50 FLOTATION AND STABILITY 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 the ship is concerned the weight seems to be located at the suspension point. 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 not always possible. For instance, when a weight is being lifted by a shipboard crane, as soon as the weight is lifted clear of the deck or quayside 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, could cause a stability problem. This is important to the design of heavy lift ships. 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 with liquids. 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 slowly through a small angle to the vertical the liquid surface will move so as to remain horizontal. In this discussion a quasi-static condition is considered so that slopping of the liquid is avoided. Different considerations would apply to the dynamic conditions of a ship rolling. For small angles, and assuming the liquid surface does not intersect the top 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 of transfer of volume is: where I\ is the second moment of area of the liquid, or free, surface. The moment of mass moved =p f f»/ 1 , where p f is the density of the liquid in the tank. The centre of gravity of the ship will move because of this shift 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. 52 FLOTATION AND STABILITY 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 G 2 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 = p f /i/pV Another way of looking at this is to draw an analogy with the loss of stability due to the suspended weight. The water in the tank with a free surface 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 its transverse position. If the loss is sufficiently large, the metacentric height becomes negative and the ship heels over and may even capsize. It is important that the free surfaces of tanks should be kept to a minimum. One way of reducing them is to subdivide wide tanks into two or more narrow ones. In Figure 4.17 a double bottom tank is shown with 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 free surface effect to a quarter of its original value. Using two bulkheads to divide the tank into three equal width sections reduces the free surface to a ninth of its original value. Thus subdivision is seen to be very effective and it is common practice to subdivide the double bottom of ships. The main tanks of ships carrying liquid cargoes must be designed taking 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 where unavoidable must be taken into account in the design. The operators must be aware of their significance and arrange to use the tanks in ways intended 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 by calculation 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 and centre of gravity of the light ship. For particular conditions of loading the weights of all items to be carried must then be added at their appropriate centres of gravity to give the new displacement and centre of gravity. It is difficult to account for all items accurately in such calculations 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 known distances tranversely across the deck and observing the angles of inclination. The draughts at which the ship floats are noted together with the water density. Ideally the experiment is conducted when the ship 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 assessed and 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 54 FLOTATION AND STABILITY Figure 4.18 Inclining experiment alongside sets 3 and 4. G moves to GI as the ship inclines to a small angle 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 weight are 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 by set 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 unit applied moment. When the metacentric height has been obtained, the height of the centre of gravity is determined by subtracting GM from the 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 be found using the recorded draughts. To obtain accurate results a number of precautions have to be observed. First the experiment should be conducted in calm water with little wind. Inside a dock is good as this eliminates the effects of tides and 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. All weights 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 position when the inclining weights are restored it is an indication that a weight has moved in the ship, or that fluid has moved from one tank to another, possibly through a leaking valve. The number of people on board must be kept to a minimum, and those present must go to defined positions when readings are taken. The pendulum bobs are damped by immersion in a trough of water. The draughts must be measured accurately at stem and stern, and must be read at amidships if the ship is suspected of hogging or sagging. The density of water is taken by hydrometer at several positions around the ship and at several depths to give a good average figure. If the 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 the longitudinal and vertical positions of B. In this case detailed calcula- tions should be carried out to find these quantities for the inclining waterline. The Merchant Shipping Acts require every new passenger ship to be inclined upon completion and the elements of its stability determined, Stability when docking or grounding When a ship is partially supported by the ground, or dock blocks, its stability will be different from that when floating freely. The example of a ship docking is used here. The principles are the same in each case although when grounding the point of contact may not be on the centreline and the ship will heel as well as change trim. Figure 4,19 Docking 56 FLOTATION AND STABILITY Usually a ship has a small trim by the stern as it enters dock and as the water is pumped out it first sits on the blocks at the after end. As the water level drops further the trim reduces until the keel touches the blocks 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 critical stability 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 at draughts of 2.7m forward and 4.2m aft. The length between perpendiculars is 150m and the water has a density of 1025kg/ m 3 . Assuming the blocks are horizontal and the hydrostatic data given are constant over the variation in draught involved, find the force on the heel of the sternframe, which is at the after perpendicular, when the ship is just about to settle on the dockblocks, 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 actual waterline 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 moment acting 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 a dry dock the 'launch' is like an undocking except that the ship is only partially complete and the weights built in must be carefully assessed to establish the displacement and centre of gravity position. Large ships are quite often nowadays built in docks but in the more general case the ship is launched down inclined ways and one end, usually the stern, enters the water first. The analysis may be complicated by the launching ways being curved in the longitudinal direction to increase the rate of buoyancy build up in the later stages. An assessment must be made of the weight and centre of gravity position 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 specially strengthened 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 58 FLOTATION AND STABILITY Figure 4.20 Launching enters the water the waterline at various distances down the ways can be noted on the profile. From the Bonjean curves the immersed sectional areas can be read off and the buoyancy and its longitudinal centre computed. The ship will continue in this fashion until the moment of weight about the fore poppet equals that of the moment of buoyancy about 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; the buoyancy which increases as the ship travels down the ways; the moment of weight about the fore poppet which is also effectively constant; the moment of buoyancy about the fore poppet; the moment of weight about the after end of the ways; and the moment of buoyancy about the after end of the ways. Figure 4.21 Launching curves FLOTATION AND STABILITY 59 The maximum force on the fore poppet will be the difference between the weight and the buoyancy at the moment the ship pivots about the fore poppet which occurs when the moment of buoyancy equals the moment of weight about the fore poppet. The ship becomes fully waterborne when the buoyancy equals the weight. To ensure the ship does not tip about the after end of the ways, the moment of buoyancy about that point must always be greater than the moment of weight about it. If the ship does not become waterborne before the fore poppet 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 be sufficient depth of water to allow the ship to drop freely allowing for the dynamic 'overshoot'. The stability at the point of pivoting can be calculated in a similar way to that adopted for docking. There will be a high hogging bending moment acting on the hull girder which must be assessed. The forces acting are also needed to ensure the launching structures are adequately strong. The ship builds up considerable momentum as it slides down the ways. This must be dissipated before the ship conies to rest in the water. Typically chains and other energy absorbing devices are brought into action during the latter stages of travel. Tugs are on hand to manoeuvre the 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 greater than, say, 4 or 5 degrees, the point, M, at which the vertical through the inclined centre of buoyancy meets the centreline of the ship, can no longer 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 and buoyancy with W 0 Lo 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 the centreline 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 of buoyancy, is given by: This expression for GZ is often called Atwood 's formula. [...]... in Table 4.2 By plotting GZ against inclination the range of stability is found to be 82° 66 FLOTATION AND STABILITY Table 4.2 Inclination (") sin (p 0 15 30 45 60 75 90 0 0,259 0.500 0.707 0.866 0.966 1.000 SG sin . to be 82°. 66 FLOTATION AND STABILITY Table 4.2 Inclination (") 0 15 30 45 60 75 90 sin (p 0 0,259 0.500 0.707 0.866 0.966 1.000 SG sin <p m 0 0.0 93 0.180 0.255 0 .31 2 0 .34 8 0 .36 0 sz in 0 0.11 0 .36 0.58 0 .38 -0.05 -0.60 GZ m 0 0,2 03 0.540 0. 835 0.692 0.298 -0.240 Transverse . 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. distance h to a position 54 FLOTATION AND STABILITY Figure 4.18 Inclining experiment alongside sets 3 and 4. G moves to GI as the ship inclines to a small angle and B moves to Bj