship Buoyancy and Stability 35

36 FSM at Large inclinations: approximate method v maximum volume of fluid of the tank (m 3 ); b maximum beam of the tank (m); ρ density of the fluid inside the tank (t/m 3 ); l maximum length of the tank (m); h maximum height of the tank (m); δ block coefficient of the tank; k nondimensional coefficient. 09/11/2017 Ship Buoyancy and Stability 36

37 FSM at large inclination, actual method 09/11/2017 Ship Buoyancy and Stability 37

38 The effect of suspended cargo/mass on transverse stability r Stability moment without suspended weight δm s = Δ M 0 δφ M ulk Heeling moment due to the suspended weight l δm hw = mglδφ W M 0 Z m P = m g Effective stability moment δm corr = Δ M 0 mgl δφ B 0 K B Metacentric height corrected for the suspended weight M corr = M 0 mgl Δ δφ 09/11/2017 Ship Buoyancy and Stability 38

39 Cargo hoisted from a quay r a) b) r l M 0 m W M' 0 l Z m T B 0 T' B' 0 B' P = m g K K ' Cargo is hoisted from a quay using ship s own crane As soon as the cargo is free from the quay the weight of the ship will increase: W 1 = W + m g = 1. Anyway the cargo is small enough to be neglected P << W and 1. 09/11/2017 Ship Buoyancy and Stability 39

40 Resulting heeling angle r a) b) r l M 0 m W M' 0 Z l m T B 0 T' B' 0 B' P = m g K K ' M ulk = m g l + m g r M 0 = m g l + m g r = m g r M 0 m g l = m g r M korj 09/11/2017 Ship Buoyancy and Stability 40

41 Ship s heeling due to the turning maneuver The ship rudder is deflected: the ship starts to turn with a constant velocity V S The ship trajectory is circular and the relative diameter is called steady diameter: D S In this condition the ship is affected by a centrifugal force in turning. F kp = mω 2 R S = ρ ωv S = Δ g V S 2 R S Centrifugal force Angular velocity ω = V S R S Steady radius R S = D S 2 L WL 5 09/11/2017 Ship Buoyancy and Stability 41

42 Moment due to turning maneuver M 0 W F kp R y = F kp + F R F h = 0 R z = W Δ = 0 B T/2 M x = M ulk M S = 0 D S /2 K T/2 F h Steady turn means system in equilibrium it is not accelerating Heeling moment M ulk is made up by two forces: centrifugal and hydrodynamic reaction forces The rudder force F R is neglected, but it would have a restoring effects The hydrodynamic reaction force is assumed to be applied in the middle of the centerboard at T/2. 09/11/2017 Ship Buoyancy and Stability 42

43 Moment due to turning maneuver K W M 0 B F kp T/2 T/2 D S /2 M ulk = F h K T 2 F h Having neglected the rudder forces, for the equilibrium the hydrodynamic reaction and the centrifugal force have the same magnitude. V S 2 F h F kp = Δ g R S The heeling moment, choosing as pole the center of gravity is: 2 Δ V S K T g R S 2 ΔM 0δφ = 0 δφ = V S 2 gr S 1 M 0 K T 2 Ship stability moment for small angles assumptions: M S = ΔM 0 δφ 09/11/2016 Ship Buoyancy and Stability 43

44 The effect of the height of C on steady heel due to turn, fast vessels 2 Δ V S K T g R S 2 ΔM 0δφ = 0 δφ = V S 2 gr S 1 M 0 K T 2 Fast vessels will experience an higher heeling angle due to turn. The center of gravity plays an important role: its vertically reduction means higher stability and lower turning heeling moment. Fast vessels tend also to have lower steady diameter in maneuvering 09/11/2017 Ship Buoyancy and Stability 44

45 Other causes of heeling moments Heeling moment due to wind, gust and waves; Heeling moments due to ice formation; Tug: heeling moment due to transversal tow; Bulk carrier: heeling moments due to grain shifts; Passenger ship: heeling moment due to passenger crowding on a ship side; Firefighting vessels: heeling moments due to monitors spraying actions and bow and stern thrusters actions if present; Ecc 09/11/2017 Ship Buoyancy and Stability 45