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UNDERSTANDING SHIP STABILITY

1.1 EQUILIBRIUM AND STABILITY


In general, a rigid body is considered to be in a state of equilibrium when the resultants of all forces and moments acting on the body are zero. In dealing with static floating body stability, we are interested in that state of equilibrium associated with the floating body upright and at rest in a still liquid. In this case the resultant of all gravity forces (weights) acting downward, and the resultant of the buoyancy forces, acting upward on the body, are of equal magnitude and are applied in the same vertical line.

(A) Stable equilibrium.

If a floating body, initially at equilibrium, is disturbed by an external moment, there will be a change in its angular attitude. If upon removal of the external moment, the body returns to its original position, it is said to have been in stable equilibrium and to have positive stability.

(B) Neutral equilibrium.

If, on the other hand, a floating body that assumes a displaced inclination because of an external moment remains in that displaced position when the external moment is removed, the body is said to have been in neutral equilibrium and has neutral stability. A floating cylindrical homogeneous log would be in neutral equilibrium.

(C) Unstable equilibrium.

If a floating body, displaced from its original angular attitude by an external force, continues to move in the same direction after the force is removed, it is said to have been in unstable equilibrium and was initially unstable.

A ship may be inclined in any direction. Any inclination may be considered as made up of an inclination in the athwartship plane and an inclination in the longitudinal plane. In ship calculations the athwartship inclination, called heel or list, and the longitudinal inclination,
called trim, are usually dealt with separately.

1.2 Weight and Center of Gravity


​This topic deals with the forces and moments acting on a ship afloat in calm water, which consist primarily of gravity forces (weights) and buoyancy forces. Therefore, equations are usually developed using displacement weight,  W, and component weights, w.


The total weight, or displacement, of a ship can be determined from the draft marks and Curves of Form. The position of the center of gravity may be either calculated or determined experimentally. Both methods are used when dealing with ships. The weight and center of gravity of a ship that has not yet been launched can be established only by a weight estimate, which is a summation of the estimated weights and moments of all the various items that make up the ship.

After the ship is afloat, the weight and center of gravity can be accurately established by an inclining experiment. To calculate the position of the center of gravity of any object, it is assumed to be divided into infinitesimal particles, the moment of each particle calculated by multiplying its weight by its distance from a reference plane, the weights and moments of all the particles added, and the total moment divided by the total weight. The result is the distance of the center of gravity from the reference plane. The location of the center of gravity of a system of weights, such as a ship, may be calculated by multiplying the weight of each component by the distance of its center of gravity from a reference plane, and dividing the total moment of the components by the total weight. The location of the center of gravity is completely determined when its distance from each of three planes has been established.

1.3 Displacement and Center of Buoyancy


​Force of buoyancy is equal to the weight of the displaced liquid, and that the resultant of this force acts vertically upward through a point called the center of buoyancy, which is the center of gravity of the displaced liquid. Application of these principles to a ship or submarine makes it possible to evaluate the effect of the hydrostatic pressure acting on the hull and appendages by determining the volume of the ship below the waterline and the centroid of this volume. The submerged volume, when converted to weight or mass of displaced liquid, is called the displacement, W.

1.4 Interaction of Weight and Buoyancy


​The attitude of a floating object is determined by the interaction of the forces of weight and buoyancy. If no other forces are acting, it will settle until the force of buoyancy equals the weight, and will rotate until two conditions are satisfied:


(a) The centers of buoyancy B and gravity G are in the same vertical line
(b) Any slight rotation from this position will cause the equal forces of weight and buoyancy to generate a couple tending to move the object back to float on stable equilibrium.

The center of gravity may be either above or  below the center of buoyancy.

An exception to the second condition exists when the object is a body of revolution with its center of gravity exactly on the axis of revolution When such an object is rotated to any angle, no moment is produced, since the center of buoyancy is always directly below the center of gravity. It will remain at any angle at which it is placed (neutral equilibrium).

A submerged object that is clear of the bottom can come to rest in only one position. It will rotate until the center of gravity is directly below the center of buoyancy. If its center of gravity coincides with its center of buoyancy, as in the case of a solid body of homogeneous material, the object would remain in any  position in which it is placed.

A ship or submarine is designed to float in the upright position. This fact permits the definition of two  classes of hydrostatic moments.

Righting moments.

A righting moment exists at any angle of inclination where the forces of weight and buoyancy act to move the ship toward the upright position.

Heeling moments.

A heeling moment exists at any angle of inclination where the forces of weight and buoyancy act to move the ship away from the upright
position. The center of buoyancy of a ship or a surfaced submarine moves with respect to the ship, as the ship is inclined, in a manner that depends upon the shape of the ship in the vicinity of the waterline. The center of buoyancy of a submerged submarine, on the contrary,
does not move with respect to the ship, regardless of the inclination or the shape of the hull, since it is stationary at the center of gravity of the entire submerged volume.

The moment acting on a surface ship can change from a righting moment to a heeling moment, or vice versa, as the ship is inclined, but this cannot occur on a submerged submarine unless there is a shift of the ship's center of gravity.

Lowering of the center of gravity along the ship's centerline increases stability. When a righting moment exists, lowering the center of gravity along the centerline increases the separation of the forces of weight and buoyancy and increases the righting moment. When a heeling moment exists, sufficient lowering of the center of gravity along the centerline would change the heeling moment to a righting moment. Similarly, sufficient lowering of the center of gravity along the centerline could change the initial stability in the upright position from negative to positive.

if the ship's center of gravity were to rise along the centerline, the ship would capsize transversely long before there would be any danger of capsizing longitudinally. However, a surface ship could, theoretically, be made to founder by a downward external force applied toward one end, at a point near the centerline and at a height near or below the center of buoyancy, without capsizing. It is unlikely, however, that an intact ship would encounter a force of the required magnitude.

Surface ships can, and do, founder after extensive flooding as a result of damage at one end. The loss of buoyancy at the damaged end causes the center of buoyancy to move so far toward the opposite end of the ship that subsequent submergence of the damaged
end is not adequate to move the center of buoyancy back to a position in line with the center of gravity, and the ship founders, or capsizes longitudinally. 

In the case of a submerged submarine, the center of buoyancy does not move as the submarine is inclined in a fore-and-aft direction. Therefore, capsizing of an intact submerged submarine in the longitudinal direction is possible, and would require very nearly the same moment as would be required to capsize it transversely. If the center of gravity of a submerged submarine were to rise to a position above the center of buoyancy, the direction, longitudinal or transverse, in which it would capsize would depend upon the movement of liquids or loose objects within the ship. 

1.5 Upsetting Forces


The magnitude of the upsetting forces, or heeling moments, that may act on a ship determines the magnitude of moment that must be generated by the forces of weight and buoyancy in order to prevent capsizing or excessive heel.External upsetting forces affecting transverse stability may be caused by:
​
  1. Beam winds, with or without rolling.
  2. Lifting of heavy weights over the side.
  3. High-speed turns.(d) Grounding.
  4. Strain on mooring lines.
  5. Towline pull of tugs.
  6. Internal upsetting forces include:
  7. Shifting of on-board weights athwartship.
  8. Entrapped water on deck.

When a ship is exposed to a beam wind, the wind pressure acts on the portion of the ship above the waterline, and the resistance of the water to the ship's lateral motion exerts a force on the opposite side below the waterline. Equilibrium with respect to angle of heel will be reached when:
  1. The ship is moving to leeward with a speed such that the water resistance equals the wind pressure,and
  2. The ship has heeled to an angle such that the moment produced by the forces of weight and buoyancy equals the moment developed by the wind pressure and the water pressure.

There are numerous other situations in which external forces can produce heel. A moored ship may be heeled by the combination of strain on the mooring lines and pressure produced by wind or current. Towline strain may produce heeling moments in either the towed or towing ship. In each case, equilibrium would be reached when the center of buoyancy has moved to a point where heeling and righting moments are balanced. In any of the foregoing examples, it is quite possible that equilibrium would not be reached before the ship capsized. It is also possible that equilibrium would not be reached until the angle of heel became so large that water would be shipped through topside openings, and that the weight of this water, running to the low side of the ship, would contribute to capsizing which otherwise would not have occurred. Upsetting forces act to incline a ship in the longitudinal as well as the transverse direction. Shifting of weight saboard in a longitudinal direction can cause large changes in the attitude of the ship because the weights can be moved much farther than in the transverse direction. When very heavy lifts are to be attempted,as in salvage work, they are usually made over the bow or stern, rather than over the side, and large longitudinal inclinations may be involved in these operations.

Stranding at the bow or stern can produce substantial changes in trim. In each case, the principles are the same as previously discussed for transverse inclinations. When a weight is shifted longitudinally,or lifted over the bow or stern, the center of gravity of the ship will move, and the ship will trim until the center of buoyancy is directly below the new position of the center of gravity. If a ship is grounded at the bow or stern, it will assume an attitude such that the moments of weight and buoyancy about the point of contact are equal.

In the case of a submerged submarine, the center of buoyancy is fixed, and a given upsetting moment produces very nearly the same inclination in the longitudinal ​direction as it does in the transverse direction. The only difference, which is trivial, is due to the effect of liquids aboard which may move to a different extent in the two directions. A submerged submarine, however, is comparatively free from large
upsetting forces. Shifting of the center of gravity as the result of weight changes is carefully avoided. For example, when a torpedo is fired, its weight is immediately replaced by an equal weight of water at the same location.

1.6 Submerged Equilibrium


Before a submarine is submerged, considerable effort has been expended,both in design and operation, to ensure that:
  1. The weight of the submarine, with its loads and ballast, will be very nearly equal to the weight of the water it will displace when submerged.
  2. The center of gravity of these weights will be very nearly in the same longitudinal position as the center of buoyancy of the submerged submarine.
  3. The center of gravity of these weights will be lower than the center of buoyancy of the submerged submarine.

These precautions produce favorable conditions which are described, respectively, as neutral buoyancy,zero trim, and positive stability. A submarine on the surface, with weights adjusted so that the first two conditions will be satisfied upon filling the main ballast tanks, is said to be in diving trim.The effect of this situation is that the submarine,insofar as transverse and longitudinal stability are concerned, acts in the same manner as a pendulum. This imaginary pendulum is supported at the center of buoyancy, has a length equal to the separation of the centers of buoyancy and gravity, and a weight equal to the weight of the submarine.It is not practical to achieve an exact balance of weight and buoyancy, or to bring the center of gravity precisely to the same longitudinal position as the center of buoyancy. It is also not necessary, since minor deviations can be counteracted by the effect of the bow and stern planes when un

TRANSVERSE STABILITY IN SHIPS

Transverse stability largely determines the ship's seaworthiness.  Among other things, the location of points G, B and M can determine if a ship, for example, will experience problems in bad weather.
 
The shipbuilder can influence the location of:
-   'B', through  the shape and volume of the hull
-   'M:  by the area and shape of the water-plane area.
 
For every ship B and M are in a fixed position for each particular draft. The crew only has control of G by loading discharging or transferring weight.

Transverse stability can be divided into:
  • initial stability  (angles up to about  5°)
  • stability with angles of heel greater than 5°
 
Important for transverse stability are the distances:
  • KM (and if given, BM), indicative for form stability (see table below).
  • KG is indicative of weight stability. (must be calculated)
Hydrostatics of a drill ship
Hydrostatics of a drill ship
Hydrostatics-Draft vs displacement
Hydrostatics-Draft vs displacement
hydrostatics- Draught vs LCB & LCF
hydrostatics- Draught vs LCB & LCF
hydrostatics- Draught vs VCB & VCF
hydrostatics- Draught vs VCB & VCF
An extract from the hydrostatic data of drillship are shown above. This clearly shows that with the different draft values of displacement,LCB,LCF,VCB,VCF and many other details changes:  This is due to the change o/the surface area and form of the waterline.

 3.2   Stability of form  and weight

3.2.1    Form  stability

If you have a canoe  and  a rowboat  sitting next to each other, the rowboat  will appear more stable in the transverse direction  than the canoe because  the rowboat  is beamier. Justifiable, because the wider the vessel, the more stable it is as a rule. Why? As soon as a boat lists slightly, buoyancy shifts to the lower side.

The more beam a vessel has, the more buoyancy (located at B) will shift. The distance over which B moves to the lower sir is proportionate    to the square of the beam of the ship.
Transverse   stability   is largely dependent upon:            
  • beam,                                                       
  • draft   (prevents   the  bilge  rising  out the water),                                                 
  • Freeboard (prevents the deck edge from going under water).                                  
This is directly related to the shape of the hull. Here, we are referring to the influence of form on stability.
Definition of form stability.

For a given angle of heel, point B changes position   such that a moment   (= force) distance) is created. This moment is equal to the up thrust x the distance between work lines   of the buoyancy and displacement force. This moment is called the stability moment and works against the angle of heel i.e. tries to return the vessel to initial position of equilibrium.

3.2.2    Weight stability

how can a relatively narrow ship still be stable? As soon as the ship lists, the position of gravity, G, also moves to the lower side
from its original   vertical position.  However B at the same angle of heel moves more than G moves from its original position.
Then the up thrust return the ship back to her original state of equilibrium. With a relatively narrow ship, B will closer, to G than with a beamy ship.

Afloat, in which there is almost no stability of form, G lies under B. With small list, G moves from the vertical to the is higher side. We are referring now to weight stability.
Weight Stability in SubmarinesWeight Stability in Submarines
Definition of weight stability
 
With regard to weight stability, only the center of gravity moves from the vertical position. Depending on the location of support (location of the responding force) the body is then in stable, unstable or neutral equilibrium.

Summary

In practice, a ship's seaworthiness  is based on  both  principles   of  stability,  whereby both extremes are also reached.An offshore pontoon that carries its cargo solely on its deck, depends mainly on its form stability, while  a submarine  depends  on its weight stability to remain upright. The submarine and the empty pontoon will  experience a short rolling effect and move swiftly,will experience an unpleasant short rolling effect and move stiffly. With other ships, a combination of weight and form stability exists.

3.3   Location of CENTER OF BUOYANCY (B) AND STABILITY

​B (Center Of Buoyancy = COB) indicates the location   of the resulting buoyancy of the displaced seawater. The location of B is dependent   upon the hull's form. B is the volumetric center of the hull and consequently, the center of buoyancy. Buoyancy is equal to the weight   (displacement)   of the ship. Without   the equilibrium of these forces, the ship would capsize. 
The resultant   (vector) of all upward pressure is always perpendicular   to the momentary waterline.  This means that the alignment of this vector in calm water is different from that in waves and thus, changes continuously.
The results of the forces that move to G and B position   themselves in a line.  The ship must,  therefore,  lie neutral  in the water  and  not  move  around   one  of its axes (rolling or pitching,  etc.),
The  resultant  of G is downward  and is perpendicular   to  the  horizontal   surface  area The resultant  of B  is upward  and is perpendicular  to the transitory  waterline. Both results are equal.
When   a ship is sailing in waves, point   B continually   changes its location.   The   resulting moment will therefore, continue to attempt to line up B with G.Note:  point  G remains  (with an equivalent weight  distribution)    a fixed point.  If it is unstable the ship will capsize. If the ship loses its equilibrium   due to an external force (for example a gust of wind), the distribution   of upward force on the hull changes. The outcome is that the resulting buoyancy (B) will move in the direction the   movement    of B with   an increased list. If the ship’s list continues   to increase, for whatever reason, then B will also move still further from the vertical plane of symmetry.
Depending   on the shape of the hull, B will continue   to return to the vertical plane of symmetry as the ship continues to move. Depending   on the shape of the superstructure, this will occur as soon as the ship turns 180° (upside down).

3.4   Location of the Metacenter, M AND STABILITY


The importance   of M's location to transverse (initial)   stability is great.  The location ofM depends on the location of B. The location of G in relation to M is mainly decisive for the stability.

Stability can be:
  • Positive  (G under  M)
  • Neutral  (G at M)
  • Unstable  (G above M) 
Here ship heel is 0 degree. So distance between center of buoyancy (B) and Metacenter (M) will depend on water line breadth at that draft. For Higher beam,BM will be large and for narrow beam vessel BM will be smaller
As the vessel starts to heel,waterplane area increases. Increased waterplane area helps to increase KN values. So B1M will be more compared to BM.
As vessel heels further,its deck edge submerged into water and bilge area comes out of water,which will significantly reduce water plane area at the draft. So KN values will start reducing and B1M will also start reducing.
As ship heels more,waterplane area further reduces. And distance B1M will reduce further
​Up  to about  5° list and/or   trim,  it can be assumed  that  B forms  a circle with  M  as  center  point.  BM is, therefore, a fixed distance. The drawings demonstrate   that with an increase in heel,  the location  of M depends on: the shape of the submerged  part  of the
ship  (thereby,  the  location  of B, which changes continuously  in a moving ship) the surface area of the waterline  whereby the width  of the waterline  is the most important   factor (this also changes continuously  in a moving ship). Consequently, the BM becomes large initiaIIy with  an  increasing List. If the list further increases, the waterline surface area, decreases along with the BM distance (depending on the draft).
 The   distance   of BM is decisive for amount of righting moment. With larger list, there is a false metacenter N. The value BM for transverse angles is significantly different from M for longitudinally angles with BML being much larger.

3.5   Model data AND STABILITY

 
The lines plan accurately represents the hull's form. Amongst other lines the water lines are drawn at constant distances from base line to the design waterline. With  the Simpson Rules the  area  of the  water plane   as well  as the area of the ordinates  (sections)  can be calculated

The   following   can  also  be  derived  from these rules:
-   the statically moment  of the waterlines in relation  to the keel
-   the ordinates  in relation  to the aft perpendicular   so  that   the  volume   of the hull can be determined.
KB and LCB can be found in this way.

Bonjean curves ROLE IN STABILITY

​
A line representing the area below the water lines at each ordinate gives the possibility to calculate displacements coefficient for a given trim at each ordinate (section).
A graph is drawn, whereby the draft can be eadon the vertical axis and the area of the 1 frame on the horizontal axis. The 6 calculated areas are then given and combined in the graph. This graph for all the sections is called the ‘Bonjean Curves’. With the Bonjean curve of each ordinate and the Simpson's Rules: the  volume  of the  hull  at each  draft  and trim can be calculated.
The so calculated displacement represents the submerged part of the ship without the shell plating, rudder, propeller etc.
The  displacement   has  to  be  adjusted  accordingly. The  buoyancy  at WL  4 can be  found  by measuring  horizontally  the Bonjean  curves
at the level of waterline  4 for each section. By combining   the areas formed below the waterline drawn on these section values represents the buoyancy.
In the third figure, the draft is 7 meters, the resulting value for an area of 120.
Bonjean Curve
Bonjean Curve
The  Bonjean  data  appear  in table form  in the ship's specific stability  booklet  (hydrostatic particulars).
The  following  is given  for  each  ordinate and waterline:
  • The  distance  from basis to waterline  for each ordinate  (section).  For each water- line, the area of the submerged part of the ordinate.
  • The distance from each ordinate to ordinate 0 (is aft perpendicular).
  • The statical moment of area in relation to the baseline.
The statical moment of area in relation to ordinate O. Each arbitrary  draft  and trim as well as the volume  of the  hull,  LCB  and  KB  can  be calculated  from these data.
Many more sections can be used to determine not only the ship’s displaced volume with each trim but also with each list, preferable with a computer’s assistance. At the same time, it is possible to calculate B for the ship on a series of waves. The accuracy of the calculations rely on the accuracy of the Bonjean curves. 

​New methods of calculation


The newest methods   involve dividing the shell surface in small elements. For each element the hydrostatic pressure is calculated.
To find the sum, by means of the direction and strength of the hydrostatic force on all calculated surface elements, the position of B is determined.   The position of B can, in this manner, be calculated for each transitory waterline,   as well as in complicated wave systems.

3.6   Center of gravity  'G' ROLE IN STABILITY

 
The  assumption  is that  the total weight  of the ship (weight of structural  parts, machinery,  outfit,  cargo, fuel, etc.) is concentrated  at a point G  ( Center of Gravity=COG). The  summation   and  alignment   of which, are  represented   by a vector,  the  so-called resultant  of all weights for the ship. G is the only point  that  can be directly  influenced by the ship's crew. The crew determines where the different weights (cargo, ballast, fuel, supplies)  are placed onboard.
In above cargo weight distribution,it has been shown that ship's center of gravity depends highly on distribution height of cargo. As cargo is placed near bottom,G goes down and vice versa as cargo goes up,location of G also goes up.
Vector:
Quantity    that   indicates   magnitude    as well as direction.  With   regard  to stability, this means that  the vector indicates:
  • a force in tons
  • the working direction  of this force
Resultant:
One  vector  that   replaces  a number   of functioning  vectors on the same body without  changing  the result. 
 
g    = center of gravity of component
G   = center of gravity of the entire ship.

Amongst  other  things, the crew decide:
  • the amount  and sequence  in which  the loaded/ discharged  goods are placed
  • where they are placed
  • which  fuel,  ballast  and  drinking   water tanks are filled or emptied G is thus, dependent on the magnitude of the ship's weight, and  even moreimportant, where it is located.

3.6.1   Determination   of the location of Center of gravity (G) of Ship

The  shipyard  estimates  the  location  of G for the 'empty ship'.
The 'light ship weight'  is the weight  of the ship with  only the compulsory   inventory  onboard 'compulsory   equipment':   equipment that  is part  of the completed  ship, such as anchors, life-saving apparatus,  etc.

The   shipyard  can  fairly  accurately  determine  the  light  weight  of the  empty  ship based on the materials  used. Above  the calculations  for G (light  ship) compulsory  inclining  experiment  has to be performed.   The location  of G is found from the inclining  experiment and  adjusted  for  possible  known  weights  to  be  added  or  removed. This is the starting point for the calculation of G for any other loading condition Additionally,   the  "lightship  weight"  is determined at the same time as the inclining experiment.

The  location  of  G  can be calculated:
  • by the summation  of all weight,  multiplied  by their relative distance  from each weight  to baseline or aft perpendicular  divided  by the total weight
  • by moving of substantial  weights  (inclining experiment)
  • empirically by measuring the rolling period

3.6.La    Law of Moment   Equilibrium

The location  of G above the keel is determined by all weights of ship and cargo, but also on the position  of these weights above the keel (VCJ.
In order  to calculate the VCG the following information   is necessary:
  • displacement  and VCG  of the empty ship
  • the weight  of each added load or cargo
  • the weight  of each added load or cargo
  • weight to a point  of reference, usually the keel
Formula:

Moment= force x lever
units:        tm x   m

For a simple explanation  of 'moment:  consider a seesaw. A parent exerts a moment  in relation  to the pivot point  of:
80 x 4 = 320 kilogram-meter   (kgm)
and the child: 20 x 4 = 80 kgm
Outcome:  equilibrium   one end of the seesaw is suspended  and cannot  be moved. The moments are not equal. The  moment   that  both  now exert relative to the pivot point,  is the same and equilibrium is achieved. The parent  exerts a moment  relative to the pivot  point at 80  x  1 = 80 kgm  and  the child,  the same moment,  namely 20 x 4 =80 kgm.
In  principle,  the same occurs on a ship . If there   is  equilibrium    between   all  moments to port  and starboard  there is no list. The  axis of rotation   is located  in the cen- terline plane. This  also applies in the longitudinal   sense. The   ship  then   turns   on  the  lateral  axis through the waterplane Center Of Floatation  (COF).

Law of Moment   equilibrium:
​

Total   moment    of   a  number    of   forces (weight) x levers in relation  to a fixed point  or area is equal  to the  summation of all the individual  moments  in relation to the same point  or area.
 
The  magnitude   of moment  exerted on the ship thus dependant upon  the:
  • magnitude   of weight  (in tons)
  • lever distance  (in meters)
Center of gravity Reference
Center of gravity Reference
Explanation  of different  moments  that weight  (g) exerts on a ship with  the points of  reference,  distances   and  abbreviations used:

Direction         Abbreviation         Abbreviation                    Point  of             Distance    Explained
                           Momentum           explained                        Reference
Vertical                  MH                    Vertical Moment                Keel                    VCg           Vertical 
Transverse             MT                    Transverse Moment          Centerline         TCg
Longitudinal          ML                    Longitudinal Moment      Aft perpen         LCg           Longitudinal
Momentum balanceMomentum balance
Points of reference can be:
  1. aft perpendicular
  2. fore perpendicular
  3. amidships
  4. centerline
  5. keel (base line).
With a  few illustrations and simple calculations,  from the left illustrations it appears that a large weight moved a short distance has the same effect as a small weight moved  a long distance. The moment for both is equal. On a ship, the location of G is determined by dividing the total moment  by the total weight.

3. 6.1.c INCLINING TEST

The shipbuilder can estimate the displacement, KG and GM based on the materials (mostly steel) used in building. In order to calculate the correct GM of the empty ship, the ship must undergo  an inclining experiment (stability test) to determine KG. The results of the test serve as the basis for all stability calculations. Should the results of the stability test deviate from the ship Builder's calculations, then it could be that the weights of the materials were incorrectly calculated.
 
The weight of the ‘empty ship’ (=  fI. ) must be as accurate as possible. After a substantial conversion, a new inclining experiment can be requested.

During the test:
  • the ship must be free to roll   (mooring wires  slack,  etc.)
  • it must be calm with no wind
  • no disturbance waves.
 
The test must be conducted multiple times both starboard and portside with consistent outcome to ensure an accurate result.
 
A known weight (1) is moved transversally across a known distance (2) as a result of which the ship lists. (1) The weight must be so large that:
  • the ship remains within an initial range of stability (max.  List 50)
  • equal to about 2 % displacement
(2) approximately V2 the  breadth.
G’s change of location is indicated in the figure. The ship's list due to relocating the weight is accurately measured. This can be done by means of a plumb line. If a plumb line is used, it is usually suspended in a hold where the weight hangs in a tank of water to stabilize the plumb line.
The result is determined by measuring the distance the pendulum  moves on a tape line (QR). In practice,a special instrument is currently used that registers the list in fractions of degrees. After a number of weight moves GM can be calculated.
Picture
 QR = plumb line movement at tapeline
PQ = length of pendulum
​The next step is to determine KG. In the hydrostatic tables of the ship, or From the drawing with  the hydrostatic curves, for the specific draught T, KM can be found.
KG is then KM - GM
3. 6.1.d Oscillation test

By making the ship roll when in port a rough estimate of GM can be achieved with the formula below.
fx B2T2<p the factor f lies between 0.5 and 0.7. This factor is equal to (2kt)2 ,in which kt is the radius  of gyration  (see box) of the ship, while rolling.
-   B represents the breadth of the ship  in meters
-   T <p is the oscillation period (time)
 
Oscillation period: the time, measured in seconds, for a complete rolling from port to starboard and back again to port (or vice versa).
Note:
  • the rolling action (T) can be achieved, using the ship's crane, by lifting something heavy from shore and afterwards putting it quickly down again as a result of which the ship will roll.
  • Conditions:   calm water, slack mooring wires, no wind, no waves.

 Radius of gyration

Every ship has a:
  • mass  gyration counter to each motion
  • an apparent mass gyration momentum  (I",) against  rolling (a  rotating motion  on the x-axis).
 
This moment consists of a total of all mass gyration moments of each individual mass (Ixx) and the additional mass gyration moment resulting from the water's motion during rolling (mt ).
The following formulation is often used:
  • kcp the radius of gyration of the ship while rolling
  • the remaining symbols introduce the specific mass of the sea water Ckg/m3) and the displacement (rn"), respectively.
The exact formula for the rolling period (in seconds) is indicated by:

g the acceleration  gravity in m/ S2.
 
If the ship's rolling period is measured and the value of GM is known (for example from the stability calculation) then the radius of gyration while rolling can be determined, or the value of kt is known, then the GM can be determined (often used as a means to monitor GM of a ship).

3.6.2    Stiff and tender ship

The formula used with the oscillation test says something important  about the rolling period (T)  namely:  that a relationship exists between the rolling period and GMo. If 'T' is short, (the rolling period short) ,GM (initial stability) is large as is the righting arm. This is referred to as a stiff ship. This usually concerns ships with a heavy cargo (steel or ore) in the lower hold. A ship that is too stiff can be dangerous. The cargo may shift and forces on the ship structure can be to high.

NB: Possibly more ships have been lost by too great a GM than ships with a GM to small. (See IMO recommendations: GM maximum 3% of the breadth).
Besides the difficulties indicated above, rapid rolling in bad weather is also unpleasant for the people on board.
 
Preventing a ship from being too stiff:

the shipbuilder can design a ship for heavy cargo where the center of gravity of the cargo is relatively high by:
  • a high double bottom
  • fortified tween decks (where part of the cargo can be placed)
 the crew can:
  • fill high tanks with their center of gravity above G and if the maximum displacement is not yet exceeded
  • the crew can partly fill tanks in order to create free surface effect. As a result of which G will increase virtually and the ship will be less stiff 

If 'T' is long, (the rolling period long and GM small), the ship is referred to as tender. An increasingly longer rolling period may indicate that the situation onboard may be unsafe and the ship could even capsize.
A typical example of this is the phenomena of icing, freezing of spray in arctic waters. Due to the increase of ice on deck and mast G moves upward and the ship can eventually capsize.
 
Several advantages of a tender ship:
  • It  is more  pleasant   for  the  passengers; therefore, passenger ships are usually tender.
  • The load on the structure of a slow-rolling ship is less. The cargo will shift less quickly, the forces on any lashings will be lower.

The condition of a vessel which is too tender, can be improved by discharging weight located above G or loading weight below G. Most ships are generally referred to as 'tender'  or 'stiff'. This does not necessarily indicate an unsafe situation. The terms 'too  tender'  or 'too  stiff'  are dependant upon so many factors.
Ice formation on decks increases ship's vcg,thus making it more tender and ship becomes prone to capsize.
3.8  Level of Capacity
Picture
3.8  The Righting Arm

As seen in  the  illustration,   the  moment consists of two  equal forces (vectors):
  • gravity, located at G
  • buoyancy, located  at B
the lever or the righting  arm is the shortest  distance (perpendicular) between both vectors.
 
The magnitude  of the  righting  arm  (momentum) is the product  of:
  • exerted pressure (displacement )
  • GZ (lever of statical stability)  
The  magnitude   of  the  righting   moment is:
moment = Displ x GZ
 
If this formula is further  divided,  then  (see illustration above):
righting moment  = Displ x GZ
moment = Displ x GN  sin ¢
moment = Displ x (KN  -  KG) sin ¢
moment = Displ  x (KN sin ¢ - KG sin ¢)
 
From the righting  moment  it appears  that only the crew has influence  over the location of G. Namely, the  crew  can  decide  how  much weight (up to a certain level)  can be loaded discharged and where it will be located. crew can only change the remaining factors (Displ, M,  KN sin ¢) a little  or not  at all.

 3.8.1  Height of G above the keel
​As stated earlier, the location   of G is dependent   upon   the distribution    of weight
on the ship. In each of the illustrations, the added weight (cargo) is placed in a somewhat exaggerated position. The result is that G also shifts to a few extreme positions. It is clear to see that the righting arm changes in direction as well as magnitude. The ship in the drawings lists, due to an external force (a wave, for example). From  the  illustrations, it appears that G's position is determined to a large extent:
  • by the direction of the moment's rotation and thus, the ship
  • the magnitude of the righting arm (GZ) and thus, the degree of form stability.
It can be seen in the above pictures that as the (vertical) distance between G and M becomes greater, the righting arm is also greater and thus, the righting moment. KM is always larger than KG with a positive or aligned righting moment.

3.8.2    Horizontal displacement of G
G can move horizontally during loading, discharge or transfer of weight. This causes a short term (a few seconds) listing moment because the buoyancy and gravity vectors no longer line up. As a result, the ship has a list and/or trim. Point B (buoyancy) moves to the lower side. 
As soon as the vectors line up again, the lever of the righting moment becomes zero, and the ship remains in position. If the horizontal displacement of G is so great that the buoyancy vector does not come in alignment of the vector of G, then the ship will capsize.

3.8.3    Consequences of incorrect loading
​In figures 1 up to 4 is shown a normal stable position, as well as an unstable position:
  • moored    (hawsers/mooring lines firm)
  • unmoored     (hawsers loose)
 Figures 1 and 2 is normal. G is at an adequate distance below M. This also causes no problems in bad weather.
The situation in figures 3 and 4 is dangerous. G lies above M. The ship is not allowed to go to sea since G must lie a minimal 0.15 meter under M. 
As long as the ship is moored, with the hawsers tight, little can happen, but as soon as the ship releases her mooring lines she will list. How far will it list? The breadth of the waterline will increase due to the list. Subsequently M rises. When M lies at G, there is no longer a (turned) arm and the ship will reach maximal list. ​The only solution is to lower point G. At sea, it is usual to fill the (lower positioned) ballast tanks with seawater.
3.9 Curve of Statical stability
 
In order to illustrate the extension of the righting arm's at each angle, these levers are displayed as a curve, the stability curve (see 3.9.3).
The curve shows the stability of the ship at all heeling angles (usually up to 90°). The curve of the righting arms applies to a specific draft and weight distribution. The righting arm must be sufficiently large at each angle for the ship to right itself various conditions such as bad weather. 
The curve must be derived and evaluated: during loading or discharging and before sailing changes of weight distribution during the voyage due to use of fuel and/or drinking water have to be considered.

The calculation and appraisal of the curve can be produced by computer. With each new entry the ships loading computer immediately recalculates the righting arms and thus, the curve. The responsible officer must be thoroughly aware of the basis of this calculation. He/she must know which rules to apply, if the ship's stability is decreasing to such extent that action is required.

As long as the gravity of the ship does not change, the shape of the statical stability curve is determined by the ship's form and the water plane area. For example with a small freeboard or small draft the water plane area can change considerably. The curve is only applicable if the ship lies in calm water. If the ship sails in calm water or waves, the water plane area changes and thus, a continuously changing curve results. 

The curve shows:
  • The righting moment or the righting arm at each angle of list
  • The energy produced by the righting momentum to resist a list from 0° to any chosen angle.
The magnitude of the levers and thus, the area under the curve, will decrease if:
  •  Weight (cargo, ballast, etc.) is placed above G
  • Weight is discharged under G
Changes in the direction in the GZ curve are caused by the following reasons:
  • The deck is submerged
  • The coaming is submerged
  • The bilge rises above water
All due to the waterline breadth changes, positively or negatively. The horizontal scale is the distance calculated from 0° to the point where the righting arm is negative (more than 83°). That is, at which point the ship will dynamically overturn. The ship will overturn at the top of the curve with a statical load.

Cross curves of stability (Stability curves)

For every ship or barge the results of the values of KNsinq as function of the displacement, can be given in table or graph form. Lateral curves of stability are referred to in a graph. These values play an important role in determining GZ. Stability books are not legally required to provide values above 60°. Therefore, a curve with values above 60° is usually not used.

Statical and Dynamical
 
that is immediately absorbed by that object. "Dynamical" refers to the force exerted on an object that is absorbed gradually by that object.

Statical examples:

This is a statical motion because the swing immediately absorbs the pressure exerted on it. A crane loads a heavy weight on a ship from the quay. While the cargo runner (cable) takes weight, the ship experiences a slowly increasing list. This is a statical motion because the force necessary to hoist the weight is directly absorbed by the ship.

Dynamical examples:
 
The same swing is pushed higher with considerable force. The swing's gravity cannot absorb the sudden force and shoots upwards. The swing has a dynamic motion in this case.

The same ship has hoisted a heavy weight number of meters. The weight suddenly falls back to the quay and the ship lists to the other side. The ship is in no position to absorb the sudden change in gravity and undergoes a dynamic motion.
Take a ship that lies in equilibrium in calm water without list and trim. With a very small list, the waterlines W oLo and WfLf intersect each other at centerline. Because the ship's form is symmetric in relation to the longitudinal median plane: the increase in displacement to starboard is equal to the decrease in displacement to port. The mean draft remains unchanged.

There is, however, in comparison with a list of 0°, another distribution of displacement over the ship's length. The volume, which results in the fore ship as a result of the list, is not always as great as the volume aft. This is due to the difference in the form of the frame above the waterline fore and aft. Through this new distribution of displacement over the ship's length, there is a new mean draft (which deviates little, however, from the initial draft). At the same time, the initial center of buoyancy 'B' shifts similarly to a list of 0°. A trim moment develops because B and G are no longer lined-up vertically. The ship will trim until this occurs. A new equilibrium is found whereby the ship has trim.
Because the transitory waterline WfLf differs with W oLo' the CO F, with this new waterline, will be different from W oLo'
 
As a result of the list, the ship has a different:
  • Mean draft (however slight)
  • Trim (fore and aft draft) 
Using the same arguments, a ship in a state of equilibrium in calm water without list and trim will encounter a change of trim if the mean draft changes and G remains unchanged. Because of the difference in shape of the fore and after bodies the longitudinal position of B will change, causing a trimming moment.

3.9.2 Determining the righting arm GZ

The righting arm (GZ) can be calculated for a specific heeling angle explained below. GZ can be calculated for different heeling angles from which the GZ curve can be drawn.

Determining GZ (drawing below left):
  • GZ is equal to PQ
  • PQ==KQ-KP
  • KQ is equal to KNsin¢.
  • KP is equal to KGsin¢
 
Abbreviations:
  • G = Center of Gravity, point of application of the results of the ship's total weight
  • ¢ = angle of list
  • sino: relationship between the opposite side and the oblique side 
  • KNsin¢ is calculated by the shipbuilder and can be found in the hydrostatics particulars with the draft amplitude (T) or displacement:
  • KG is known; thus KGsin¢ can be calculated.
 
If the righting arms (GZ) are known, a vector is drawn in which vertically: GZ are given in centimeters or meters horizontally: the list angle of the ship for example, from 0° until 60°.

The data of the ship shown here are: If

Draft ==3.90 meters
Displacement = 3500 tons.
KG is known, in this example, 4.9 meters.
Example of calculation KGsin¢:
4.9 meters x sin 10° = 0.851 meter, 
4.9 meters x sin 20° = l.676 meters, etc.

To draw the GZ curve's initial path accurately works as follows:
  • place GMo vertically at a heeling angle of 57  (If GMo is negative then it should be drawn below)
  • Connect the peak of the vertical with the 0 coordinate
  • Draw the curve tangent to the line up to approximately 5°.

Displacent                                         Heeling angles
merit              5°           10°        20°         30°         40°         50°         60°
4500           0.447     0.894      1.782     2.611     3.453     4.103     4.552
4250           0.445     0.891      1.791     2.630     3.489     4.150     4.552
4000           0.444     0.890      1.793     2.658     3.521     4.192     4.584
3750           0.445     0.892      1.799     2.699     3.551     4.228     4.614
3500           0.458     0.919      1.862    ,2.853     3.672     4.257    
3250           0.451     0.905      1.831     2.804     3.626     4.277     4.666                            \
3000           0.447     0.896      1.811     2.751     3.586      4.288     4.642

Heeling  angle   degrees        5°         10°        20°        30°        40°         50°        60°
KNsin¢              meters    0.458     0.919     1.862    2..853    3.672     4.309     4.684
3.9.4 Stability regulations

Since it is important to know if the righting arm are sufficiently large, they are placed on a quadrant, with on the horizontal axis the list and on the vertical axis the length of the righting arm. The curve provides an immediate picture of the ship's stability.
 
Due to its importance, international (IMO) regulations are applied to the area under the GZ curve, the so called Dynamic Stability
(cm.rad). These regulations are set forth in IMO resolutions and MSC publications.

Exceptions to stability criteria

The stability regulations for all ships longer than 24 meters are stated in resolution A 749. 
 
Nevertheless, there are vessels that, because of their construction, nature of their activities or cargo they transport (grain, wood), deviate to such from the normal standard that supplementary regulations have been assigned. Relatively new, are the regulations for containerships longer than 100 meters. With this, the standard regulations, along with a specially adjusted factor for the dimensions of the ship, were tightened. Passenger ships also have supplementary regulations.

Additionally for fishing boats, special purpose ships (factory and expedition ships), supply ships, floating oil rigs, (unsinkable) pontoons, 'dynamically supported craft' such as. Hovercraft, separate regulations are issued.
 
Regulations applied to 'High speed craft' are published in the 'International Code of Safety for High Speed Craft'. Additionally, there are still no international supplementary rules for sailboats and professionally utilized pleasure craft, only the classification has some regulations.
a.1 The area under the righting arm curve (GZ curve) up to 30° angle of heel should not be less than 0,055 metre-radians.
a.2 The area under the righting arm curve (GZ  curve) should not be less than 0,09 metre-radians up to 40° angle of heel or up to:
a.3 The angle of flooding  (CPf) is less than 40°.
b.l The area under the righting arm curve  (GZ curve) between the angles of heel of 30° and 40° should not be less than 0.03 metre-radians  
b.2 Between 30° and CPf if this angle is less than 40°.
c. The righting arm GZ should be at least 0,20 metres at an angle of heel equal to or greater than  30°.
 d. The maximum righting arm should occur at an angle of heel preferably exceeding 30° but not less than 25°.
​e. The initial metacentric height GoMo should not be less than 0,15 meter.
The ship with a List. This means that with this List, the righting arm is nil. If the List increases due to rolling, then a righting moment deveLops; if the list decreases, a heeling momentum results. In both cases, the ship will return to its initial list, whereby the lever of the right arm is again nil.
3.9.5 Determining the area under the curve

The area under the curve represents the energy involved in the dynamic stability. In order to calculate the area under the curve, the problem is, that the levers are given in cm or meters on the vertical axis and the angles are given in degrees on the horizontal axis. To find the area below the curve, the heeling angles are also given in terms of distance.
​
3.9.6 Examples of unacceptable stability curve
The curve rises too slowly and does not meet the requirements of regulation a.l. This occurs with ships having either a small KM in relation to KG or a small GM.
GMo' (initial stability) is great, the curve rises quickly but has already reached its maximum at an angle of approximately 10°.The curve is not sufficient for regulation d. This can occur with ships that have more beam and little freeboard.
This is done to create radians for the number of heeling angle degrees. To give heeling angles in radians, the number of heeling degrees must be divided by 57.3°. When the area below the curve is calculated, centimeter or meter radials are obtained.

The area under the curve can now be calculated in the following ways:

Make a triangle with the same area as the area under the curve.
This is less accurate, but still a good way to obtain a quick impression of the area
(Surface area = h x base x height).
This can be applied to regulations a I, a2 and a3. Make a rectangle of the area.

(Surface area = h x base x height).
This can be applied to regulations bl and b2.
Using the Simpson's Rules

3.9.7 Comparison of ships' forms

When during the design of a ship it is decided to change the main dimensions (draft and beam) but maintaining the same displacement and KG, this results in different curves of static arms as follows:

3.9.7.a The black curve (ship 1)

The ratio of the centerline's draft beam of a cargo ship is 2: 1 to 3: 1. For the base model drawn here, it is 3: 1. The black curve, shown in the stability curve, has an initial stability GMo of approximately 15 cm, the maximum righting arm, 40 cm and the stability's range, 68°. The curve's point of inflection is at 32°, the point at which the deck is immersed. At that point, the waterline width does not increase further. In this case, the ship meets the minimal stability requirements.

3.9.7.b The green curve (ship 2)
​

The larger freeboard has a noticeable effect on the curve. The initial stability GMo is the same as the base model; the point of inflection, however, lies further away. In this case, with a 34° list when the bilge rises out of the water.
The maximum righting arm is also larger, approximately 63 cm, but more remarkable
is the increase of the stability's range. Up to a 90° list, the ship still has a positive GZ. The area under the curve, representing dynamical stability, is much greater 

3. 9. 7.c The red curve (ship 3)

It can be concluded from the previous chapters, that the ship's beam has a positive effect on form stability. This is immediately apparent in the red curve.
The GMo is now about 96 cm and the ship is more rigid. The maximal lever is 90 cm, but the point of inflection is reached earlier at 26° .
The deck and bilge rise up from the water at nearly the same time. The waterline width decreases more quickly than in the base model. The range of stability is only slightly larger than that of the base model.The dynamical angle of capsizing is now approximately 73° and the area below the curve is much greater. The area beneath the curve of ship 3 is, however, almost the same as that of ship 2 with the larger freeboard.

3.9.7.d Conclusion

For ships with the same displacement and KG, compared to the base model, the following conclusions can now be drawn.

If the ship has a higher freeboard:
  • there is no effect on initial stability
  • the range of stability rises considerably
  • the area below the curve increases and with this, the dynamical stability is nearly the same as the broader ship it can resist a greater heeling momentum than the base model the effect in swells is the same as that of the base model.

Increasing the ship's beam:
  • large effect on initial stability
  • has a negligible effect on stability range the area beneath the curve increases and with this, the dynamical stability subsequently, it can withstand a much greater statistical heeling moment than the base model. Thus, the larger the beam, the more rigid (stiff) the ship is.
3.9.8 Statical and dynamical stability 

The explanation regarding the difference between statical and dynamical stability can be found on page 48.Statical and dynamical stability are further explained in the following examples. The heeling moment is produced by a beam wind.

The following situations are further examined:
  • A steady wind pressure (statical). The ship experiences a constant list.
  • A gust of wind at 00 list whereby exerted pressure remains the same during the entire rolling period (dynamical).
  • A situation such as mentioned in a. where a lateral force caused by a sudden gust of wind is exerted on the ship (dynamical). 

3.9.8.a Statical wind pressure

In this example, there is steady wind pressure on the ship. This pressure has built up so slowly that the moment caused by the exerted force is completely absorbed by the righting moment. When the wind pressure has reached its maximal value and remains constant ,there is equilibrium between the moment caused by wind pressure and the hydrostatic righting moment. Consequently ,The ship has a constant list called a statical list, CPs'. This is expressed in the curve by the vertical levers of the wind moment. The stability regulations are based on the static arm values decreasing with a fixed wind lever value over the whole stability range. This result in a substantial decrease of the area below the curve. The statical heeling angle is where the horizontal line intersects the GZ curve.

The magnitude of the energy built up by the ship during rolling relates to the location of the center of buoyancy (B) and the location of G. The location of these two points (B and G) relates to the size of the righting arm. If a ship rolls due to swells, it will make a specific oscillating motion to both sides of the statical heeling angle.That will depend on draft and GM.

To determine wind heeling lever of any ship following information will be required.

  • The magnitude of the profile area above the waterline
  • The magnitude of the wind-force per m2
  • The point of application of the resulting forces on, the profile area above water and profile area below water

3.9.8.a Wind gust (dynamical)

The ship in this example has a list of 0 degrees. If struck by a wind gust of longer duration,The ship is no longer in a state of equilibrium and rolls to starboard. The force from the gust of wind is greater than the ship can immediately absorb. The result is that the list does not remain confined to static equilibrium angle such as in the previous example, but rolls to approximately 25 degrees. It can be seen in the graph that the 'static equilibrium angle is at 15 degree.

To what extent the ship rolls, is wholly dependent upon the size of area beneath the curve,the dynamical stability. Prior to static equilibrium angle, heeling lever are more than righting arm and after this angle righting arms are more than heeling arm.

In the above drawing, the same figures are used as on the previous page. The lever of the wind arm is once again 0.113 meters. The result is that all righting arms are 0.113 meters smaller at each heeling angle. What remains are the reduced righting arms and accompanying leftover areas. The leftover area or remaining dynamical stability is sufficiently large to absorb the excess energy of the gust of wind. In the example above, the remaining area is large enough. The area that the excess energy absorbs is same as area ADE.
By regulation, wind heeling lever shall be assumed constant throughout all heeling angles. So gust wind heeling arm is drawn as straight line.

Summary:


From 0 to 15 degrees the moment caused by the wind is greater than the static stability moment, subsequently:
  • An excess of energy is exerted on the ship.
  • The ship rolls beyond 15
 The stability moment is greater from 15, subsequently:
  • The excess energy built up by the wind decreases as the list increases.
  • The ship rolls until about 26 degrees, dynamical equilibrium . The excess energy built up by the wind is gone in the case of dynamic equilibrium.
 
Ultimately the area of ABC is same as DCE.
3.10 List momentum
 
The ship's center of gravity moves parallel to the transferred weight or expended force. Across a distance of:

GG'= p x a / .6. ---+ with the transfer of weight (p)
GG'= P x a / (.6. + p) ---+ by loading weight (p)
 
This weight is a force that causes a moment on the ship. This moment can come either from inside the ship or from outside.
 
Examples of listing moments:

  • Movement of liquids
  • Shifting of cargo
  • wind moment
  • change of wave moment rudder moment
  • tugboat (via hawser)
  • collision
  • paid out gear (e.g. fishing nets)
 
Equilibrium is disturbed in all of the above situations, either:

  • by transfer or loading of weight
  • besides gravity and buoyancy, by a third force exerted on the ship.
 
The result is that the righting arm (GZ) can change. The above-mentioned situations can be hazardous for the ship.

For each list moment, it must be ascertained whether the moment is a:
  • statical moment (continuous force)
  • dynamical moment
  • combination of both.

 Examples of static and dynamic:

  • A steady wind is static, a gust of wind, dynamic
  • slowly shifting grain gives a statical moment
  • weight dropped from a crane has a dynamical moment
  • a slowly turning rudder is static, but if it turns quickly it can deliver a combination of statical and dynamical moment.
 
The examples here show an exaggerated idea of the consequences to the stability curve and thus, to stability, when weight is shifted (with loading, for example).
 
Observations regarding these situations:
​
  • In practice, a similar asymmetric distribution of weight may not occur and definitely not at sea.
  • For clarity's sake, G moves parallel to GZ. G usually moves parallel to the displaced weight.
​ 
Drawings 1,2 and 3 show respectively:

  • A partially loaded ship (1)
  • The ship takes a list from external causes. (2)
  • The distribution of weight doesn't change; G's position doesn't change. (2)
  • B moves to the low side (2)
  • A righting arm with lever G oZ (2)
  • The list increases (3)
  • Because B moves further towards the low side, the lever of the righting arm becomes considerably larger (3)

Drawings 4, 5 and 6 show respectively:

  • A 150 ton weight is moved horizontally resulting in an asymmetric distribution of weight (4)
  • The center of gravity, G, moves parallel to the displaced weight. The interval 01 G0G1 is, for example, 0.10 meters. (4) The ship takes a list due to the asymmetric distribution of weight
  • External cause (5)
  • The lever of the righting arm becorng5 smaller with the interval of GoG1. The lever's extent is now G1Z1. (5)
  • The list increases (6)
  • The lever of the righting arm (G1 Z1) is smaller than in figure 3.

3.10.1 Free liquid correction

Liquid (or a substance that acts like liquid), that can freely move on a ship, can be extremely hazardous. Annually, many ships capsize, resulting in many casualties.
 
The seriousness of the situation is largely dependent upon:
  • The area in which the liquid can freely move
  • The amount of liquid (in relation to the ship's weight)
  • The height above the keel where the liquid is placed (Kg)

Liquid in a tank, hold or on deck, is in fact, weight that can move freely. As soon as the ship lists and/or trims, the liquid moves to the low side whereby the righting capacity decreases or even becomes negative. If liquid can move over the full beam of the ship, the effect is maximal. In particular, a dangerous situation can quickly develop on the (auto) deck of a Ro- ro ship. Because water rushes quickly to the low side, a dynamical effect results, and the ship takes more list.
If the liquid is confined in a tank which is less in width than the beam of the ship, the liquid can only move over the width of the tank and the effect is less serious. This is referred to as the 'free liquid effect'. This causes 'list moment' on the ship, the so called free liquid moment. In order to retain equilibrium, the ship must respond with equal stability moment (the lever of the stability moment is equal to the lever of the list moment).

The following terms and abbreviations are used:

  • free surface correction (FSC)
  • free surface moment (FSM).
  • Shifting of liquid (weight), affects the center of gravity G, which then moves away from the centerline. Subsequently the righting arm decreases by G0G1, The righting capacity is thus, reduced.
  • As soon as the decline of the lever (GoG1) becomes greater than the maximum righting arms (GZ), the ship will capsize.

The following figures show the cross section of a ship with an exaggerated large double bottom. In each drawing:

  • The ship's list is caused by an external force, such as a wave
  • The water in which the ship lies, is fresh water
  • The liquid in the double bottom tank is also fresh water.
​The  double  bottom   tank  in figure  1 is empty.
  • The center of gravity Go remains in the same place.
  • The center of buoyancy  moves to the low side parallel to ZuZj
  • The  magnitude   of the righting  arm = Ll x GoZ
​The  double  bottom   tank  is completely   filled  in figure 2.
 
Comparison   with figure 1:
-   Increased  draft
The center of gravity G is lower due to the increase in weight  under  G. The lever of the righting  arm is now G] Z
-   The center of buoyancy  moves the same distance
-   The magnitude   of the righting  arm = Ll x G] Z
The double bottom tank is half full in figure 3.
 The liquid flows to the low side. G moves Parallel to line Z II Z. due to the transfer of weight. The result is:
 
- The magnitude of the righting arm is diminished by approximately GoG1
The righting arm is now G1Z: the resulting lever.
A force can be moved over its own vector. G1 is transferred virtually along its own vector and is virtually located in point G' at now virtually decreased by GoG'.The remaining initial stability is MaG'.
 
The distance GoG' can be calculated as follows: 12 x Ll .​As presented in chapter 3.7, a box-shaped body: =1
t is the surface moment of inertia expressed in m" (Lt x B/). In the hydrostatic data, GoG' is generally noted in rn",
To be expressed as moment (mt), it must be multiplied by the density (p) of the liquid in the tank or divided by the stowage factor.
 
Remarks:
 
- The formula can only be applied to a rectangular tank. Since most holds are not rectangular, this formula only gives an approximation.
- Density plays a small role, namely:
  • the contents of the tank must be multiplied by p (t/m3) of the transferred liquid
  • ship displacement must be multiplied by p (tl m-) of the seawater.

aLegend:
L, = length of tank
B, = breadth of tank
V = ship displacement in rn" Ll = displacement in tons
 
 In figure 4, a lateral bulkhead has been placed in the same double bottom tank, creating 2 similar tanks, in which the breadth is equal to half the width of the whole tank. Both tanks are half full.
 
By this intervention, the reduction of MoGo (GoG') will be considerable, namely,4 of the original decrease.
 
The evidence of this is as follows:
There are now 2 tanks, thus 2 corrections
GoG:
However, the tank is halved in breadth, thus B is a 12B.
The breadth is calculated to the 3ed power, thus 12B x 12B x 12B = Ys B3.

​The correction per tank is now Ys smaller.
However, there are 2 tanks, thus the total correction is 2 x Ys = \4 of GoG:
 
If 2 partitions are inserted in the double bottom tank, (creating 3 tanks) then the correction is ';.th of GoG'.
As should be evident, the effect of free flowing water on decks/holds of a Ro- Ro IA
ship can have serious consequences. This is also because on these ships, liquid is then found at a relatively high level.
 
Example:
Suppose there is a fire on one of the highest auto decks of a Ro-ro ship. By means of an automatic sprinkler system, water spreads on the auto deck. If the water isn't drained quickly enough (for example, with a clogged drain), it will ultimately flow to one side. In this case, the heeling moment will be greater than the righting moment.
(GZmax xzs).
 
As a result, another force caused by water flowing to one side, develops.
The solution is to install transverse and/or longitudinal bulkheads. However, the placing of these can interfere with trucks being driven on or off the ship.
 
The problem is also applicable to multi- purpose and heavy load carriers.
On multi-purpose ships, the law requires a (minimum) number of transverse bulk- heads in order to, among other things, reduce the 'free liquid problem'. The dis- advantage is that these bulkheads impede flexible loading and unloading.
Free-flowing water in the lower hold is not as disastrous as water found in higher
holds.
Example:
The effect of free liquid in a large hold, is clarified in the following drawings:
drawings 1: 10 em fresh water stands in
the hold
drawings 3: idem example 2, the hold is now split in 2 compartments by a longitudinal bulkhead.
Displacement (hold not yet filled with water): 1100 tons
The illustrations and curves exemplify real situations.
A dry cargo ship with measurements of:
L = 85.00 meters B = 11.80 meters TswTIme=r 4.95 meters
Displacement (summer) = 4300 tons
Dimensions of hold =
The amount of rn' fresh water in the hold of 1A is:
63 m. x 9.5 m. x 0.10 m. = 60 m ', equal to 60 tons.
In drawing 1B, 60 tons of water has moved to port, a distance of approximately half of the
hold's breadth (9.5 /2)
The center of gravity G moves parallel to the transferred water at a distance of GOG2.
The decrease in the righting arm is already worked out in curve 1D.
Conclusion: the ship takes (WIth 10 ern water 111 the hold) a [ist of 5.5 , a negative 1I11t stability and a smaller area under the curve.
3.10.2  Heavy lift
The  use of 'heavy lift' is relative.
What   is heavy  for  one  type  of ship,  may not be for another  ship. This  chapter  deals with ships that  load or unload  heavy cargo with their  own gear.
 
Heavy cargo can roughly be classified as follows:
_  'easy heavy lift': pieces from 50 to about
250 tons
_  'medium  heavy lift': pieces from 250 to approximately   1000 tons
_  'difficult  heavy lift':  pieces above  1000
tons.
 
The  handling  of heavy cargo  must  always be  carried  out  with  the  required   caution in   connection    with   the   risks   incurred. Loading,  discharging  and the voyage itself, require   extensive   preparation,    especially with  regard to ship's strength  and stability. Also  securing   of  the  cargo  needs   to  be looked  at in advance.
 
Multi-purpose   ships, normally have two cranes up to  120 tons per crane lifting  ca- pacity,  which,  when  working   in  tandem, can lift pieces of240  tons.
Special designed  'heavy cargo' ships can be fitted  with tWOcranes of 1200 tons each. The  modern   cranes  are wireless  (remote) controlled   from  a person  on deck or even ashore, close to the piece of cargo.
 
During  the actual  cargo operation,   i.e. the loading or unloading  of one (heavy) parcel,
the intention   is to limit  a maximum  list at
2°.
 
A larger list is not advised, as the cranes aredesigned  to work  upright.  When  working
!            I                                                                                  under  an angle, stresses in the construction may occur, which  are not foreseen.
Also, if the vessel has a larger list, the risk
exists that  the cargo moves sideways when coming  free from its rest. The  moment  the cargo is free of the  deck,  for the ship  as a system, its center  of gravity  moves imme- diately  from  the  original  location   at  rest,in this case the ranktop,  to the location  of suspension,  in this case the top of the crane jib. When   at that  moment   the  ship  has a list, this list may increase, and the situation is out of control
 
moment   of  lift-off.  It  is therefore   of  ut- most importance  that before the actual hoisting  starts, the 'hook'  is vertically above the center of gravity of the parcel.
During  the tightening  of the slings this can be controlled,  and adjusted  if necessary.
As soon  as the winch  of the  loading  gear rums,  and  the heavy load  is lifted  off the ship, quay or lighter,  the weight  will be at the top of the crane's arm.
 
During  the turning,  peaking  and/or   slack- ening of the crane, G will change position continually. As soon as the load is lifted,  G will move  in  the  direction   of the  loading gear's hoist  point   across  a distance  equal
to:
pxa
 
L1
 
The stability  can be improved  by tempo- rarily increasing  the waterline  by utilizing a pontoon   at the waterline  level firmly at- tached to the  ship. This  increases  the wa- terline width,  and so the height  of M
Often a tween deck pontoon   hatch cover is provided with  special attachment   devices. The ships shell has attachments   at various levels,depending   of the draft at the time of cargohandling.
During  slewing and  topping  of the  crane,
the position  of G will change continuously. The ships list has to be kept under  control DY  pumping  ballastwater.
 
The voyage, including  loading and unload- ing operations,  is calculated  once more  by the shipping   company's  specialists.  After chat,the entire  operation   is discussed with the ship's captain.  The captain  is ultimately responsible for the ship's safety.

The list is determined   as follows:
GM=pxa/L1x tan <p
It can also be written  as:
 

tan cp   =pxa/L1xGM
 
If the value of tan cp  is known,  the list cp  can be determined.
 
The formula includes:p:
a:
 
 
 
Ll:

the cargo's weight
the (maximal)  distance  of the crane's hoist point  in relation  to the centerline.
the cargo's weight including  the heavy load
the interval  between  G and M

​NB·. theessiituauon  ponraye  d in the first set ofdr aw.mgs is only an example.
In.  practice, the  list will increase  no  more than about 2°.
When  a heavy cargo weight
has  to  be  lifted   from   the quay, the slings are tightened first, making  sure  the  hook is vertically  above the center of gravity of the weight. The cargo is then lifted by chang- ing the  ballast  condition   of the ship. When   the  cargo  is lifted off the quay and trans- ferred to the ship's hold,  bal- last has to be adjusted  con- tinuously during  the transfer
to  keep the ship upright.
Lifting by pumping  ballast prevents    the    ships    cargo from bouncing   and/or   slip- ping when  listing  to the op- posite  side  or  standing   on end, paired  with  dynamical forces and acceleration,  with all inherent  damage.
3.10.3   Bulk  cargo

3.10.3a   General
 
and flow to the low side on a tilted  surface.

and  also  after  settlement   (see d), remain
 
iscould  be built  that  won't  crumble.  If too much  water is added  to the wet sand, then it will collapse.Various  bulk  cargos  crashing  around  in  a hold will behave in roughly  the same man- ner. This  chapter  concerns  grain and its ef-
3.10.3b  Behavior of grain in a hold

Loose grain  acts more  or less like a liquid.
If a hold  or a number  of holds are partially
 
I                 filled with grain, it has the same effect as a free liquid  surface (see chapter  3.10.1).
The  grain shifts  to one side causing a haz-
 
extreme case, the ship could capsize. There
I                               ardous  situation  onboard  and  in the most
 
is also the problem  of settling  of the cargo.
I                   For  example,  if  ground   coffee  is poured into a can, it has a specific volume.
I                      If the can is struck on the table a few times,
the volume decreases through  settling.
 
 
Grain   refers  to  all agricultural   prod- ucts  having  the  same  characteristics, such  as wheat,  corn,  oats, rye, barley, rice, dried  peas  and  beans,  seeds and other  derivatives with  the same quali- ties.
The  same  occurs  with  grain  in  the  hold. Due  to  vibrations   and  movement   of  the ship, the grain cargo settles and the volume decreases, but of course, the weight remains the same.
The  center  of gravity  of the  grain  in  the hold will drop somewhat.
 
Every ship is required  to carry a book  con- taining  a number   of specific cargo  condi- tions.  The  book  is one  of the  obligatory certificates  and other  particulars  on board. Amongst  other  conditions  it contains   reg- ulations  for a grain cargo.
When   these  regulations   are  fullfilled  the ship  may  load   a  grain   cargo.  The   ship then    meets   the   requirements     laid   out in the IMO  'International   Grain Code: (MSC.23(59)  ).
3.10.3c   loading the holds
If the  holds  are entirely  filled  with  more or less full, the grain will not shift. It and  also  after  settlement   (see d), remain
In practice,  the  problem   often  is that the amount   of grain  to be transported,   is not
sufficient  to fill the entire hold.Additional     bulkheads    and   'tween decks keep the grain in place.Once  the grain  is loaded,  the stowage fac- tor provided  by the shipper  may cause ad-ditional   problems   (see  section   2.9).  Theume of space it occupies.

Example:
A specific stowage  factor  is provided  for a
cargo  of grain.  During  loading  it  appears smaller   than   indicated:    the   quantity    of
grain  is delivered  as per  contract   but  the

stated volume is smaller.
 
Subsequently,   the  specified  hold  can't  be completely  filled and there  is space for the grain  to  move  with  the  afore  mentioned stability  concequences.
A  possible  solution   is to  load  additional grain.  However,  it  will  not  be  feasible  if the ship has already  reached  its maximum draft.
 
3.10.3d   The position of G after the grain has shifted
It can be seen in the drawing  that G moves parallel  to line ZlZ2   as the grain  shifts. For clarification  purposes,   the  angle  at which the grain has shifted,  as well as the position of B, en G] is exaggerated.
 Suppose  the  grain  has  shifted.  What   are the  consequences   for  stability?   Actually, the  weight  moves  from  port  to  starboard (or vice versa) as described  in the previous chapters.   The  center  of gravity  G  moves 

parallel  to the movement
a distance  equal to:
 
 
pxa
 
 
 
p   = weight  of the grain
a   = the distance  over which  the grain shifts
f1 = displacement

 GoG] =    volume x a

(m3/t)    x t         mJ     '

How   are  the  various  data  determined

calculated:
f1: the displacement   of the ship includ- ing the grain to be loaded.
-   The    ship's   management    obtains    the stowage  factor   (m3/ton    or  ft3/ton)    of the grain from the shipper
The   shipbuilder    calculates   the   maxi-'volumetric   heeling  moment'   expressed            in  rn", This
is to  be looked
up  in  the
 grain data.

the  level area of the  grain  is lowered  and
 
tance:
GOGI must  be  multiplied   with  factor  'K'
due  to  settling  of the  grain.  As it  settles,

the grain moves laterally over a greater dis-
 
K = l.00  ---+  completely  filled hold  as- suming that the center of gravity is equal to the volumetric  center of gravity
-   K = l.06  ---+ completely  filled hold with
below deck holds partially  filled.
-   K = 1.12 ---+ partially  filled hold
 
After   the  grain  has  shifted,   a  new  G is found  (G I) and the new lever of the right·
ing  arm  can  be determined    (GIZI).  This can be figured  out by reducing  the lever of
the original  arm (GoZo) with: (GoG I x cosine heeling angle). Formulated  as:
GIZI  = GOGI x cos]
 
3.10.3e   Testing the conditions
The   table  with   the  grain   data  onboard ship can quickly indicate  whether  the ship meets the requirements  for the transport of
grain.
The table maximum  allowable grain heeling
 
 moment,  is used to test the maximal  allow'

able heeling moment  against the calculated

heeling moment.
Arguments:  VCG  and displacement.
Note:   this  concerns  the  heeling  moment (mt)  of the  cargo. The  volumetric  heeling moment   (rn") must  then  be divided by the stowage factor  (m3 It).

 
 

The    above    requirements stricter than  the standard  crit
are eria
 
NB:  Ore,   in  general,   does  not shift as quickly as grain.
​However,if the top layer df one of      G
the in between  layers contains  too much moisture,  there  is a change ofshifting.
Asore is much  heavier than  grain the consequences  for the ship can be worse.

3.10.4   Effect of wind  on the ship
 
I'                 General
Wind  force:
_  causes waves  that,  depending   on  their
size have effect on the ship's motion,
_  causes (extra) heeling depending  on:
•   the ship's profile area above water
•   wind  direction  and velocity
These factors have an effect on the stability and are discussed in this chapter.
 
The   following   stability   calculations    are provided  by the IMO,  aruong others:
_  Wind  pressure coming from abeam.
_  The  wind's  effect  is determined   by the profile area above the waterline. This depends  upon:
•   the mean draft
•   if applicable  deck cargo such as containers  or project  cargo
3.10.4.a  Statical forces

\1

If the ship experiences steady wind pressure such as trade winds, it will take a small list. This list originates from the equilibrium between   the   lever  caused  by  wind   and righting  arm. (See section  3.9.8 also.)
3.10.4.b  Dynamical forces
The  rules  contain   extra  requirements   re-
garding  a rolling  ship  which  undergoes  wind gust oflonger  duration.
The gust of wind is considered  a dynamical force since the ship is unable  to absorb  the force immediately.
This  situation  has an effect on the magni-
tude  of the  (remaining)   lever of the right- ing arm and thus, on the curve (see section
3.9, 'Statical  and Dynamical').
3.10.4.(   Rolling amplitude  (¢J
The   rolling  amplitude   (¢)   is a continu- ous motion  of the ship as a result of waves around   a point   of  equilibrium   (¢).   The rolling  amplitude   is, among  other  things, dependant  upon the form of the submerged hull, the size of bilge keel(s) and GM.
 
Two examples where  it is assumed  that  ¢a is 10°:
If there  is no transverse  moment   (point of equilibrium  ¢s is 0°), the ship will roll in swells from  10° to port  to 10° to star-
board.
If transverse moment  exists and the ship
takes a static list (¢),  of about  5° to port, the ship will roll from  15° to port  to 5° to starboard  in swells.
 
The  rolling  period/time    (also determined for the extent  of ¢),  is the time needed  for a complete  rolling  from  port  to starboard
and back to port.
This  can  give an  indication   of the  initial stability  (GMo)'
3.10.4.d  Determining  maximum   heel after a wind gust (¢J
(See drawing on the following page).
The  curve  to port  is reversed  (mirror  im- age), so that  the heel ¢c can be determined simply.
 
This works as follows:
a.  The   ship  experiences  continues   trans- verse wind pressure (heel ¢s = new equi- librium).
b. The  ship rolls in swells.
c.  The  ship  reaches its most  extreme  roll- ing position  at a certain  moment.  From
¢s' the rolling  amplitude   extends  to the high side (to port  in the illustration).
d. In this  position,   the  ship  experiences  a strong gust of wind  to starboard.
e.  At the same time, the ship begins to roll
f. The  forces that  make the ship roll back to starboard  are a run-down  of the forc- es mentioned   in a. + d. + e.
g. The  ship  is unable  to  roll further  than
500 in these circumstances.
 
3.10.4.e  Explanation   (for determining  ¢J From  10° heel to port  to 0° the ship experi- ences the following forces to starboard:
ABC:   the  energy  built  up  by the  ship during  rolling from starboard  to port
BCDE:   the  statical  wind  pressure;  thelever of this arm is 0.08 meters
DEFG:    wind   pressure   from   gust  of
wind
The  lever of this arm is 0.08 + (0.08 x
50%) = 0.12 meters
 
From  0°, the  rolling  to  starboard   experi- ences counter-pressure   from the excesswa-      F
ter pressure  (the  area below  the  curve) to
starboard.
D
Subsequently:0.08 meters

at 12° (¢)  the lever of the statical wind
pressure  equals  the  righting  arm of the
ship
at 15° the combination   of both wind le-  B
vers (BD  + DF)  equal the righting  arlll
(KH)
_  after KH,  both  wind levers (BD + DF)
are smaller than  the righting  arm of the     A
ship
_  KLM   shows   the   remaining,    the  dif-
ference  between   the  area HKMN   annHKLN.
staticawl  ind lever, for example:
In short, triangle AFK indicates  the eneray 
b

Il'lhlakinatnI e shiIp to roll to starboard.
e area that  the remaining  righting  arms raetf'iect' must b e surLnCc:'ient  1y large to contain
[[anglewith  the size of AFK.
wind lever after gust of wind:

0.08 m + (0.08 x 50%) = 0.12 meters 
In this example, KLM  (energy directed  to port)  is equal to AFK.  The  farthest  line of triangle  MN  cannot  pass beyond  the  50° angle. The ship then satisfies the wind con- ditions.
If the ship accomplishes  this before  50° is
reached,  then  this angle is considered   the
maximum.

3.10.5   Determining  a list caused  by shifting moment.  Summary
3.10.S.a.2 Larger lists (<»>SO)                                                                                                   .                 . Scribanti's  formula  can be used from  10° to l S", (3.10.3)  This formula  cannot  be used With larger lists.
To find the list for larzer shifting  moments,  a graphic solution  is needed.
The  stability  curve is the accumulation   oflevers  of the righting  moment  (GZ).  In the line from 0° to 60°:                    .  ,.        .
To draw the lever of the listing moment  in the curve, the angle that the ship takes is found  at the intersection  of the ship s list, influenced
by an asymmetric  distribution   of weight or a list moment  caused by an external forc~.    .    .,                     .
A shift moment  decreases with the cosine of the heeling angle through  an asymmernc  distribution   of weight.
NB: list moments  to starboard  move to the top; in list moments  to port,  they move below.
3.9.S.a.l  A small list (<»<SO)
To calculate  a small list, the same formula 
asymmetric  cargo, heavy load

3.10.5.c  Heavy load
The  list caused by a heavy load placed  on one  side or  a load  still hanging  from  the

can be applied  as for  the  inclination   test However,    not  to  calculate  GMQ,   but  in- stead to figure out the expected  list caused by the shifting  moment.  Similar small lists are  normally   neutralized   by  counter   bal-
last.
 
Nevertheless,  the formula  is:
tan cp =

Herewith:  p x a = list moment
 
 
pxa
And     --=GG1
/:,.
 
This  can even be calculated  without   a cal- culator.  For small angles,  tan<» = sin<»= ¢ radials apply.
The  formula  can then be easily applied in:
pxaxS7",3                                                   <»=          /:"xGM
Note:  An  asymmetrically   loaded   cargo  not  only
causes a lists, but also has a limited  range of
stability.

3.10. 5.b Determining  the list caused by the shifting of grain
The  shifting  lever is calculated  by dividing
pxa
/:,.xGMQ


 
or  <»=arctan






3.10. 5.b Determining  the list caused by the shifting of grain
The  shifting  lever is calculated  by dividing the  volume   of  shifting   moment   (VKM)      GZ
by the product   of stowage  factor  (sf)  and displacement   (/:"). That  brings the shifting lever to a heeling angle of zero degrees.
 
 
VKM Formulated  "Yo ITrain    =
sfx  /:,.
The lever 40° is multiplied  by 0.8, the value of cosine 40°.
The  curve "Yocoscp     can now be drawn in the


 
Formulated  "Y40-grain        = 0.8 x "YO-grain
 
A straight  line is drawn  through  the tip of both   levers. The  intersection   of  this  line with  the  curve  must  then  be  lower  than
12°.

3.10.5.c  Heavy load
The  list caused by a heavy load placed  on one  side or  a load  still hanging  from  the
The heeling lever is then the product  of the cargo's weight  (p)  and  the horizontal   dis- tance  between  G and  the cargo's center  of gravity (a) divided by the displacement  (/:,.)
pxa
Formulated  "Yo  =--The  curve "Yocoscp     can now be drawn in the
stability  curve.
Or  in simplified  terms, just  as with  grain, the lever at 40° list, namely, 0.8"yo-
The point  of intersection  indicates  the list.

Cargo of large ships

3.10. 5.d 'External  forces
The  law is based  on  the  assumption   that the  heeling  force  in  the  entire  range  re- mains steady during  the wind  moment  for normal  ships. This  is done  to simplify  the calculation   and  provides  extra  safety  (see
With   sailing  vessels,  the  wind   lever  de- creases with cos2cp     (see section  6.1).
The  curve "YOCOS2cp  is then  drawn  in the sta- bility curve in order to determine  the angle with  a specific wind  force given  a certain sail area. With  listing forces that  tug boats, can exert on a ship, the lever of the listing moment  decreases again with coscp.
3.11 Various topics
 
3.11.1  KG max
The  form, values and range of the stability curve are determined   by formula:
GZ(O° -  60°) = KNsinY"-  KGsinY", whereby the values for KNsinY'"just as those of KM,  are determined   by the form  of the hulL They  are given for the mean draft and trim or displacement  in sea water.
 
For a specific displacement,  the maximum KG can be verified  so that  the conditions pertaining   to stability  are satisfied. Particularly  for ships having no loading and
stability  computer  onboard,  the maximum possible KG is calculated  for each displace- ment. These are then checked against all requirements.
Small drafts  are indicated  at the top of the curve  (preferably  Y"> 30°, but  not  smaller than  25°).

KM  is large for small drafts,  but  the bilge

will rise quickly  out  of the water  and  the

top will be easily reached.
The  wind  requirement   plays a larger  role with smaller drafts than with larger drafts.
If the draft  increases, the minimal  GM (>O.ISm)  or the area under  the curve will playa   role.  To  determine   which  require-KG  can be calculated  for  each draft  (dis- placement).  These are placed on a chart for small ships.
The  displacement  is indicated  on the hori- zontal  axis and  the  KGmax,  on  the verti- cal  axis. Whether    the  ship  meets  all  the requirements,   can now be seen at a glance. However,  it does  not  indicate  why a ship doesn't  meet the requirements.
The  same calculations  are also provided  in tables with accompanying  stability  data.
For  each  10  ern  difference   in  draft,   themaximum   possible   KG  is  indicated    for each stability  requirement.
Thus  for the:
-    maximum  GZ at Y">30°(req. 1) position   of the top of the curve (req. 2) area under  the curve up to 30° (req. 3) increase from 30° to 40° (req. 4)
area from 0° -  40° (req. 5)
minimal  GMo   (req. 6)
wind condition   (req. 7).
 
The   last  column   (Max.  VCG)   indicates which  condition   is representative
(always  the  condition    with   the  smallest
KG).
​There  may be another  column with further trim  details  and also the maximal  value of KG  in  relation  to  stability  in  the  case of damage.
 
The   tables  show  why  the  ship  does  not meet the requirements.
 
Neither  of the methods,  however, provides information   or an indication  about  statical or dynamical  range of stability or the actual position  of the top  of the curve, as shown in the stability  curve.
 
3.11.2  Calculation  of the area under  the curve
The  area under  the stability  curve, representing  the energy a ship can exert against a mo-
ment  caused by a list, can be represented  as 2:GZ¢, whereby ¢ is reported  in radian.
 
As long as the curve runs evenly, the calculation  of the area can be estimated  by regarding the area 0- 30° as a triangle  and the area 30-40°, a trapezium
 
Area 0°_30° is then:  0.5 x  (300/5r,3)    x  GZ30°  mrad = 0.262  x  GZ30° mrad
 
Area 30°-40° is then:  0.5(10°/57°,3)   x (GZ30°  + Gz400) mrad =
0.873  x (GZ30°  + GZ400) mrad
 
For a more  accurate  calculation  of the area, especially if the curve is less regular, such as ships with  a low freeboard  and  a high  coaming,  Simpson's  Rules can be used assuming that  the curve  is of the second  degree. The  substantiation   of this will not  be addressed
here. The  resulting area is a reasonably  accurate estimate.
 
The  practical  application  is (see figure):
The  curve is divided  in a number  of equal parts (h) on the X axis.
In this case, 4, resulting  in an uneven number  of vertical ordinates  (Yo-Y 4)'
According  to Simpson's  Isr rule, the area is now
V3x h  x  (Yo + 4 *y 1 + 2 x Y 2 + 4 x Y 3 + Y 4)
The  calculation  of the area under  the curve using the  1st rule of Simpson  works  as fol- lows:
The  area from  0°_30°, 0°-40° and the increase from  30°_40° must  be calculated  There  is
one problem  in calculating  the area 0-30°. Because this part of the curve cannot  be divided in an equal number  of parts  on the X axis, (from which  the GZ value is determine an
intermediate   step is used.
 
First, the area 0°-10° has to be calculated  and this part  of the curve regarded  as a triangle
 
(area shaded  in blue).
Next, the area from  10°-30° (in red) has to be calculated;  together,  they indicate  the area
 
0°_30°.
 
The  area from 0°-40° can be directly figured using Simpson's  lsr Rule.
To reduce the area from 0°_30°, an increase from 30°-40° is obtained  (in green)
​10° is equal to 10°/5r,3  = 0.175 radials
 
The  calculation  now includes the following steps: Area 0° _10° = 0.5 x 0.175  x GZ1  = A
Area 10° _ 30° = V3x 0.175  x (GZ1 0° + 4*GZ20° + GZ300) = B
Area 0° - 30° = A + B
Area 0° _40° = V3x 0.175 x  (GZO° + 4 x GZlO° + 2 x GZ20° + 4 x GZ30° + GZ400) = C
Increase 30° - 40° = C -  (A + B)
3.11.3  Mathematical   explanation  of  the stability   curve
The  formula  to calculate  stability  levers is
GZ = KNsin)1>- KGsin)1>
 
KGsin)1>has a pure sine form and is 0 by 0°
and a maximum  of 90°.
 
KNsin)1>however, has no pure sine form be- cause KN is a continually  changing value. The  form  of the  curve  then  is also solely determined   by KNsin)1>.
 
The  irregularities  are caused  by the  inter- section  of the waterline  and hull  at differ-
ent angles )1>.
 
The  start of the curve (origin to ca. 3°)
For the range from 0° to ca. 3° the calcula- tion GZ = GM  sin()1»is used.
 
This  formula   can  also  be  represented   by
GZ=GMx)1>
with   the  angle  )1>in  radian   equal  to  the
first straight  (linear)  part  of the GZ curve. Thus,  the  intersecting   angle  of the  curve with  the  source  can  also be  found  by ex- tending  GMo  at 57°,3.
 
Scribanti's  formula
From ca. 3°, the GZ curve is non-linear  and the form  can be reasonably  well described
by Scribanti  's Formula.
GZ = (GM  + 0.5 BM tan2()1») sin()1».
 
Scribanti's formula applies to ships with straight   side walls, so far the  deck  is not submerged  nor the bilge rise out of the wa-
ter.
Scribanti  is useful until about  a 10° heeling angle  for  normal  ships  with  straight  side walls. Beyond  that,  so many  irregularities play  a role,  that  a straightforward    math- ematical equation  is no longer possible. For ships with pronounced   outward  or inward thrusting  frames, Scribanti's  formula  is not recommended   as the  results  are not  accu-
rate enough.
 
Explanation  of Scribanti's  formula
 
 
Basis
_  ship has vertical side walls
the deck edge is not submerged bilge does not rise out of waterFrom the above figure of a.ship  with a heeling angle ¢,
GZ<I! = GZ~   + Z~ZcjJ
 
 
making use of the displacement  rule, the following applies:
 

f

L
12 y2 tano 23  y tanc dx
_ Vu 23  Y tan<p          o
Zcp                                V


f

12 tan2<p      23
o


y3 dx

Filling in the above,                                   GZ$ = GMsin<j)    +Z~  sin  ]
 
 
 
 
results in Scribantis     formula
 
 
 
 
 
 
Application  of Scribanti:
Suppose a ship has negative initial  stability, like the lever of statical stability  below.

How large is the heeling angle (cp)  if
GZ=O?
 
 
 
GZ" = (GM + ~BMtan2<j) )sin<j)   = 0 sino  =0     v   GM + ~BMtan2<j) = 0
 
According  to Scribanti,  GZ = 0:
 
 
 
 
 
GZ = 0 if:                                                                      ,I
 
<j)= 0      or
 
 
From which:
The   first  solution   applies  to  the  upright ship  and  the  second,  to  the  ship  with  a heeling angle or loll.
 
Sample calculation:
 
 
 
<ilL    _-   arctan ~-2-G-- M   = ~-2X(-0.1) = M-.2   = 0.2 radialen
BM                 4.8              4.8
<ilL    = 0.2 radialen x 57.3 graden/radiaal  = 11.5 graden
Given GM  = -0.1 m and BM = 4.8 m
For a ship with assigned GM and M values,
the ship  appears  to be unstable  (to GM  <
0). In flat water,  the ship will not  capsize, but will turn  on its longitudinal   axis until it  reaches  a fixed  heeling  angle  equal  to
11°.5.

How  high  is initial  instability  then  at this
statical heeling angle?
Initial  stability  in this situation  can be de- termined  using the following data.
​The   tangent   to  the  GZ   curve  at  point GZ=O creates an angle with the axis whose tangent  is equal  to  the value of initial  in- stability.
This is made visible in the figure as GML•

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