Vessel Stability
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What is Stability? Stability is the tendency of a vessel to rotate one way or the other when forcibly inclined whether by winds and/or seas so as to resist capsizing by returning to an upright position after being heeled over. Many forces influence the stability of a vessel in the water and each vessel will reacts differently to heeling forces. Vessel operators should be aware of how the design of their vessel interacts with external forces of nature and affect its stability. If the vessel was built and loaded correctly, outside forces may cause the vessel to temporally roll from side to side, however the vessel will return to the upright by itself. A vessel stays afloat by offsetting two forces of nature. The force of gravity attempts to push the boat down into the water, while the force of buoyancy of the water will be pushing the hull up. As long as the buoyancy is greater than the gravity, the vessel will float. How resistant it is to capsizing is dictated by how these two forces act on the vessel. EXAMPLE OF GRAVITY -VS- BUOYANCY
1 ton of steel
1 ton of steel
If the cube of steel is placed in water it sinks. There is not enough displaced volume for the forces of buoyancy to act upon. If the cube of steel is formed into a boat’s hull and is placed in the water it will float. The larger volume of the boat's hull allows the forces of buoyancy to support the hull's weight. The boat's hull will sink to a draft where the forces of buoyancy and the forces of gravity are equal. THE LAWS OF BUOYANCY 1. All floating objects possess the property of buoyancy. 2. A floating object displaces a volume of water equal in weight to the weight of the body. 3. An object immersed (or floating) in water will be buoyed up by a force equal to the weight of the water displaced. Buoyancy The buoyancy of a vessel is determined by the shape of the immersed hull form. The larger the hull, the more weight it can support. Stability, however, is dictated by the distribution of that hull volume. For example, beam has a much larger impact on stability than length. As a general rule, the wider the hull, the more stable it is. A deeper hull will also be more stable than a shallow one. The center of buoyancy is the point at which all the vectors of the floating forces of the hull can be said to act vertically upward. The designer usually has a great deal of control of a vessel's stability characteristics while it is still being designed. Good practice includes designing a stability margin into the hull before the vessel is built. Unfortunately, features that make a vessel more stable are often in direct conflict with the other aspects of the design. While it may be tempting to simply enlarge the beam to increase stability, this will also increase construction cost as well as increase the propulsion resistance of the hull. 2 of 28
Increased resistance in turn drives up fuel consumption and operating costs over the life of the vessel. As with all good designs, a balance between the design criteria and operational requirements must be reached. Weight or Force of Gravity The hull, machinery, outfitting, and cargo load determine vessel weight. As vessel cargo load is increased, the hull will settle deeper in the water until the buoyancy equals the weight. While this may intuitively seem to increase stability, adequate Equally important to the overall weight of the loaded vessel is how that weight is distributed. The center of gravity is the point at which the vector of the whole weight of the vessel can be said to act vertically downward. As a general rule, a lower center of gravity means a more stable vessel. A vessel with a high center of gravity is said to be "top heavy." When a vessel lists or heels to one side, the center of gravity pushes down in the direction of the list. The vessel weight and center of gravity change constantly as vessel loading changes. For example, a heavy object placed high on a deck will produce a higher center of gravity - and less stability - than a load stored below deck. Similarly, removing a load from low in the vessel, such as burning fuel oil, will cause an increase in the vessel's center of gravity, thus reducing stability. Additionally, vessels gain weight over their lifetimes as equipment is added or other changes are made to the arrangements. A good design will allow for some weight growth, but careful attention must be paid to modifications to the vessel to ensure that it continues to meet the applicable stability requirements.
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Definitions Center of Buoyancy (B) is the point at which all vertical upward forces (buoyant forces) are said to act. It is the center of the volume of the immersed portion of the vessel. When a hull is rotated in the water the center of buoyancy will move as the shape of the underwater portion of the hull changes. 1. When the boat rolls to starboard, "B" moves to starboard, and when the boat rolls to port, "B" moves to port. 2. When the boat's hull is made heavier, the drafts increase as the boat sits deeper in the water. "B" will move up. 3. When the boat's hull is lightened, the drafts decrease as the boat sits shallower in the water. "B" will move down. 4. The Center of Buoyancy moves in the same direction as the boat’s waterline.
Center of Gravity (G) is the center of the total weight of the loaded vessel. It is the point where the entire weight of the boat and its contents are concentrated. If additional excess weight is added to the boat then this point “G” will be located higher or lower depending upon if the weight was added above or below G. The position of "G" is dependent upon the distribution of weights within the boat. As the distribution of weights is altered, the position of "G" will react as follows: 1. "G" moves towards a weight addition 2. "G" moves away from a weight removal 3. "G" moves in the same direction as a weight shift
When both of these forces, Buoyancy and Gravity are equally oppose to each other from directly opposite directions then the boat is said to be “At Rest” or upright. Heel is the temporary tilting of a vessel from side to side. If this tilt becomes permanent it is then called a List List is a permanent tilting of a vessel to one side or the other. Freeboard is the distance between the water and the working deck of the vessel. If the deck edge goes under water when the vessel heels, the danger of capsizing is increased. An overloaded vessel will have too low a freeboard, and the deck may submerge with even a light heel caused by wind or water conditions.
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Displacement is the weight of the volume of water that is displaced by the underwater portion of the hull is equal to the weight of the boat. This is known as a boat's displacement. The unit of measurement for displacement is the Long Ton (1 LT = 2240 LBS). Force is a push or pull that tends to produce motion or a change in motion. Units are expressed in tons, pounds, Newtons, etc. Parallel forces may be mathematically summed to produce one "Net Force" considered to act through one point. Weight is the force of gravity acting on a body. This force acts towards the center of the earth. Units are expressed in tons, pounds, kilograms, etc. Moment is the tendency of a force to produce a rotation about a pivot point. This works like a torque wrench acting on a bolt. Units are expressed in foot tons, Newton meters, etc. Free surface effect influences the stability of a vessel. When a vessel with full tanks heels over, the tank’s center of gravity does not change, so it does not affect the vessel’s stability. However, water on deck, liquids in holds, bilge water, and partially filled tanks will cause a shift of the liquid with the movement of the boat. When this happens, the center of gravity also shifts, making the vessel less stable. This "free surface effect" reduces stability and increases the danger of capsizing. Initial stability is the stability of a boat in the range from 0% to between 7% and 10% of inclination. Overall stability is a general measure of a boat's ability to resist capsizing in a given condition of loading. Dynamic stability is the work done in heeling a boat to a given angle of heel. Keel (K) is the base line reference point from which all other reference point measurements are compared. Metacenter (M) is a point where the lines of buoyant forces intersect as the boat is inclined through small angles of heel. As the boat is inclined, the center of buoyancy moves in an arc as it continues to seek the geometric center of the underwater hull body. The position of the metacenter is a function of the position of the center of buoyancy, thus it is a function of the displacement of the boat. The position of "M" moves as follows: 1. As the Center of Buoyancy moves up, the Metacenter moves down. 2. As the Center of Buoyancy moves down, the Metacenter moves up.
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Stability at Work If an outside force such as a wave caused this boat to heel over to one side, And if the weight of the boat and its contents has not changed, or shifted, then the center of gravity (G) will remain in place. But look at the “B”, it moved in the direction of the lower side. This point is now the center of Buoyancy and the upward forces of Buoyancy will seem to push harder on this lower side. The result will be to return the hull to its upright position. The Metacenter “M” is a point near the centerline of the boat at rest. It is a point that will normally be stationary and directly over the point of buoyancy regardless of which side “B” moves. If you consider “B” as the ball at the bottom of a pendulum then “M” is the connecting point from which that pendulum swings. The Metacentric Height then is the distance between the center of gravity “G” and the metacenter “M” and is simply called “GM”. The GM is crucial to stability. The further apart that “G” and “M” become, the more stable the boat and the quicker it will right itself. A long GM also causes an uncomfortable quick snapping roll. To provide a more comfortable ride, most passenger boats are built with a shorter GM. This will allow the vessel to slowly recover. Too little GM results in a vessel with a long, slow roll that, while comfortable, could lead to capsizing. As long as the weight of a vessel, and the location of that weight, remains constant, and then the center of gravity would not move. THE STABILITY TRIANGLE When a boat is inclined, the center of buoyancy shifts off centerline while the center of gravity remains in the same location. Since the forces of buoyancy and gravity are equal and act along parallel lines, but in opposite directions, a rotation is developed.
This is called a couple, two moments acting simultaneously to produce rotation. This rotation returns the boat to where the forces of buoyancy and gravity balance out. The distance between the forces of buoyancy and gravity is known as the boat’s righting arm. As shown above, the righting arm is a perpendicular line drawn from the center of gravity to the point of intersection on the force of buoyancy line. For small angles of heel (0° through 7° to 10°, the metacenter doesn’t move), the value for the boat’s righting arm (GZ) may be found by using the trigonometry equation of: With initial stability (0° through 7° to 10°) the metacenter does not move, and the Sine function is almost linear (a straight line.) Therefore, the size of the boat’s Righting Arm, GZ, is directly proportional to the size of the boat’s Metacentric Height, GM. Thus, GM 6 of 28
is a good measure of the boat’s initial stability. RIGHTING MOMENT (RM) The Righting Moment is the best measure of a boat's overall stability. It describes the boat's true tendency to resist inclination and return to equilibrium. The Righting Moment is equal to the boat’s Righting Arm multiplied by the boat’s displacement. As long as the G remains below the M then there will be available Righting moment to right the vessel. Once G is centered over the M then the vessel will capsize. Righting energy is the term used to describe a vessel's ability to right itself after being heeled over. A properly-loaded vessel should have positive righting energy to a heel of at least 50 degrees. The magnitude of the largest righting arm is also an indication of a vessel's stability.
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STABILITY CONDITIONS The positions of Gravity and the Metacenter will indicate the initial stability of a boat. Following damage, the boat will assume one of the following three stability conditions: POSITIVE STABILITY The metacenter is located above the boat’s center of gravity. As the boat is inclined, Righting Arms are created which tend to return the boat to its original, vertical position.
NEUTRAL STABILITY The metacenter and the boat’s center of gravity are in the same location. As the boat is inclined, no Righting Arms are created (until M starts to move below G after the boat is inclined past 7º10º). NEGATIVE STABILITY The boat’s center of gravity is located above the metacenter. As the boat is inclined, negative Righting Arms (called upsetting arms) are created which tend to capsize the boat.
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Weight Movements Because of the relationship between weight and buoyancy of a given hull shape, both GM and righting energy vary significantly with the weight and center of gravity of the loaded vessel. This means that how a vessel is loaded has the largest impact on the stability of the vessel. Adding a load Adding weight above a boat’s center of gravity will change its stability. If the center of gravity is raised too much, the boat will become unstable. As a result, less force is required to capsize the vessel. Removal of a load Removal of weight from below the center of gravity also decreases stability. The center of gravity will rise to a higher level decreasing the GM of the vessel resulting in a vessel with a quicker roll period. Shifting a load Shifting weights vertically, no matter where onboard it is, will always cause the boat’s center of gravity to move in the same direction as the weight shift. When you raise a weight it will cause a rise in G, decreasing your stability. This is the reason passenger limits are put on upper decks of public vessels. The more passenger weight kept on the main deck the better the stability of the boat. Shifting weight horizontally, adding a weight off centerline, will always cause the boat’s center of gravity to move in the same direction as the weight shift. The boat’s center of gravity will move off centerline, the boat will develop a list. A weight shift causing the boat’s center of gravity to move off centerline will always reduce the stability of the boat. Whenever weight is added, moved or removed a boat, the boat’s center of gravity rarely moves in only one direction. Fortunately, the effects are cumulative
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Damage Control The previous pages have discussed vessel stability characteristics in the intact state. They also apply to a damaged vessel. However, the buoyant force and center of buoyancy of the damaged hull will differ significantly from that of the intact hull, depending on hull compartmentation as well as the location and extent of damage. As the unexpected excess weight of flooding water is added, the boat’s center of gravity will initially decrease and the vessel will seem to become stable. At some point in time the center of gravity will start rises. Once it reaches the metacenter and goes beyond the boat will capsize. This new location of the downward force of gravity will now fight against the righting tendency of the new point of buoyancy that had shifted previously. The result is that the boat will no longer return to the upright.. If an area flooded with the water is only partially filled, and there is no restriction to the side-to-side movement of the water, the result can be devastating This free movement or sudden shifting of water is called “Free Surface Effect” and has a major effect on a vessel’s stability. If the area is completely flooded or if there are restricting boundaries to act as baffles then the water is no longer subject to Free Surface Effect. Also, as the boat develops a list, its center of gravity may move in the same direction as the point of buoyancy if there is also a corresponding shift in weight as in the diagram. A good operational practice is to minimize free surface effect by dividing tanks with baffles and fluid cargo holds with bulkheads and by keeping the number of partially filled tanks and holds to an absolute minimum.
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Vessel Stability – Warning Signs, Precautions The most important factors in preventing a boat from capsizing are a welldesigned, maintained, and loaded vessel and an experienced operator and crew. Preventing an unstable vessel condition and being able to recognize the warning signs when such a condition does occur can save lives. You should be on constant watch for loss of stability.
A well-designed vessel will resist capsizing or foundering in severe conditions if it is operated properly. To reduce the likelihood of these incidents, keep these rules in mind: Be aware of external forces – wind, waves, and water depth. Always check the weather forecast before departure. Avoid rough weather conditions. Don’t overload your vessel. Be aware of the amount of weight added to your vessel and available freeboard. Distribute the passengers and cargo evenly. Partially filled water ballast and fuel tanks contribute to instability. Free surface liquids must be contained so their influence will not upset the balance of your vessel. Prevent water from entering the interior of your vessel by keeping hatches, doors, and windows closed, as practicable, when underway. Regular maintenance of gaskets and fastening devices will help to ensure water tightness. Any water shipped on board must be removed as quickly as possible. Scuppers and drains must meet design criteria and be kept in good working order. Adjust course, speed, or both as practicable to minimize vessel motion, rolling in particular. Avoid sharp turns or turns at high speed when loss of stability is possible.
Stability Warning Signs Observe the stability and roll of your boat. Make sure the vessel’s movement and reaction to sea conditions is normal, steady, and safe. Check to make sure your boat is visibly stable before you leave the dock. It should not be listing to port or starboard or trimmed excessively by the bow or stern. Observe freeboard and check for flooding. A flooded vessel may appear stable when it is in fact not. Make sure the passengers remains seated during the voyage. Make sure that bilge level alarms are operational. Unusual operation of bilge pumps may indicate an excessive amount of water is entering the interior of the vessel. A combination of prevention efforts and awareness of the warning signs of instability, along with operator knowledge, can accomplish a great deal in reducing instability and capsizing.
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SUBDIVISION AND DAMAGE STABILITY If the shell of the ship is damaged so as to open one or more internal spaces to the sea, flow will take place between the sea and these spaces until stable equilibrium is established or until the ship sinks or capsizes. The loss of watertight integrity could be due to collision, grounding or internal accident such as an explosion. It is impractical to design a ship to withstand any possible damage. The degree to which a vessel approaches this limit is the true index of its safety. To reduce the probability of loss, the hull is divided into a series of watertight compartments by means of transverse watertight bulkheads extending from side to side of the ship. It is true that the more severe the standard adopted for subdivision and stability, the greater the probability that capital and operating costs will be increased. For example, too close spacing of bulkheads may unnecessarily increase both the first cost and operating costs and may also seriously restrict the vessel’s usefulness. In addition, it might be expected that the more bulkheads the safer the ship. But damage may occur entirely between adjacent bulkheads or may involve one or more bulkheads. Hence, for a given length of damage, any increase in the number of bulkheads may actually increase the likelihood of bulkhead damage, which would reduce rather than increase the chances of survival. The General Effects of Flooding (1) Change of Draft The draft will change so that the displacement of the remaining unflooded part of the ship is equal to the displacement of the ship before damage less the weight of any liquids which were in the space opened to the sea. (2) Change of Trim The ship will trim until the centre of buoyancy of the remaining unflooded part of the ship lies in a transverse plane through the ship’s centre of gravity and perpendicular to the equilibrium waterplane. (3) Heel If the flooded space is unsymmetrical with respect to the centerline, the ship will heel until the centre of buoyancy of the remaining unflooded part of the ship lies in a fore-and –aft plane through the ship’s centre of gravity and perpendicular to the equilibrium waterline. If the GM in the flooded condition is negative, the flooded ship will be unstable in the upright condition , and even though the flooded space is symmetrical, the ship will either heel until a stable heeled condition is reached or capsize. Trim and heel may result in further flooding through immersion of openings bulkheads, side shell or decks (downflooding). (4) Change of Stability Flooding changes both the transverse and longitudinal stability. The initial metacentric height is given by: GM = KB + BM - KG Sinkage results in an increase in KB. If there is sufficient trim, there may also be an appreciable further increase in KB as a result. BM tends to decrease because of the loss of the moment of inertia of the flooded part of the waterplane. However, sinkage usually results in an increase in the moment of inertia of the undamaged part of the waterplane, thus tending to compensate for the loss. Also, trim by the stern usually increases the transverse moment of inertia of the undamaged 12 of 28
waterplane, and vice versa. For most ocean-going ships the combined effect of these factors is usually a net decrease in GM.
(5) Change of Freeboard The increase in draft after flooding results in a decrease in the amount of freeboard. Even though the residual GM may be positive, if the freeboard is minimal and the waterline is close to the deck edge, submerging the deck edge at small angles of heel greatly reduces the range of positive righting arm GZ, and leaves the vessel vulnerable to the forces of wind and sea.
(6 ) Loss of Ship Where changes in draft, trim and/or heel necessary to attain stable equilibrium are such as to immerse non-watertight portions of a ship, equilibrium will not be reached because of progressive flooding and the ship will sink either with or without capsizing. Where the maximum GZ in the damaged condition is adequate and where the immersion of non-tight portions of the ship only results in slow extension of flooding, sinking may be quite slow. In such cases, control measures aimed at stopping progressive flooding, either by reducing heel, pumping leakage water or fitting emergency means of checking the flow of water or a combination of such measures may be successful. Therefore, it has been realized that providing the master with an instruction manual outlining damage control measures available to minimize flooding would be a valuable contribution to safety.
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Floatation Calculations In order to assess the ship’s ability to withstand damage, it is necessary to determine: (a) The damaged waterline, i.e. the new draft, trim and heel. (b) The damage stability, i.e. after flooding. The floatation calculations can be carried out by either one of two methods, the lost buoyancy method or the added weight method. 1. The Lost Buoyancy Method In this method the lost buoyancy due to a compartment or compartments being opened to the sea is calculated. This lost buoyancy and its moments are equated to the buoyancy gain and moments accompanying sinkage, trim and heel of the remaining intact part of the ship. In this method, it is assumed that the displacement and the position of the centre of gravity are unchanged. This procedure is convenient and simple to use if the form of the vessel and configuration of the flooded space are such that the resulting sinkage, trim and heel do not involve extreme or discontinuous changes in the remaining undamaged part of the waterline. Consequently, this procedure is often used for merchant ships. Compartments of ships open to the sea do not fill totally with water because some space is already occupied by structure, machinery or cargo. The ratio of the volume which can be occupied by water to the total gross volume is called the permeability μ. For cargo spaces it is taken as 60%, for accommodation spaces as 95% and for machinery spaces as 85%. The steps of this method are as follows:
1. Calculate the permeable volume of compartment up to the original waterline. 2. Calculate TPC, longitudinal and lateral positions of CF for the waterplane with the damaged area removed. 3. Calculate revised second moments of areas of the waterplane about the CF in the two directions and hence new BMs. 4. Calculate parallel sinkage and rise of CB due to the vertical transfer of buoyancy from the flooded compartment to the layer. 5. Calculate new GMs. 6. Calculate angles of rotation due to the eccentricity of the loss of buoyancy from the new CFs. 14 of 28
This is best illustrated by an example.
Example A compartment having a plane area at the waterline of 100 m 2 and centroid 70 m fore of midships, 13 m to starboard is bilged. Up to the waterline obtained before bilging, the compartment volume was 1000 m3 with centres of volume 68.5 m fore of midships, 12 m to starboard and 5 m above keel. The permeability was 70%. Before the incident the ship was floating on an even keel draft of 10 m at which the following particulars are given: Displacement ∆ 30000 tonnes KM ( longitudinal) 170 m KG 9.40 m WP area 4540 m2 KM ( transverse) 11.40 m CF fore of midships 1m KB 5.25 m LBP 220 m Calculate the heel and trim when the compartment is bilged. 1. Permeable volume = 0.70 x 1000 = 700 m3 2. Damaged WP area = 4540 - 100 = 4440 m2 Movement of CF aft = Movement of CF port =
= 1.55 m = 0.29 m
3. Original IT IT =
m4
=
Damaged IT (ignoring I of the compartment about its own axis) I’T = 180000 – 100 x 132 – 4440 x (0.29)2 = 162727 m4 Damaged BMT =
5.56 m
Original IL m4
IL =
Damaged IL I’L = 4.822 x 10 6 - 100 x 69 2 - 4440 x (1.55)2 = 4.335 x 106 m4 Damaged BML = 4. Parallel sinkage s =
Rise of B =
= 148.1 m 0.16 m
0.121 m 15 of 28
5. Damaged GMT = 5.25 + 0.121 + 5.56 - 9.40 = 1.531 m Damaged GML = 5.25 + 0.121 + 148.1 - 9.40 = 144.07 m 6. Angle of heel φ =
=
Angle of trim θ =
=
= 11.0 0
= 0.0115 rads
Change of trim t = θ x LBP = 0.0115 x 220 = 2.53 m TF = TFo + s + t =
= 11.43 m
TA = TAo + s - t =
= 8.90 m
2. The Added Weight Method In this method the water entering the damaged compartment can be regarded as an added weight. Therefore, the resulting displacement varies with the increase in draft, trim and/or heel. Consequently, the position of the centre of gravity of the ship changes as well. Since this method involves a direct integration of volumes and moments up to the damaged condition waterplane, by direct use of Bonjean Curves, it is just as well adapted to dealing with complex flooding conditions as with simple ones. Also for greater accuracy this method is recommended. When the new drafts are calculated, more water would enter the compartment because the ship is now deeper in the water in way of the damage. A second calculation is therefore necessary to take this additional weight of water into account so that a second approximation is obtained to the waterline in the damaged condition. This new waterline will now involve a further addition of weight so that a third calculation would be necessary to obtain yet a closer approximation to the correct waterline. It will be seen that this is an iterative process stopped when the desired degree of accuracy is obtained. Example A vessel of constant rectangular cross-section is 60 m long and 10 m wide. It floats at a level keel draft of 3 m and has a centre of gravity 2.5 m above the keel. Determine the fore and aft drafts if an empty, full-width, fore-end compartment 8 m long is opened to the sea. Lost buoyancy method Area of intact waterplane, A = 52 x 10 = 520 m2 Volume of lost buoyancy, v = 8 x 10 x 3 = 240 m2 Parallel sinkage, s =
= 0.46 m
The vessel will now trim about the new centre of floatation, F1. The position of F1 can be found by taking moment about midships: (60 x 10 x 0) – (8 x 10 x 26) = (60 x 10 – 8 x 10) F1 16 of 28
F1 = - 4 m or 4 m aft
KG = 2.5 m GML = 1.73 + 65.1 - 2.5 = 64.33 m
Added weight method Mass added at 3 m draft = 8 x 10 x 3 x 1.025 = 246 tonne Parallel sinkage, s = New displacement = 60 x 10 x 3.4 x 1.025 = 2091 tonne
A second calculation considering the mass of water entering at 4.45 m draft will give drafts forward and aft of 5.14 m and 2.05 m respectively. Third and further calculations would come progressively closer to the lost buoyancy draft values.
Damage Stability Calculations (a)Initial Stability Consider a ship of displacement ∆ admitting a weight w having a free surface i as shown in figure. We shall now show that, whether the flooding be considered an added weight or a lost buoyancy, the result will be the same.
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Added Weight Method
Lost Buoyancy Method
(b) Stability at Large Angles It is now required to determine GZ values after damage. To do that cross curves for the volume of flooding water are constructed as follows:
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1. For each angle of heel, a series of parallel waterlines W1L1, W2L2, W3L3 and W4L4 is drawn. 2. For each of these waterlines, the weight of flooding water w and its lever Gz are calculated. 3. Intact displacement up to the same waterline must also be calculated. 4. The weight of flooding water w, up to each waterline is added to the original displacement ∆ to give a damaged displacement ∆ 1 (∆1 = ∆ + w). This is plotted on a base of WL together with the corresponding w.Gz (step 2) and the intact displacement (step 3).
5. Where the two curves of ∆1 and intact displacement cross each other is the point of vertical equilibrium for that angle of heel and w.Gz can be read off. 6. The restoring lever GZ for the intact ship at displacement ∆1 (= ∆ + w) can be read directly from the ship cross curves of stability. 7. The damaged righting moment is then given by: 8. A righting lever corresponding to the original displacement ∆ is given by dividing the last expression by ∆. 19 of 28
9. This procedure is repeated for all angles. The GZ curve after damage will, in general, appear as in the following figure:
The point at which the curve crosses the φ-axis, at angle φ 1, represents the position of rotational equilibrium; this will be zero when the flooding is symmetrical about the middle line. With symmetrical flooding, the upright equilibrium position may not be one of stable equilibrium. In this case, the GZ curve will be as the one shown below:
Where φ2 is the angle of loll. If the GZ curve does not rise above the φ-axis, the ship will capsize; if little area is left above, capsize will occur dynamically. Subdivision and Damage Stability Criteria The regulations must provide the answers to the following three key questions: (a) What is the extent of damage to which the ship is assumed to be subjected? (b) Where is the damage assumed to be located? (c) What condition is the ship permitted to be in after the assumed damage? What is the permissible sinkage, trim, and heel; the residual GM and/or GZ, righting energy and range of stability? Standards for subdivision and damage stability have been established by international conventions, by recommendations of IMO, by national regulations and by classification society rules. In this chapter we shall confine ourselves to the standards related to the International Convention for the Safety of Life at Sea “SOLAS” and the International Convention for the Prevention of Pollution from Ships “MARPOL”. The “SOLAS” Standards The basic philosophy of these standards is that the true index of safety is the probability of survival after damage occurring anywhere along the length of the ship, between or on a bulkhead. Fundamentally, three probabilities relate to subdivision and damage stability requirements: (a) Probability that a ship may be damaged. (b) If the ship is damaged, the probability as to the location and extent of damage. 20 of 28
(c) Probability that the ship may survive such flooding. Chapter II-1 of the SOLAS shall apply to ships the keels of which are laid on or after 1 January 2009. These requirements apply to cargo ships of 80 m in length (L) and upwards and to all passenger ships regardless of length. The degree of subdivision shall vary with the subdivision length (Ls) of the ship and with the service, in such manner that the highest degree of subdivision corresponds with the ships of greatest subdivision length (Ls), primarily engaged in the carriage of passengers.
Definitions 1. Subdivision Length (Ls) : The greatest projected length of that part of the ship at or below deck or decks limiting the vertical extent of flooding with the ship at the deepest subdivision draft. 2. Mid-Length : The mid-point of the subdivision length of the ship. 3. Aft and Forward Terminals : The aft and forward limits of the subdivision length. 4. Length (L) : The length as defined in the International Convention on Load lines. 5. Deepest Subdivision Draft (ds) : The waterline which corresponds to the summer load line draft of the ship. 6. Light Service Draft (dl) : The service draft corresponding to the lightest anticipated loading and associated tankage. 7. Partial Subdivision Draft (dp) : The light service draft plus 60% of the difference between the light service draft and the deepest subdivision draft. 8. Permeability (μ) : The permeability of a space is the proportion of the immersed volume of that space which can be occupied by water. 9. Bulkhead Deck : In a passenger ship means the uppermost deck at any point in the subdivision length (Ls) to which the main bulkheads and the ship’s shell are carried watertight. The bulkhead deck may be a stepped deck. In a cargo ship the freeboard deck may be taken as the bulkhead deck. 10. Amidship : At the middle of the length (L). The Required Subdivision Index R The subdivision of a ship is considered sufficient if the attained subdivision index A, is 21 of 28
not less than the required subdivision index R and if, in addition, the partial indices As, Ap and Al are not less than 0.9 R for cargo ships. 1. In case of cargo ships greater than 100 m in length (Ls):
2. In the case of cargo ships not less than 80 m in length (Ls) and not greater than 100 m in length (Ls):
Where Ro is the value R as calculated in accordance with the formula in 1. 3. In the case of passenger ships:
Where: number of persons for whom lifeboats are provided number of persons (including officers and crew) the ship is permitted to carry in excess of N1 The Attained Subdivision Index A 1. The attained subdivision index A is obtained by the summation of the partial indices As, Ap and Al, (weighted as shown) calculated for the drafts ds, dp and dl in accordance with the following formula:
Each partial index is a summation of contributions from all damage cases taken in consideration, using the following formula:
Where: i represents each compartment or group of compartments under consideration, pi accounts for the probability that only the compartment or group of compartments under consideration may be flooded, si accounts for the probability of survival after flooding of the compartment or group of compartments under consideration. 2. The summation indicated by the above formula shall be taken over the ship’s subdivision length (Ls) for all cases of flooding in which a single compartment or two or more adjacent compartments are involved. 3. In the flooding calculations carried out according to the regulations, only one breach of the hull and only one free surface need to be assumed. The assumed vertical extent of damage is to extend from the baseline upwards to any 22 of 28
watertight horizontal subdivision above the waterline or higher. However, if a lesser extent of damage will give a more severe result, such extent is to be assumed.
Calculation of the factor pi
The factor pi for a compartment or group of compartments shall be calculated according to formulae given in Regulation 7-1 of Chapter II-1 of the SOLAS. These formulae are based on the following parameters:
•
j = the aftmost damage zone number involved in the damage starting with No.1 at the stern.
•
n = the number of adjacent damage zones involved in the damage.
•
k = the number of a particular bulkhead as barrier for transverse penetration in a damage zone counted in direction towards the centre line. The shell has k = 0
•
x1 = the distance from the aft terminal of Ls to the aft end of the zone in question.
•
x2 = the distance from the aft terminal of Ls to the forward end of the zone in question.
•
b = the mean transverse distance in metres measured at right angles to the centerline at the deepest subdivision loadline between the shell and the vertical barrier extending between the longitudinal limits of the space. In any case, b is not to be taken greater than B/2.
Calculation of the factor si 1. The factor si shall be determined for each case of assumed flooding, involving a compartment or group of compartments, in accordance with Regulation 7-2 as follows:
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θe is the equilibrium angle of heel in any stage of flooding, in degrees; θv is the angle , in any stage of flooding, where the righting lever becomes negative, or the angle at which an opening incapable of being closed weathertight becomes submerged; GZmax is the maximum positive righting lever, in metres, up to the angle θv; Range is the range of positive righting levers, in degrees, measured from the angle θe. The positive range is to be taken up to the angle θv; Flooding stage is any discrete step during flooding process, including the stage before equalization (if any) until final equilibrium has been reached. The factor si for any damage case at any initial loading condition, di, shall be obtained from the formula: si = minimum { sintermediate,i or sfinal,i · smom,i } Where: sintermediate,i is the probability to survive all intermediate flooding stages until the final equilibrium stage, and is calculated in accordance with paragraph 2; sfinal,i is the probability to survive in the final equilibrium stage of flooding. It is calculated in accordance with paragraph 3; smom,i is the probability to survive heeling moments, and is calculated in accordance with paragraph 4.
2. The factor sintermediate,i is applicable only to passenger ships (for cargo ships sintermediate,i should be taken as unity) and shall be taken as the least of the s-factors obtained from all flooding stages including the stage before equalization, if any, and is to be calculated as follows:
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Where GZmax is not to be taken as > 0.05 m and Range as ≯ 7o. sintermediate,i = 0, if the intermediate heel angle exceeds 15 o. Where cross-flooding fittings are required, the time for equalization shall not exceed 10 min.
3. The factor sfinal,i shall be obtained from the formula:
Where: GZmax is not to be taken as > 0.12 m; Range is not to be taken as > 16 o; K = 1 if θe ≤ θmin K = 0 if θe ≥ θmax otherwise, Where: θmin is 7o for passenger ships and 25 o for cargo ships; and θmax is 15o for passenger ships and 30o for cargo ships. 4. The factor smom,i is applicable only to passenger ships (for cargo ships smom,i shall be taken as unity) and shall be calculated at the final equilibrium from the formula:
Where: Displacement is the intact displacement at the subdivision draft; Mheel is the maximum assumed heeling moment as calculated in accordance with paragraph 4.1; and smom,i ≤ 1
4.1.
The heeling moment Mheel is to be calculated as follows:
Mheel = maximum {Mpassenger or Mwind or Msurvivalcraft} 4.1.1. Mpassenger is the maximum assumed heeling moment resulting from movement of passengers, and is to be obtained as follows: Mpassenger = (0.075 . Np) . (0.45 . B) (tm) Where: Np is the maximum number of passengers permitted to be on board in the service condition corresponding to the deepest subdivision draft under consideration; and 25 of 28
B is the beam of the ship. Alternatively, the heeling moment may be calculated assuming the passengers are distributed with 4 persons per square metre on available deck areas towards one side of the ship on the decks where muster stations are located and in such a way that they produce the most adverse heeling moment. In doing so, a weight of 75 kg per passenger is to be assumed. 4.1.2.
Mwind is the maximum assumed wind force acting in a damage situation:
Mwind = (P. A . Z) / 9806 (tm) Where: P = 120 N/m2 A = projected lateral area above waterline Z = distance from centre of lateral projected area above waterline to T/2 T = ship’s draft, di 4.1.3. Msurvivalcraft is the maximum assumed heeling moment due to the launching of all fully loaded davit-launched survival craft on one side of the ship. It shall be calculated assuming that all lifeboats and rescue boats fitted on the side to which the ship has heeled after having sustained damage shall be assumed to be swung out fully loaded and ready for lowering. 5. Unsymmetrical flooding is to be kept to a minimum consistent with the efficient arrangements. Where it is necessary to correct large angles of heel, the means adopted shall, where practicable, be self-acting, but in any case where controls to equalization devices are provided they shall be operable from above the bulkhead deck. In all cases, si is to be taken as zero in those cases where the final waterline, taking into account sinkage, heel and trim, immerses the lower edge of openings through which progressive flooding may take place and such flooding is not accounted for the calculation of factor si. Such openings shall include air-pipes, ventilators and openings which are closed by means of weathertight doors or hatch covers.
Permeability
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For the purpose of the subdivision and damage stability calculations of the regulations, the permeability of each general compartment or part of a compartment shall be as follows:
Spaces
Permeability
Appropriated to stores
0.60
Occupied by accommodation
0.95
Occupied by machinery
0.85
Void spaces
0.95
Intended for liquids
0 or 0.95
For the purpose of the subdivision and damage stability calculations of the regulations, the permeability of each cargo compartment or part of a compartment shall be as follows:
Spaces
Permeability at draft Permeability at draft Permeability at draft ds dp dl
Dry cargo spaces
0.70
0.80
0.95
Container spaces
0.70
0.80
0.95
Ro-ro spaces
0.90
0.90
0.95
Cargo liquids
0.70
0.80
0.95
Special Requirements Concerning Passenger Ship Stability
A passenger ship intended to carry 36 or more persons is to be capable of withstanding damage along the side shell. Compliance with this regulation is to be achieved by demonstrating that si is not less than 0.9 for the three loading conditions. The damage extent is to be dependent on both N and Ls such that:
1. The vertical extent of damage is to extend from the ship’s moulded baseline to a position up to 12.5 m above the position of the deepest subdivision draft. 2. Where 400 or more persons are to be carried, a damage length of 0.03 Ls but not less than 3 m is to be assumed at any position along the side shell, in conjunction with a penetration inboard of 0.1B but not less than 0.75 m 27 of 28
measured inboard from the ship side, at right angle to the centerline at the level of the deepest subdivision draft. 3. Where 36 persons are carried, a damage length of 0.015Ls but not less than 3 m is to be assumed, in conjunction with a penetration inboard of 0.05 B but not less than 0.75 m.
Double Bottoms in Passenger Ships and Cargo Ships A double bottom shall be fitted extending from the collision bulkhead to the afterpeak bulkhead, as far as this is practicable and compatible with the design and proper working of the ship. The inner bottom shall be continued out to the ship’s sides in such a manner as to protect the bottom to the turn of the bilge. Such protection will be deemed satisfactory if the inner bottom is not lower at any part than a plane parallel with the keel line and which is located not less than a vertical distance h measured from the keel line, as calculated by the formula: h = B/20. However, in no case is the value of h to be less than 760 mm, and need not be taken as more than 2000 mm. Initial Testing of Watertight Bulkheads, etc. 1. Testing watertight space not intended to hold liquids and cargo holds intended to hold ballast by filling them with water is not compulsory. When testing by filling with water is not carried out, a hose test shall be carried out where practicable. Where a hose test is not practicable because of possible damage to machinery, electrical equipment insulation or outfitting items, it may be replaced by a careful visual examination of welded connections, supported where deemed necessary by means such as a dye penetrant test or an ultrasonic leak test or an equivalent test. 2. The forepeak, double bottom and inner skins shall be tested with water to an appropriate head. 3. Tanks which are intended to hold liquids, and which form part of the watertight subdivision of the ship, shall be tested for tightness and structural strength with water to a head corresponding to its design pressure. The water head is in no case to be less than the top of the air pipes or to a level of 2.4 m above the top of the tank, whichever is the greater.
Peak and Machinery Space Bulkheads 1. A collision bulkhead shall be fitted which shall be watertight up to the bulkhead deck. This bulkhead shall be located at a distance from the forward perpendicular of not less than 0.05L or 10 m, whichever is the less, and not more than 0.08L or 0.05L + 3 m, whichever is the greater. 2. No doors, manholes, access openings, ventilation ducts or any other openings shall be fitted in the collision bulkhead below the bulkhead deck. 3. Bulkheads shall be fitted separating the machinery space from cargo and accommodation spaces forward and aft and made watertight up to the bulkhead 28 of 28
deck. In passenger ships an afterpeak bulkhead shall also be fitted and made watertight up to the bulkhead deck.
Subdivision and Damage Stability of Oil Tankers This topic is dealt with in Chapter 4 of the International Convention for the Prevention of Pollution from Ships “MARPOL”. Location of Damage a) In tankers of more than 225 m in length, anywhere in the ship’s length. b) In tankers of more than 150 m, but not exceeding 225 m in length, anywhere in the ship’s length except involving either after or forward bulkhead bounding the machinery space located aft. The machinery space shall be treated as a single floodable compartment. c) In tankers not exceeding 150 m in length, anywhere in the ship’s length between adjacent transverse bulkheads with the exception of the machinery space. Extent of Damage •
Side Damage
1
Longitudinal Extent
2
Transverse Extent
3
Vertical Extent •
or 14.5 m, whichever is less or 11.5 m, whichever is less From the moulded line of the bottom shell plating at centerline, upward without limit
Bottom Damage For 0.3L from F.P. of the ship
1
Longitudina l Extent
2
Transverse Extent
3
Vertical Extent
or 14.5 m, whichever is less or 10 m, whichever is less or 6 m, whichever is less,
Any other part of the ship or 5 m, whichever is less or 5 m, whichever is less or 6 m, whichever is less, 29 of 28
measured from the moulded line of the bottom shell plating at centreline
measured from the moulded line of the bottom shell plating at centreline
If any damage of a lesser extent would result in a more severe condition, such damage shall be considered.
Damage Survival Oil tankers shall be regarded as complying with the damage stability criteria if the following requirements are met: 1. The final waterline, taking into account sinkage, heel and trim, shall be below the lower edge of any opening through which progressive flooding may take place. 2. In the final stage of flooding, the angle of heel due to unsymmetrical flooding shall not exceed 25 o, provided that this angle may be increased up to 30 o if no deck immersion occurs. 3. The stability in the final stage of flooding shall be investigated and may be regarded as sufficient if the GZ curve has satisfied the criteria shown on the following figure.
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The range should be at least 20 o beyond the position of equilibrium in association with a maximum residual righting lever of at least 0.1 m within the 20o range; the area under the curve within this range shall not be less than 0.0175 m.rad. Unprotected openings shall not be immersed within this range unless the space concerned is assumed to be flooded.
4. The Administration shall be satisfied that the stability is sufficient during intermediate stages of flooding.
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