INTACT STABILITY INTRODUCTION A rigid body is said to be in a state of equilibrium when the resultant of all forces and moments acting on the body are zero. In case of a floating ship in static condition, all vertical forces (all components of lifting forces – buoyancy etc. and weight) must cancel each other and should act in the same vertical line so that no moment acts on the ship. Even if a ship is in forward motion at steady speed, horizontal forces cancel each other (thrust and resistance) and the body is said to be in equilibrium. Let us suppose an external force or moment is applied on a rigid body displacing it from its original position of equilibrium. If the body comes back to its original position of equilibrium after the external force/ moment is removed, the body is said to be in stable equilibrium. If, on the other hand, the body stays in the altered position even after the force/ moment is removed, the body is said to be in neutral equilibrium. A third possibility is that the body may continue to move away from its original position of equilibrium even after the external force/ moment is removed. The original equilibrium of the body is then said to be unstable equilibrium. A ship is a floating body in equilibrium in static condition or in steady forward motion. It can and does experience momentary forces/ moments due to various reasons such as a gust of wind athwartship, due to steady turning or due to passenger movement on deck to one side. Due to these forces, the vessel does move away from its original position of equilibrium and takes an angular displacement in the transverse vertical plane (heel or list) and/ or longitudinal vertical plane (trim). Once the external force is removed, the vessel should come back to its original position if it is in stable equilibrium. It should stay in the displaced position of some heel and trim in neutral equilibrium. If the vessel is in unstable equilibrium, it should exhibit increased heel and trim after the force/ moment is removed ultimately causing the vessel to capsize. Fig. 1(a) shows a floating vessel in static equilibrium condition without any heel where the centre of gravity and centre of buoyancy are in the same vertical. However, when the vessel is disturbed from its original position
due to an external force/ moment, the under water geometry changes, thus changing the position of centre of buoyancy from B to B1. Fig. 1(b) shows the position of B1 indicating stable equilibrium when a righting moment acts on the ship uprighting the vessel. Fig. 1(c), on the other hand, shows a positions of B1 indicating unstable equilibrium when a heeling moment acts on the ship taking it further away from its original position of equilibrium. Fig. 2 shows the case of neutral equilibrium where, due to shape of the under water body, the position of B1 is in the same vertical as G and therefore no righting moment or heeling moment acts on the vessel. The vessel stays in the new position in equilibrium. Stability related to motion in the transverse plane, i.e., related to heel or list, is called transverse stability and that in the longitudinal plane is termed longitudinal stability. Normally a vessel experiences some amount of heel and trim simultaneously since forces/ moments generally act both in transverse and longitudinal direction. However, it is common to treat transverse stability and longitudinal stability separately to study their effect on ship under review. External forces and moments can act on the ship in a steady or unsteady manner causing the vessel to have translatory and rotational velocity and acceleration. Examples are wave encounter at sea where there could be a steady drift force in addition to an oscillatory wave force, wind forces acting on the ship, turning of the vessel, collision and grounding, heeling moment due to grain shifting etc. In this note, the dynamic effects on the vessel are ignored and the stability of the vessel in a dynamic environment is considered assuming the vessel to be statically poised.
(a) Equilibrium in static position
(b) Stable
(c) Unstable
Fig. 1 Equilibrium of a Floating Body
(a)
(b)
Fig. 2 Neutral Equilibrium of a Floating Body
STABILITY AT SMALL ANGLES Transverse Metacentric Height Consider a symmetric ship heeled to a very small angle, δ φ shown, with the angle exaggerated, in Fig. 3. The centre of buoyancy has moved off the ship’s centreline as the result of the inclination, and the lines along which the resultants of weight and buoyancy act are separated by a distance GZ , the righting arm. A vertical line through the centre of buoyancy will intersect the original vertical through the centre of buoyancy, which is in the ship’s centreline plane, at a point M , called the transverse metacentre, when δ φ → 0 . The location of this point will vary with the ship’s displacement and trim, but, for any given draft, it will always be in the same place. δφ
SHIP
Fig. 3 Metacentre and Righting Arm
Unless there is an abrupt change in the shape of the ship in the vicinity of the waterline, point M will remain practically stationary with respect to the ship as the ship is inclined to small angles, up to about 7 or, sometimes, 10 deg. As can be seen from Fig. 3, if the location of G and M are known, the righting arm for small angles of heel can be calculated readily, with sufficient accuracy for all practical purposes, by the formula GZ GM sin δφ
The distance GM is therefore important as an index of transverse stability at small angles of heel and it is called the transverse metacentric height.
Since GZ is
considered positive when the moment of weight and buoyancy tends to rotate the ship toward the upright position, GM is positive when M is above G , and negative when M is below G . Metacentric Height
(GM )
is often used as an index of stability when
preparation of stability curves for large angles has not been made. Its use is based on the assumption that adequate GM , in conjunction with adequate freeboard, will assure that adequate righting moments will exist at both small and large angles of heel. When a ship is inclined to a small angle, as in Fig. 4, the new waterline will intersect the original waterline at the ship’s centreline plane if the ship is wall-sided in the vicinity of the waterline, since the volumes of the two wedges between the two waterlines will then be equal, and there will be no change in displacement. If v is the volume of each wedge, ∇ the volume of displacement, and the centres of gravity of the wedges are at g1 and g 2 , the ship’s centre of buoyancy will move: (a)
in a direction parallel to a line connecting g1 and g 2
(b)
a distance, BB1 , equal to v . g1 g 2
(
)
∇.
As the angle of heel approaches zero, the line g1 g 2 , and therefore BB1 , become perpendicular to the ship’s centreline. Also, any variation from wallsidedness becomes negligible, and we may say BM =
BB1 v . g1 g 2 = tan δ φ ∇ tan δ φ
If y is the half-breadth of the waterline at any point of the ship’s length at a distance
x from one end, and if the ship’s length is designated as L , then, since the area of section through the wedge is 2×
1 ( y ) ( y tan δ φ ) and its centroid is at a distance of 2
2 y from the centroid of the corresponding section on the other side 3
1 2 ( y ) ( y tan δ φ ) ⎛⎜ 2. 2 ⎝ 3 0
L
v . g1 g 2 = ∫
⎞ y⎟d x ⎠
or
v . g1 g 2 2 = tan δφ 3
L
∫y
3
dx
0
L
1 But, ∫ y 3 d x is the moment of inertia of a figure bounded by a curve and a straight 30
line with the straight line as the axis about which moment of inertia is computed. If we consider the straight line to be the ship’s centreline, then the transverse moment of inertia of the entire water plane (both sides of C.L.) about the ship’s centreline (about the longitudinal axis) designated as IT , is IT =
2 3
L
∫y
3
dx =
0
v . g1 g 2 tan δ φ
and, therefore, when δ φ → 0 , BM =
IT ∇
The calculation of the height of the transverse metacentre above the keel, usually called K M , is a part of standard hydrostatic calculation for ships. This distance is the sum of BM , or IT ∇ , and K B , the height of the centre of buoyancy above the keel. The height of the centre of gravity above the keel, K G is found from the weight estimate or inclining experiment. Then, GM = K M − KG = K B + BM − KG
Some geometric properties including moment of inertia for geometric shapes are given in Table 1.The righting moment at small angles of heel can be written as Δ. GZ = W.GMT. SinΦ For small angles of heel, say less than 70, max. upto 100), the righting moment reduces to Δ. GZ = W.GMT. Φ
δφ
δφ
Fig. 4 Locating the Transverse Metacentre
Thus if angle of heel is known and displacement or W is known, the transverse metacentric height can be determined as
GMT = Righting moment / Φ
where Φ is in radians. This principle is used during inclining experiments when the ship is heeled to an angle Φ by a known heeling moment (= righting moment), by moving a known weight w through atransverse distance d causing a heeling moment w.d. Then, GMT = w.d / Φ Needless to say that GMT should be positive or more than zero for the vessel to have positive initial stability. But having a very high GMT is also not attractive since it would generate a small rolling period leading to faster rotational motion. This is also not a desirable feature.
Table 1. Useful Formulae for geometrically definable Bodies 1st moment = Ax = ∫ ax
2nd moment = I = Ak 2 = ∫ ax 2 For moment of inertia about any axis oo at a distance h from centre of area-I 0 = I G + Ah 2
Moment of inertia of rectangle about base =
bd 3 3
Moment of inertia of rectangle about centre-line = Moment of inertia of circle about diameter =
bd 3 12
πd4
64 BH 3 Moment of inertia of triangle about base = 12 Moment of inertia of triangle about an axis through centre of area =
Distance of centre of area of semi-circle from diameter =
BH 3 36
4r 3π
Distance of centre of area of semi-circle arc from diameter =
2r
π
Distance of centre of area of surface of a hemisphere from diameter =
r 2
4 3 πr 3 1 Volume of paraboloid = × volume of circumscribing cylinder 2 1 Volume of cone = × volume of circumscribing cylinder 3 4 Surface area of sphere = π r 2 3
Volume of sphere =
Longitudinal Metacentric Height The longitudinal metacentre is similar to the transverse metacentre except that it involves longitudinal inclinations. Since ships are usually not symmetrical forward and aft, the centre of buoyancy at various even-keel waterlines does not always lie in a fixed transverse plane, but may move forward and aft with changes in draft. For a given even-keel waterline, the longitudinal metacentre is defined as the intersection of a vertical line through the centre of buoyancy in the even-keel attitude with a vertical line through the new position of the centre of buoyancy after the ship has been inclined longitudinally through a small angle. The longitudinal metacentre, like the transverse metacentre, is substantially fixed with respect to the ship for moderate angles of inclination if there is no abrupt change in the shape of the ship in the vicinity of the waterline, and its distance above the ship’s centre of gravity, or the longitudinal metacentric height, is an index of the ship’s resistance to changes in trim. For a normal surface ship, the longitudinal metacentre is always far above the centre of gravity, and the longitudinal metacentric height is always positive. Locating the longitudinal metacentre is similar to, but somewhat more complicated than locating the transverse metacentre. Since the hull form is usually not symmetrical in the fore and aft direction, the immersed wedge and the emerged wedge usually do not have the same shape. To maintain the same displacement, however, they must have the same volume. Fig. 5 shows a ship inclined longitudinally from an even-keel water line W L , through a small angle, δθ
to
waterline W1 L1 . Using the intersection of these two waterlines, point F , as the reference for fore-and-aft distances, and letting: L
=
length of water plane
Q
=
distance from F to the forward end of water plane
y
=
breadth of waterline W L at any distance x from F
the volume of the forward wedge is Q
v=∫ 0
( y ) ( x tan δθ )
dx
and the volume of the after wedge is L−Q
∫ ( y ) ( x tan δθ ) d x
v =
0
Equating the volumes
Q
L−Q
0
0
tan d θ ∫ x y dx = tan δθ Q
∫ x y dx =
or,
0
∫
xydx
L−Q
∫
xyd x
0
These expressions show respectively, the moment of the area of the water plane forward of F and the moment of the area aft of F , both moments being about a transverse line through point F . Since these moments are equal and opposite, the moment of the entire water plane about a transverse axis through F is zero, and therefore F lies on the transverse axis through the centroid of the water plane, called the centre of flotation. In Fig. 5 AB is a transverse vertical plane through the initial position of the centre of buoyancy, B , when the ship was floating on the even-keel waterline, W L . With longitudinal inclination, B will move parallel to g1 g 2 , or as the inclination approaches zero, perpendicular to plane AB , to a point B1 . The height of the metacenter above B will be BM L =
BB1 v . g1 g 2 = tan δθ ∇ tan δ θ
The distance of g1 , the centroid of the after wedge, from F is equal to the moment of the after wedge about F divided by the volume of the wedge, and a similar formula applies to the forward wedge. If the moments of the after and forward wedges are designated as m1 and m2 , respectively, then the distance
g1 g 2 = Or
m1 m2 m1 + m2 + = v v v
v . g1 g 2 = m1 + m2
The moments of the volumes are obtained by integrating, forward and aft, the product of the section area at a distance x from F and the distance x , or
Q
m1 = ∫ ( y ) ( x tan δθ ) ( x ) dx = tan δθ 0
Q
∫x
2
ydx
0
L−Q
m2 = tan δθ
∫
x2 y d x
0
The integrals in the expressions for m1 and m2 correspond to the formula for the moment of inertia of an area about the axis corresponding to x = 0 , or, in this case, a transverse axis through F , the centroid of the water plane. Therefore, the sum of the two integrals is the longitudinal moment of inertia, I L of the entire water plane, and
m1 + m2 = v . g1 g 2 = I L tan δθ Or,
Then, when δθ → 0
IL =
BM L =
v . g1 g 2 tan δθ
v . g1 g 2 I = L ∇ tan δθ ∇
where I L is the moment of inertia of the entire water plane about a transverse axis through its centroid, or centre of floatation.
δθ
Fig. 5 Longitudinal Metacentre
Stability of Submerged Bodies When a submarine is submerged, the centre of buoyancy is stationary with respect to the ship at any inclination. It follows that the vertical through the centre of buoyancy in the upright position will intersect the vertical through the centre of buoyancy in any inclined position at the centre of buoyancy, and the centre of buoyancy is, therefore, both the transverse and longitudinal metacenter. To look at the situation from a different viewpoint, the K M of a surfaced submarine is equal to K B plus BM , or K B plus I ∇ . As the ship submerges, the water plane disappears, and the value of I , and hence BM is reduced to zero. The value of K M becomes K B plus zero, and B and M coincide. The metacentric height of a submerged submarine is usually called G B rather than GM . Thus the height of centre of gravity, KG, is of vital importance to transverse stability for floating and submerged bodies. See Fig. 6.
POSITIVE STABILITY (a) RIGHTING MOMENT WHEN HEELED
POSITIVE STABILITY
(c) RIGHTING MOMENT WHEN HEELED
NEGATIVE STABILITY (b) HEELING MOMENT WHEN HEELED
SURFACE SHIP
NEGATIVE STABILITY (d) HEELING MOMENT WHEN HEELED
Fig. 6 Effect of Height of Centre of Gravity
SUBMERGED SUBMARINE
Effect of Trim on Metacentric Height The discussion and formulas for BM , K M and GM all assumed that the waterline at each station was the same, namely, no trim existed. In cases, where substantial trim exists or when there is substantial change in water plane shape at normal trim, values for BM , K M and GM will be substantially different from those calculated for the zero trim situations. It is important to calculate metacentric values for trim for many ship types. It may be difficult to do this in the manual or semi-automatic processes of calculation. But if the calculations are carried out using a high speed computer, the exact stability parameters for various trim conditions can be estimated.
Effect of Heel on Metacentre Metacentre is essentially the point of intersection of the vertical line through centre of gravity G and the vertical through centre of buoyancy in the new position at a small angle of inclination δΦ. In Fig. 7, M1 is the metacntre in upright condition, B1M1 being the perpendicular to waterline W1L1 at a small inclination δΦ. As the B
angle of inclination Φ increases the water plane area and the transverse moment of inertia IT increases and so does the metacntric radius, volume of displacement being constant. In Fig. 7 the waterline W2L2 is at an angle Φ and W3L3 at a small further inclination δΦ, giving a new metacentre at M2. But at a certain angle of inclination, the water plane suddenly reduces either due to deck edge immersion or keel emergence. The metacentric radius BM reduces. Fig. 7 shows the nature of the locus of M with angle of heel.
φ
δθ
Fig. 7 Change of Metacentre with Heel Angle
Moment to change Trim 1cm (MCT 1cm) Moment to trim one degree = Δ GM L sin (1deg) , where GM L is the longitudinal metacentric height. We are more interested, however, in the changes in draft produced by a longitudinal moment than in the angle of trim. The expression is converted to moment to change trim one cm by substituting one cm divided by the length of the ship in cm for sin (1 deg). The formula becomes, with mass units, MCT 1cm =
Δ GM L t −m 100 L
where L is ship length in meters As a practical matter, GM L is usually so large compared to G B that only a negligible error would be introduced if BM L were substituted for GM L . Then
IL may be substituted for BM L , where I L is the moment ∇
of inertia of the water plane about a transverse axis through its centroid, and Δ = ρ ∇ , where ρ is density. Then, moment to trim one cm:
M CT 1cm = ρ ∇ ×
IL ρ IL 1 × = ∇ 100 L 100 L
For fresh water ρ = 1.0 ; for salt water ρ = 1.025 ( t m3 ) . Since the value of this function depends only on the size and shape of the water plane, it is usually calculated together with the displacement and other curves, before the location of G is known. Although approximate, this expression is used for calculations involving moderate trim with satisfactory accuracy.
Rolling Period The period of roll in still water, if not influenced by damping effects, is
Period =
const. × k GM
=
C×B GM
where k is the radius of gyration of the ship about a fore-and-aft axis through its centre of gravity.
The factor “ const. × k ” is often replaced by C × B , where C is a constant obtained from observed data for different types of ships. The variation of the value of C for ships of different types is not large; a reasonably close estimate can be made if 0.80 is used for surface types and 0.67 is used for submarines. In almost all cases, values of C
for conventional,
homogeneously loaded surface ships are between 0.72 and 0.91. This formula is useful also for estimating GM when the period of roll has been observed. The case of the ore carrier is an interesting illustration of the effect of weight distribution on the radius of gyration and therefore on the value of C . The weight of the ore, which is several times that of the light ship, is concentrated fairly close to the centre of gravity, both vertically and transversely. When the ship is in ballast, the ballast water is carried in wing tanks at a considerable distance outboard of the centre of gravity, and the radius of gyration is greater than that for the loaded condition. This can result in a variation in the value of C from 0.69 for a particular ship in the loaded condition to 0.94 when the ship is in ballast. For most ships, however, there is only a minor change in the radius of gyration with the usual changes in loading.
Floating Multi-body Systems If the water plane can be located away from its centroid, the moment of inertia of the water plane, IT can be large giving a large metacentric radius BMT and so, a large GMT. This is possible if the water plane was distributed around the centroid. Examples of such cases are catamaran, trimaran or pentamaran vessels, SWATH vessels, Semi-submersible vessels. The water planes in such vessels can be small giving advantages with regard to motion in waves along with high degree of stability. This facilitates large deck loading without compromising on metacentric height requirement.
STABILITY AT LARGE ANGLES At large angles of inclination, righting moment is still Δ. GZ, but GZ is no more equal to GMT.sinΦ. For a wall sided vessel, GZ can be derived from geometric considerations as follows.
The Wall Sided Formula
φ ϕ
Fig. 8 The Wall-Sided Formula
A ship is said to be wall-sided if, for the angles of inclination to be considered, those portions of the outer bottom covered or uncovered by the moving water plane are vertical with the ship upright. No practical ships are truly wall-sided, but many may be regarded as such for small angles of inclination ─ perhaps up to about 10 degrees. Referring to Fig. 8 let the ship be inclined from its initial waterline W L to a new waterline W1 L1 by being heeled through a small angle φ . Since the vessel is wall-sided, W L and W1 L1 must intersect on the centre line. The volume transferred in an elemental wedge of length δ L where the beam is b is
⎛1 ⎝2
δ L⎜ ×
2 ⎞ b b ⎞ ⎛b × tan φ ⎟ = ⎜ tan φ ⎟ δ L 2 2 ⎠ ⎝ 8 ⎠
Moment of transfer of volume for this wedge in a direction parallel to W L =
b2 2b tan φ δ L 8 3
Hence, for the whole ship, moment of transfer of volume is
∫
L
0
b3 tan φ d L 12
Hence, horizontal component of shift of B , BB ′ is given by 1 L b3 tan φ d L ∇ ∫0 12 I = tan φ = BM tan φ ∇
BB ′ =
Similarly, vertical shift 1 L b2 1 tan φ b tan φ d L ∫ 0 8 3 ∇ BM I tan 2 φ = tan 2 φ = 2∇ 2
B ′ B ′′ =
By projection on to a plane parallel to W1 L1
GZ = BB ′ cos φ + B ′ B ′′ sin φ − BG sin φ ⎡ ⎤ tan 2 φ sin φ ⎥ − BG sin φ = BM ⎢sin φ + 2 ⎣ ⎦ ⎡ ⎤ BM tan 2 φ ⎥ = sin φ ⎢ BM − BG + 2 ⎣ ⎦ i.e. ⎡ ⎤ BM GZ = sin φ ⎢GM + tan 2 φ ⎥ 2 ⎣ ⎦
For a given ship if GM and BM are known, or can be calculated, GZ can readily be calculated using this formula. Cross Curves of Stability Ships are generally not wall sided and therefore the above formulation is rarely used in practice except in cases of rectangular barges. That application is also limited to angles within deck edge immersion or keel coming out of water. To determine the moment of weight and buoyancy tending to restore the ship to the upright position at large angles of heel, it is necessary to know the perpendicular distance from the centre of gravity, through which the weight force W
acts downwards, to the vertical line through the centre of buoyancy – shown as line AD in Fig. 9 – through which equal upward force buoyancy acts. This distance, GZ, is referred as the statical stability lever or the righting arm. It is difficult to determine the GZ value for a vessel at any operating displacement at any angle of heel whenever required. Therefore it would be convenient if GZ was available for various displacements and angle of heel for use when ever required. But G varies with loading condition and is not a predetermined value. Therefore this distance could be calculated from some standard reference point on ship centre line plotted as a set of curves, one for each angle of heel varying with displacement. Such a set of curves is generally referred as Cross curves of stability. In Fig. 9 one could take as standard reference, the point O or the point at intersection of keel and ship centre line, K. Then, GZ = a – OG.sinΦ Or GZ = KN – KG.sinΦ
Commonly, KN curves are referred as cross curves of stability. GZ consists of two parts: KN or ‘a’ as in Fig. 9 and the height of G above base or KG. The first part can be calculated without any reference to weight distribution if the geometry of the ship form is known. This part of stability is normally referred as Form Stability. The second part is known as Weight Stability. Fig. 10 shows a typical diagram of cross curves of stability for a Tug Boat.
φ
Ο
φ
O
Fig. 9 Transverse Righting Arm
3
2.7
40° KN
2.4
50° KN 60° KN 70° KN
30° KN
80° KN
KN m
2.1
90° KN
20° KN
1.8
1.5
1.2 10° KN 0.9
0.6 10
30
50
70
90 110 130 Displacement Tonne
150
170
190
210
Fig. 10 Cross Curves of Stability (KN) for a typical Tug Boat KN calculation essentially involves estimation of volume of displacement and the three co-ordinates of centre of buoyancy – LCB, VCB and TCB. Assuming there
is no trim in any heeled condition; LCB need not be calculated since it will not alter with heel. If an inclined water line at an angle Φ is drawn on the body plan of a ship, at each section, sectional area and its moments about base and centre line can be estimated. Then these values at each section can be integrated along the length of the ship to get the total volume of displacement and the VCB and TCB. In earlier days, this could be done by use of an instrument called the Integraph which could give the area and moment about a fixed axis by moving a cursor along the sectional outline. Now a days, this can easily be done using standard CAD software packages such as AUTOCAD. Ship form is a complex three dimensional shape, not amenable to mathematical representation. This problem becomes more complex in this case since, with angle of inclination increasing, the deck may be immersed which has also to be modelled along with ship surface. Further, if some structures, such as forecastle, hatch coamings, deck houses etc., above the upper deck are also watertight, they should be taken into account for calculation of KN since these contribute to increased stability. If it is possible to represent the entire water tight enclosure (including all the above) in numerical form as offsets at very closely spaced intervals as y = f(x,z) and as a single object, it would be possible to generate another object numerically similar in nature, by intersecting the ship object with a two-dimensional plane surface. Then the immersed body area and moments about base and ship centre plane can easily be estimated using simple trapezoidal rule or straight line integration method. If the defined offsets are sufficiently close, the numerical error in this method should be well below acceptable limits. Most of the standard software packages use this method for calculating cross curves of stability. Since ships are not symmetrical in fore and aft directions, with heeling the LCB is likely to shift from its original position. This imposes a trimming moment on the ship. In this altered position the LCB should be in the same vertical as the original LCB (or, LCG). So our earlier assumption that there should be no trim is essentially erroneous. While doing numerical computation using a computer, it is convenient to take this into account and calculate the KN values at various angles of heel at the equilibrium trim condition known as ‘free trim’ condition. This can be done be a numerical trial and error method.
The KN values calculated at initial no-trim condition will change if there is an initial trim. The initial trim would depend on loading condition. It may be necessary to estimate cross curves of stability corresponding to various initial trimmed water lines. This estimation becomes easy and accurate if computer based numerical computation is carried out. Statical Stability Lever GZ The statical stability curve is a plot of righting arm or righting moment against angle of heel for a given condition of loading. For any ship, the shape of this curve will vary with the displacement, the vertical and transverse position of the centre of gravity, the trim, and the effect of free liquids. These values of the righting arm, plotted against angle of heel, form the statical stability curve, shown in Fig. 11 This figure illustrates the general case, in which the centre of gravity is not on the ship’s centreline (creating an initial list), rather than the usual case, in which the centre of gravity is on, or very near, the centreline. Fig. 12 Is the statical stability curve for the same ship, at the same displacement, with the same K G and free liquids, but with the centre of gravity on the ship’s centre line. The curve of righting arms may be converted to a curve of righting moments by multiplying the ordinates by the ship’s displacement. The righting-arm curve may
ANGLE OF INCLINATION
Fig. 11 Typical Static Stability Curve
RIGHTING ARM, m
EQUILIB. ANGL
RIGHTING ARM, m
therefore be used as a curve of righting moment by adding a scale of moments.
ANGLE OF INCLINATION
Fig.12 Typical Static Stability Curve
- CG off centreline - CG on centreline The features of the GZ curve indicate stability characteristics of the vessel under consideration. These include the following:
(a)
At small angles, GZ =GMT.sinΦ ≈ GMT.tanΦ and therefore, initial metacentric
height GMT is equal to slope of the GZ curve at the origin (Fig. 13). Therefore, the value of the ordinate of the tangent to the GZ curve at the origin at an angle of 1 radian or 57.30 is the initial metacentric height GMT. If initial metacentric height is negative, the vessel may experience a sudden heel after which it may be stable at
RIGHTING LEVER
that angle of heel. This is called angle of loll (fig. 14).
GM
φ = 1 RADIAN ANGLE OF HEEL (DEGREES) φ
Fig. 13 Typical Statical stability Lever Curve
ANGLE OF LOLL
CURVE
1 RAD
1
NAGATIVE
Fig. 14 Angle of Loll (b)
The righting moment can be obtained by multiplying GZ with displacement or,
righting moment = Δ.GZ. The maximum heeling moment the ship can withstand is
given by the maximum righting moment the ship can impose and is, therefore, proportional to maximum value of righting arm GZ. (c)
The range of stability is indicated by the angle range between which the
righting arm is positive. (d)
Angle of deck edge immersion is normally at the point of inflexion of the GZ
curve. Though, the deck edge along the length of the ship gets immersed at different angles, normally it is within a small range and is well indicated by the point of inflexion of the GZ curve. (e) The area under the curve indicates the dynamic stability and is discussed later. (f) The ship being symmetrical port and starboard, the stability lever curve is symmetrical about the x- and y- axis as shown in Fig. 13.
Dynamic Stability Static stability of any ship in any condition can be evaluated by superimposing various heeling arms resulting from specific upsetting forces (wind, turning, etc.) on a curve of righting arms. Although the statical stability curve, as the name implies, is the representation
(
)
of the righting arm, GZ , or righting moment W GZ of a ship when in a fixed-heel attitude, the curve can be used to determine the work involved in causing the ship to heel from one angle to another against the righting moment. The area under any portion of a curve of righting moment, such as the shaded area in Fig. 15 represents the work required to heel the ship from angle A to angle B . A moment, multiplied by the angle through which it is exerted, represents work. In the case of a ship, where the moment varies with the angle, if M is the moment at any angle of heel, φ then the work required to rotate the ship against this moment through an angle δ φ is M δφ (φ in radian ) and the work required to rotate it from A to B is B
Work = ∫ M dφ A
which is the area under the curve between A and B .
RIGHTING MOMENT
ANGLE OF INCLINATION
Fig. 15 Work Required to Heel a Ship
The total area between the righting-moment curve (at zero degrees to angle D) and the horizontal axis represents the total work required to capsize the ship from the upright position. This is often referred to as dynamic stability, although it does not really
involve
dynamics
because
wave-induced
rolling
velocities
and
the
accelerations are not considered. One can also say that the total area under the curve from A to B in Fig. 16, which represents the work done in heeling the ship from 0 to B, is the potential energy, EB acquired by the ship at B. If all external heeling moments are then released, this energy will bring the ship back to the upright, zero-heel condition. But at this point the potential energy will have been transformed into kinetic energy equal to EB minus energy loss (i.e, energy expended in overcoming the resistance of the water to rolling). This kinetic energy (proportional to the square of angular velocity) will carry it to an angle such that the area under the righting-moment curve, from the upright to that angle, is equal to the ship’s kinetic energy at zero inclination minus the energy absorbed by the resistance of the water. If the heeling moments developed by the heeling forces are calculated for several angles of inclination, these moments may be plotted on the same coordinates as the statical stability curve, as illustrated in Fig. 17. Note that both curves are extended to the left to show heel in the opposite direction. The dynamic stability, can be estimated up to angle Φ by can be evaluated and is equal to area under GZ curve between 0 and Φ. This is shown in Fig. 18.
RIGHTING MOMENT
φ
ANGLE OF INCLINATION AREA = E - LOSSES B
Fig.16 Effect of Rolling on Dynamic Stability
RIGHTING MOMENT
HEELING MOMENT
ANGLE OF INCLINATION
Fig. 17 Heeling and Righting Moments
ST A BI LI TY A M IC D YN
RI GH TI N
G
M
OM
EN TS
RIGHTING MOMENT
ANGLE OF HEEL
Fig. 18 Static and Dynamic Stability Lever
Heeling moment can be caused due to various external forces on the ship. Some of the common ones are given below: (i)
If the curve labelled “heeling moment” represents the moment of a beam wind, the moment will vary with the angle of inclination because of changes in the “sail” area, projected on a vertical plane, and in the vertical separation of the centroids of the wind pressure and the water pressure acting on the hull.
WIND PRESSURE
WATER PRESSURE
Fig. 19 Effect of Beam Wind
(ii)
If the heeling-moment curve represents the effect of high-speed turning, the moment will decrease at the larger angles, since the vertical separation of the centrifugal; force and the water pressure will vary approximately as the cosine of the angle of inclination.
CENTRIFUGAL FORCE
WATER PRESSURE
Fig. 20 Effect of High Speed Turn (iii)
A heeling moment due to the crowding of passengers to one side will similarly vary as the cosine of the angle of inclination. In general, heeling moments will vary with inclination because of variations in forces, levers, or both.
(iv)
Grounding also causes a heeling moment as shown in the figure.
Fig. 21 Effect of Grounding
At points A and B in Fig. 17 the heeling moment equals the righting moment and the forces are in equilibrium. For example, if the heeling moment is caused by a lateral shift of weight (mass), the heeling moment is ω d cos φ , where ω the weight (or mass) is and d is the distance moved. At equilibrium points A and B , W GZ = ω d cos φ
If the ship is heeled to point A , an inclination in either direction will generate a moment tending to restore the ship to position A. If the ship is heeled to point B , a slight inclination in either direction will produce a moment tending to move the ship away from position B , and the ship will either come to rest in position A or capsize. The range of positive stability is decreased by the effect of the heeling moment to point B . When a heeling moment exists, as in Fig. 17, the vertical distance between the heeling-moment and righting-moment curves at any angle represents the net moment acting at that angle either to heel or right the ship, depending on the relative magnitude of the righting and heeling moments, i.e., the net righting moment for the case of a weight shift is W GZ − ω d cos φ . Coming now to energy considerations, assume that the ship has rolled to the left to angle C in Fig. 17, has come to rest, and is about to roll in the opposite direction. Between C and the origin, the heeling moment and the ship’s righting moment will act in the same angular direction, and the total moment acting on the ship will be represented by the vertical distance between the two curves. To the right of the origin, these moments will act in opposite angular directions, and the moment acting on the ship will still be represented by the distance between the two curves. Between
C and A, the shaded area, minus the energy absorbed by water resistance, corresponds to the energy imparted to the ship that will exist, as kinetic energy, when the ship rolls through point A. This energy will carry the ship to some angle D such that the area between the curves and between A and D is equivalent to the kinetic energy at point A, less the energy absorbed by the water between A and D . If there is not sufficient area between the curves and between A and B to absorb this energy, the ship will roll past point B and capsize. To reduce the danger of capsizing under these conditions, the area between the heeling and righting-moment curves and between A and B should be greater, by some margin, than that between C
and A. As a practical matter, it may be desirable to establish a limit for rolling which is considerably smaller than angle B because of the unfavourable attitude of the ship and the probability of shipping water through topside openings, i.e. down flooding, at very large angles.
SOLAS requirement The international Maritime Organisation (IMO) has formulated rules (SOLAS) for ensuring adequate stability for ships during operation. The Rules for intact stability are given in Annexure-1. The main requirements of properties of the GZ curve are as follows (heeling moment due to external forces is considered in the Annexure-1): (i)
The initial metacentric height GM 0 should not be less than 0.15 m.
(ii)
The righting lever GZ should be at least 0.20 m at an angle of heel equal to or
greater than 30o . (iii)
The maximum righting arm should occur at an angle of heel preferably
exceeding 30o but not less than 25o . (iv)
The area under the righting lever curve (GZ curve) should not be less than
0.055 metre-radian up to θ = 30o angle of heel. (v)
The area under the righting lever curve should not be less than 0.09 metre-
radian up to θ = 40o or the angle of flooding θi if this angle is less than 40 o . (vi)
The area under the righting lever curve between the angles of heel of 30o and
40 o or between 30o and θi , if this angle is less than 40 o , should not be less than
0.03 metre-radian. It can be observed that, each of the above conditions can be satisfied only up to a certain value of KG (VCG) and if the KG value crosses this limit the condition would be violated. The limiting KGmax value is likely to change based the condition considered. Fig. 22 given below shows the limiting KGmax curves satisfying various SOLAS conditions at various displacements for a feeder container vessel. The limiting envelop of the curves, defines the maximum KG value the ship should not cross at any operating condition at that displacement.
KG (max) AGAINST DISP 10.5
KG max
9.5
8.5 7.5
6.5 0
5000 Max Heel Area 40
Disp (m3) Max GZ Area 30-40
10000
15000
Area 30 Weather
Fig. 22 Limiting KGmax to Satisfy SOLAS Intact Stability Criteria
Righting Arm of a Submerged Body Since the body is completely submerged, the buoyancy force always acts through the centre of buoyancy B. Viewing this another way, in the absence of any water plane, BM must be zero so that B and M become coincident. If the body in inclined through an angle Φ, the moment acting on it is given by Δ. BG. sin Φ It is clear that this will try to bring the body to upright position if B is above G. This is the condition of stable equilibrium. The righting arm follows a sine curve where GZ = BG. sin Φ Thus the stability lever curve is typical and is dependent on the relative positions of B and G. To change the stability one has to change the CG only by flooding or emptying tanks.
π/2
π
Fig. 23 Righting Arm of a Submarine
Free Surface Effect The theoretical effect of free surface on metacentric height can be assessed by assuming that the weight of the liquid in each tank acts at the metacenter of the tank, because, at any small angle of heel, a vertical line through the actual centre of gravity will pass through this point. This is equivalent to assuming that the weight of the liquid in each tank is raised from its centroid in the upright position to its metacenter, a distance of iT v . This increases the vertical moment of the mass of the ship by (ω g ) ( iT v ) , where ω is the weight of the liquid. If the specific volume of the liquid, expressed as volume / mass, is designated as δ , then ω g = v δ and the increase in vertical mass moment becomes
v iT iT . = δ v δ an expression which is independent of the quantity of liquid in the tank. Therefore, for any condition of loading, free surface may be evaluated for small angles of heel, by adding the values of iT δ for all tanks in which a free surface exists. If this summation, which is the increase in vertical moment due to the free surface, is divided by the ship’s displacement, the result will be the rise in the ship’s centre of gravity caused by the free-surface effect. This rise, called the free-surface correction
is added to KG , the height of the ship’s centre of gravity above the keel, resulting in an equivalent reduction in the metacentric height. Hence, with displacement in mass units, GM co r = KB + BM − KG − iT δ . Δ
or with displacement in weight units, GM co r = KB + BM − KG − giT δ .W
The effect of free liquid in a tank is to cause the centre of gravity of the liquid to shift through a certain distance d , parallel to the inclined waterline. If the weight of the liquid is ω and the displacement of the ship W , the centre of gravity of the ship will move parallel to the inclined waterline through a distance ( d .ω ) W , reducing the righting arm by that amount. Hence, the reduction in GZ is
( d .ω )
W , or ( d .ω / g ) Δ
in mass units. The quantity d .ω g is known as the moment of transference. When free surface is present in a number of tanks, the summation of d .ω g for the various tanks, divided by the displacement of the ship, gives the total reduction in righting arm. If the surface of the liquid has not reached the top or bottom of the tank, the distance d is equal to the distance from the centre of gravity of the liquid with the ship in the upright position to the metacenter of the liquid, which is equal to iT υ , multiplied by the sine of the angle of inclination. Therefore,
ω ω i d . . = . T sin φ g g v or since ω g = v δ
i ω v i d . . = . T sin φ = T sin φ δ v δ g The moment of transference is thus seen to be independent of the quantity of liquid in the tank.
If the surface of the liquid has reached the top or bottom of the tank, the moment of transference will be reduced, and may be expressed by the product of
iT δ and some factor less than sin φ , or i d .ω = C. T g δ
The value of C depends on the degree of fullness, the ratio of depth to breadth of the tank and the angle of inclination, each of which has some influence on the degree to which the motion of the liquid is suppressed. Evaluation of the factor C is simplified by the fact that the tanks that contain liquid are assumed to be full, half full, 95 percent full in naval practice or 98 percent full in merchant practice. Free liquid on a ship acts in the fore-and-aft direction in the same manner as in the transverse direction. For an intact ship with normal tankage, the effect of free liquid on trim is so small that it may be ignored. Its magnitude is small in comparison to the assumption in the formula for moment to trim one cm so that the centre of gravity is at the same height as the centre of buoyancy. On unusual craft, however, free liquids may have an important effect on trim. The subdivision of large tanks into two or more smaller tanks may be an effective method of improving stability by suppressing the motion of free liquids. Grain Shifting Moment due to Carriage of Dry Bulk cargo Bulk dry cargo, such as ore, coal or grain, may redistribute itself if the ship rolls or heels to an inclination greater than the angle of repose of the substance carried (angle of repose is the angle between a horizontal plane and the cone slope obtained when bulk cargo is freely poured onto this plane). Thus, a ship may start a voyage with the upper surface of such a cargo horizontal and with the cargo evenly distributed throughout the space. But if the ship rolls sufficiently to cause a cargo shift, a list will result. A ship which has listed due to even a slight shift of cargo is open to the danger that it may later roll to increasing angles on the low side with further shifting of the cargo. Ships have been known to capsize from such progressive shifting of cargo.
Furthermore, all cargoes are directly influenced by the seaway-induced motions of the ship, which produce significant angular and lateral accelerations. In a rapidly rolling ship, such cargoes may shift even when the maximum angle of roll is less than the angle of repose of the cargo, because of the dynamic effects of rolling. Calculations using motion dynamics show that the accelerations involved in rolling produce a greater likelihood of cargo shifting when the cargo is located above the ship’s CG (as in ‘tween deck spaces) rather than below (in the hold). One design approach for ships intended to carry dry bulk cargoes is to adjust hold volumes to suit the cargo density, so that holds will normally be full when loaded. Small hatches and sloping sides at the top of the compartment will reduce the danger of shifting cargo. For general cargo ships that may sometimes carry bulk cargo it is essential to provide for fitting one or more longitudinal subdivisions in the holds and “tween decks to minimize the possibility of shift of cargo in heavy seas. Such temporary subdivision bulkheads are called shifting boards.
Usually they
consist of wooden planks laid edge to edge in steel channels or equivalent. In all cases it is essential to ascertain that adequate stability can be attained in operation to cope with any anticipated cargo, considering the restraints actually available. Of course, the ship operator is responsible for reviewing such factors prior to every voyage. Grain has long been recognized as a dangerous cargo because of its tendency to flow or shift in the hold of a rolling ship. In the past both national and international regulations relied heavily on the use of feeders from ‘tween decks to holds, which were intended to allow grain to flow downward to keep the hold full as the grain settled. Continued reports of grain cargo shifting, with some ship losses, led to a new investigation of the problem, which showed that even with feeders holds could not be assumed to be full and that shifting boards were still of great value in many cases. New grain regulations were developed that changed emphasis from attempting to prevent grain shifting to making sure that the worst possible heeling moments will not exceed acceptable limits for each ship and loading condition.
Effect of Suspended weight In the case where meat or similar cargo is suspended from a point above its centre of gravity, sometimes it is hooked into eyes under the deck above the hold. This method of stowage calls for special correction in the calculation of GM . A weight suspended from a boom is a similar case and serves as a convenient explanatory example. Such cases arise during loading/ unloading using a ship crane and the quay side. A particularly interesting effect is that on a heavy lift crane barge. The centre of gravity of a weight suspended freely from a boom will remain vertically below the end of the boom, regardless of the list of the ship. The point of suspension, therefore, is the metacenter through which the weight acts. It makes no difference in the stability of the vessel whether the weight hangs high above the deck or not, provided the point of support remains the same. A suspended weight may be treated as though its centre of gravity were at the point of support. Obviously, if a full cargo, such as meat, were suspended from several feet above its own centre of gravity, the metacentric height of the vessel would be appreciably less than it would have been with an equal weight of unsuspended cargo.
Fig. 24 Effect of Suspended Weight
Effect of Ship Poised on a Wave Waves may have a significant effect on static stability, particularly following or overtaking waves of approximately the ship’s length. Righting curves can be drawn by superimposing offsets from the wave profile on the body plan used for the calculation of cross curves. In a computer calculation, the wave profile is used for input instead of a straight waterline. Dynamic effects of rolling are excluded. Fig. 25 shows typical righting arm curves for a ship in a regular wave of the same length as the ship and height equal to LW 20 , with either wave crest or wave
RIGHTING ARM, m
trough amidships.
WAVE TROUGH AMIDSHIPS CALM WATER
WAVE CREST AMIDSHIPS ANGLE OF INCLINATION Fig. 25 Righting Arm of a Ship poised on a Wave Effect of Change of Breadth on Stability Effect of minor changes of ship parameters on ship stability can be studied using difference equations. It is well known that KB ∞ T or draught BM ∞ B2 / (T.CB) KG ∞ D or depth Then, difference equations can be written as δKB / KB = δT / T δBM / BM = 2δB / B - δT / T - δCB / CB δKG/ KG = δD / D
It can be observed that if there is an increase in breadth without changes in other ship parameters including displacement, there will be a decrease in draught causing a slight reduction in KB, but substantial increase in BM. KG remaining constant, there is a net gain on GM or initial metacentric height. Reserve buoyancy of the ship will increase due to increase in breadth and also freeboard. Thus there will be an improvement in GZ or the righting arm curve. Thus, the most important parameter affecting stability is the breadth increase of which improves stability to a large extent.
Fig. 26 Effect of increase of Beam
Effect of Change of Depth on Stability If there is an increase in depth without changing the under water body shape, BM and KB remain unchanged. However, the steel weight increases and primarily, the steel weight of the portion at and above the original depth go up by an amount equal to the change in depth. This increases the KG of the light ship. The cargo Cg also goes up due to increase in cargo volume in the upward direction. Thus there is a net increase in the KG of the loaded ship which can be approximated to be proportional to Depth change as a first approximation. Thus there is a reduction of GM or initial metacentric height. This pulls down the GZ curve at small angles. But as the angle of inclination increase the excess reserve buoyancy due to increase in depth comes into play and GZ increases. The deck edge immersion is delayed.
Fig. 27 Effect of increase of Depth
Effect of Change of Form Keeping the displacement same, if the bilge radius is increased, there will be an upward movement of under water volume which will result in an increased load water plane area. KB will go up and so will BM. Initial metacentric height will improve. On the other hand, if the under water volume is pushed down like a bulbous bow or reduction in bilge radius, KB will reduce and so will BM, reducing the net initial stability. Providing excess flare above water will lead to only marginal increase in KG but, will increase reserve buoyancy and hence, an improvement in the GZ curve. Addition of water tight erections above deck will have similar effects.
(a) Fining of the Bilges
(b) Change of Tumblehome and Flare
Fig. 28 Effect of change of Form
OPERATING CONDITIONS Stability of a ship must be assessed at various loading conditions prior to loading or during loading. Where as the form stability of a ship is fixed and is known apriori, the weight stability aspects, specifically the KG due to particular loading, the free surface effect due to liquids in tanks, grain heeling moment if any, are not known and must be assessed when required. The light ship weight and KG form the basis of this analysis. This is necessary to measured accurately and is done during inclining experiment. This experiment is conducted when the ship is nearly complete in all respects, the only liquids in the ship being the oil in the main engine pipe lines. In calm weather with no external disturbance due to wind or waves. The position and amount of weights to be removed or added to get the actual lightship weight is noted before the start of the experiment. The draught marks on fore and aft provide the basis for getting the lightship weight from hydrostatics. By moving a know weight from port to starboard (or, vice versa) and noting the heel angle, the GMT is calculated as has been discussed earlier. Knowing the KMT from hydrostatics, the lightship KG is estimated. KG of the loaded ship can then be obtained by knowing the vertical CG position of cargo either based on hold geometry (bulk/ oil/ general cargo) or unit-wise (containers, RO-RO cargo etc.). The ship can be loaded to various draughts based on operating requirements such as (i) fully loaded to maximum freeboard draught, (ii) design draught as the most common operating condition, (iii) partially loaded ship which may occur due to partial cargo availability or during loading and unloading operation or, (iv) in ballst condition with no cargo but only ballast water. Liquid carried in ships could be either cargo, ballst or consumables such as fuel oil and fresh water. Generally cargo or ballast is pressed full in a tank or the tank is left empty. Full indicates 95% to 98% of the gross capacity of the tank allowing for liquid expansion. In such compartments generally no frr surface effect is generated. In departure condition from a port all consumable tanks are also full. But during voyage, tanks are emptied of liquid due to consumption. So at the time of arrival some tank may have only partial liquids generating negative effects of free surface on stability. This also needs to be investigated and is covered by SOLAS regulations.
WEIGHT AND CENTRE OF GRAVITY One of the primary parameters of stability is the estimation of light weight and its CG position. Once the vessel is completed, one can only get the weight and CG from inclining experiment but it is difficult to change these parameters throughout its life without major modifications. Therefore it is very important to make an accurate estimate of weight and CG at the design stage and update the estimate at every stage of production and take corrective measures wherever necessary at early stages. It is necessary to have the weights updated as and when items go on board the ship and see that the design values are maintained. It is also necessary to estimate the TCG of the weights to ensure no initial heel of the vessel. If the TCG is on ship centre line, but the vessel still heels with slight disturbance to some angle, it could mean negative initial metacentric height and an angle of loll. This is also something which must be avoided. If proper CG estimate has not been done or, is erroneous causing negative metacentric height, it may be necessary to carry permanent ballast at the bottom of the ship to lower the CG. This situation is not desirable since this weight is taken away from freight earning cargo.
ANNEXURE - 1 DESIGN CRITERIA APPLICABLE TO ALL SHIPS (SOLAS Resolution A.749(18))
General intact stability criteria for all ships Recommended general criteria The area under the righting lever curve (GZ curve) should not be less than 0.055 metre-radian up to θ = 30o angle of heel and not less than 0.09 metre-radian up to θ = 40o or the angle of flooding θi if this angle is less than 40 o . Additionally, the area under the righting lever curve (GZ curve) between the angles of heel of 30o and 40 o or between 30o and θi , if this angle is less than 40 o , should not be less than
0.03 metre-radian. The righting lever GZ should be at least 0.20 m at an angle of heel equal to or greater than 30o . The maximum righting arm should occur at an angle of heel preferably exceeding 30o but not less than 25o . The initial metacentric height GM 0 should not be less than 0.15 m. In addition, for passenger ships, the angle of heel on account of crowding of passengers to one side should not exceed 10o . In addition, for passenger ships, the angle of heel on account of turning should not exceed 10o when calculated using the following formula:
M R = 0.02
V02 ⎛ d⎞ Δ⎜ KG − ⎟ 2⎠ L ⎝
where:
MR
=
heeling moment (m-t)
V0
=
service speed (m/s)
L
=
length of ship at waterline (m)
Δ
=
displacement (t)
d
=
mean draught (m)
KG
=
height of centre of gravity above keel (m)
Where anti-rolling devices are installed in a ship, the Administration should be satisfied that the above criteria can be maintained when the devices are in operation. A number of influences, such as beam wind on ships with large windage area, icing of topsides, water trapped on deck, rolliing characteristics, following seas, etc., adversely affect stability and the Administration is advised to take these into account, so far as is deemed necessary. Provisions should be made for a safe margin of stability at all stages of the voyage, regard being given to additions of weight, such as those due to adsorption of water and icing (details regarding ice accretion are given in chapter 5), and to losses of weight, such as those due to consumption of fuel and stores. For ships carrying oil-based pollutants in bulk, the Administration should be satisfied that the criteria given in such cases can be maintained during all loading and ballasting operations. The ability of a ship to withstand the combined effects of beam wind and rolling should be demonstrated for each standard condition of loading as follows: 1.
the ship is subjected to a steady wind pressure acting perpendicular to
( )
the ship’s centreline which results in a steady wind heeling lever IW1 . 2.
from the resultant angle of equilibrium (θ 0 ) , the ship is assumed to roll owing to wave action to an angle of roll (θ1 ) to windward. Attention should be paid to the effect of steady wind so that excessive resultant angles of heel are avoided;
3.
the ship is then subjected to a gust wind pressure which results in a
( )
gust wind heeling lever IW2 ; 4.
under these circumstances, area b should be equal to or greater than area a ;
5.
free surface effects should be accounted for in the standard conditions of loading.
θ0
=
angle of heel under action of steady wind
θ1
=
angle of roll to windward due to wave action
θ2
=
angle of down flooding (θ1 ) or 50o or θ c , whichever is less,
where:
θi
=
angle of heel at which openings in the hull, superstructure or deck-houses which cannot be closed weather tight immerse. In applying this criterion, small openings through which progressive flooding cannot take place need not be considered as open.
θc
=
angle of second intercept between wind heeling lever
( I ) and GZ curves. W2
( )
The wind heeling levers IW1
( )
and IW2
are constant values at all angles of
inclination and should be calculated as follows:
IW1 =
P AZ ( m ) and 1000 g Δ
IW2 = 1.5 IW1 ( m ) where:
(I ) W1
=
( ) used for ships in restricted service
504 N/m2. The value of IW1
may be reduced, subject to the approval of the Administration. A
=
projected lateral area of the portion of the ship and deck cargo above the waterline (m2)
Z
=
vertical distance from the centre of A to the centre of the underwater lateral area or approximately to a point at one half the draught (m);
Δ
=
displacement (t)
g
=
9.81 m/s2
The angle of roll (θ1 ) referred to should be calculated as follows:
θ1
=
X1
=
factor as shown in table 1
X2
=
factor as shown in table 2
k
=
factor as follows:
k
=
1.0 for a round-bilged ship having no bilge or bar keels
k
=
0.7 for a ship having sharp bilges
k
=
as shown in table 3 for a ship having bilge keels,
109 k X 1 X 2 rs (degrees)
where:
a bar keel or both
r = 0.73 ± 0.6 OG d with:
OG
=
distance between the centre of gravity and the waterline (m) (+ if the centre of gravity is above the waterline, - if it is below)
d
=
mean moulded draught of the ship (m)
s
=
factor as shown in table 4.
Table 1 – Values of factor X 1 B/d
X1
≤ 2.4
1.0
2.5
0.98
2.6
0.96
2.7
0.95
2.8
0.93
2.9
0.91
3.0
0.90
3.1
0.88
3.2
0.86
3.4
0.82
≥ 3.5
0.80
Table 2 – Values of factor s
CB
X2
≤ 0.45
0.75
0.50
0.82
0.55
0.89
0.60
0.95
0.65
0.97
≥ 0.70
1.0
Table 3 – Values of factor k Ak × 100 L× B
k
0
1.0
1.0
0.98
1.5
0.95
2.0
0.88
2.5
0.79
3.0
0.74
3.5
0.72
≥ 4.0
0.70
Table 4 – Values of factor s Ak × 100 L× B
k
≤ 6
0.100
7
0.098
8
0.093
12
0.065
14
0.053
16
0.044
18
0.038
≥ 20
0.035
(Intermediate values in these tables should be obtained by linear interpolation) Rolling period where:
T=
2C B (s) GM
C = 0.373 + 0.023 ( B d ) − 0.043 ( L 100 )
The symbols in the above tables and formulas for the rolling period are defined as follows:
L
=
waterline length of the ship (m)
B
=
moulded breadth of the ship (m)
d
=
mean moulded draught of the ship (m)
CB
=
block coefficient
Ak
=
total overall area of bilge keels, or area of the lateral projection of the bar keel, or sum of these areas (m2)
GM
=
metacentric height corrected for free surface effect (m).
