CHAPTER 6. TRANSVERSE STABILITY AT SMALL ANGLES OF HEEL In this chapter we shall consider the problem of initial stability, which correspond to very small departures from an assumed equilibrium condition. Of particular concern to the ship designer and operator is the transverse stability and consequently we apply a small angle of heel to the ship. A rigid body floating freely on the surface of a fluid has six degrees of freedom, three of translation and three of rotation. For disturbances from a state of equilibrium, the naval architect refers to the movements in the six degrees as shown in Table 6.1.
Movement Fore and aft translation Transverse translation Vertical translation Rotation about a fore and aft axis Rotation about a transverse axis Rotation about a vertical axis
Table 6.1. Dynamic surge sway heave roll pitch yaw
Quasi-static heel trim -
6.1. Heeling Forces and Moments The magnitude of heeling forces forces and moments determines determines the forces and moments moments that must be generated by the forces of weight and buoyancy in order to prevent capsizing or excessive heel. External heeling forces affecting transvers stability may be caused by;
• • • • • •
Beam winds and waves Lifting of heavy weights over the side High speed turns Grounding Strain on mooring lines Towline pull of tugs
Internal heeling forces include
• Shifting of on-board weights athwartship • Entrapped water on deck Beam winds and waves : 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. The situation is shown in Figure 6.1.
6.1
Equilibrium with respect to angle of heel will be reached when
• The ship is moving to leeward with a speed such that the water resistance equals the wind •
pressure, and The ship has heeled to an angle such that the moment produced by the forces of weight and buoyancy equals the moment developed developed by the wind pressure and the water pressure pressure Weight
Wind pressure
G
B1
Water ressure
Dis Dis lace laceme ment nt
Figure 6.1. Effect of beam winds
As the ship heels from the vertical, the wind pressure, water pressure, and their vertical separation remain substantially constant. The ship’s weight is constant and acts at a fixed point. The force of buoyancy is also constant, but the point it acts varies with the angle of heel. Equilibrium will be reached when sufficient horizontal separation of the centers of gravity and buoyancy has been produced to cause a balance between between heeling and righting moments. moments. Lifting of Heavy Weights over the Side : When a weight is lifted over the side, as shown in Figure 6.2, the force exerted by the weight acts through the outboard end of the boom, regardless of the angle of heel or the height to which the load has been lifted. Therefore, the weight of the sidelift may be considered to be added to the ship at the en of the boom. If the ship’s centre of gravity is initially on the ship’s centreline, as at G in Figure 6.2, the centre of gravity of the combined weight of the ship and the sidelift will be located along the line GA, and will move to a final position G 1, when the load has been lifted clear of the pier. Point G 1 will be off the ship’s centreline and somewhat higher than G. The ship will heel until the centre of buoyancy has moved off the ship’s centreline to a position directly below point G1. High Speed Turns : When a ship is executing a turn, a centrifugal force is generated, which acts horizontally through the ship’s centre of gravity. This force is balanced by a horizontal water pressure on the side of the ship, as shown in Figure 6.3. Except for the point of application of the heeling force, the situation is similar to that in which the ship is acted upon by a beam wind, and the ship will heel until the moment of the ship’s weight and buoyancy equals that of the centrifugal force and water pressure.
6.2
Weight
G
G1
B1
Displacement
Figure 6.2. Lifting of Heavy Weights over the Side
Weight Centrifugal force G
B
B1
Water pressure
Displacement
Figure 6.3. High Speed Turns
Grounding : If a ship runs aground in such a manner that the bottom offers little restraint to heeling, as shown in Figure 6.4, the reaction of the bottom may produce a heeling moment. As the ship grounds, part of the energy due to its forward motion may be absorbed in lifting the ship, in which case a reaction, R, between the bottom and the ship will develop. The force of buoyancy will be less than the weight of the ship, since the ship is supported by the combination of buoyancy and the reaction of the bottom. The ship will heel until until the moment of buoyancy about the point of contact with the bottom becomes equal to the moment of the ship’s weight about the same point, i.e.
(W-R).a = W.b
6.3
Weight
b G
a B B1
R Reaction force
Displacement-Reaction force
Figure 6.4. Grounding
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 vessel. In each case, equilibrium would be reached when the centre of buoyancy has moved to a point where heeling and righting moments are balanced. Movement of weight already aboard the ship, such as passengers, liquids or cargo, will cause the ship’s centre of gravity to move. If a weight is moved from A to B, as shown in Figure 6.5, the ship’s centre of gravity will move from G to G 1 in a direction parallel to the direction of movement of the shifred weight. The ship will heel until the centre of buoyancy is directly below point G1 .
6.4
w Weight A
w
G
G1
B
B
B1
Displacement
Figure 6.5. Effect of weight movement
6.2. The Transverse Metacentre Consider a symmetric ship heeled to a very small angle, δϕ , as shown in Figure 6.6. For small angles the upright and inclined waterlines will intersect on the waterline. The volumes of the emerged and immersed wedges must be equal for constant displacement. The centre of buoyancy has moved off the ship’s centreline as a 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 centreline at a point M, called the transverse metacentre, when δϕ → 0 .
δϕ
M G
Z
δϕ B B
Figure 6.6. Transverse metacentre and righting arm
6.5
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, upto about 7-10 degrees. If the locations of G and M are known, the righting arm for small angles of heel can be calculated readily, with sufficient accuracy, by the formula
GZ ≅ GM sin ϕ The distance GM is therefore important as an index of transverse stability at small angles of heel, and is called the transverse metacentric height. Since GZ is considered po sitive 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. 6.2.1. Metacentric Radius (BM)
Consider a symmetric ship heeled to a small angle (ϕ), say 2 or 3 degrees, as shown in Figure 6.7.
M
ϕ y g1
G
Z
ϕ g B
R B1
Figure 6.7. Metacentric radius
6.6
For small angles the emerged and immersed wedges are approximately triangular. If y is the half ordinate of the original waterline at the cross section the emerged or immersed section area is.
dA =
1 2 y ϕ 2
for a small length dx, the volume of each wedge is
dV =
1 2
y 2 ϕdx
The righting moment is equal to transverse shift of buoyancy.
dM = 2 ×
1 2 2 2 y ϕdx × y = y 3 ϕdx 2 3 3
The total righting moment is L
M=
∫ 0
2 3 ϕy dx 3 L
The expression within the integral sign, I x
L
2 3
= ϕ∫ y 3 dx 0
2
= ∫ y 3 dx , is the second moment of area, or the moment 0
3
of inertia, of a waterplane about its centreline. Hence the movement of buoyancy is M
= I x ϕ = v gg 1 = ∇ BB1
Referring to Figure 6.7, for small angles of heel BB1
= BM ϕ
⇒
∇BMϕ = I x ϕ
⇒
BM =
Ix
∇
Thus the height of the metacentre above the centre of buoyancy is found by dividing the second moment of area of the waterplane about its centreline by the volume of displacement. 6.2.2. Metacentric Diagram
The metacentre diagram is a convenient way of defining variations in relative heights of the centres of buoyancy and metacentre for a series of waterlines parallel to the design waterline. The position of the centres of buoyancy, B, and metacentre, M, are dependent only on the geometry of the ship and the waterplane at which it is floating. A typical metacentric diagram is shown in Figure 6.8. The vertical scale represents draught and a line is drawn at 45 degrees to this scale. For a given draught, T, a horizontal line is drawn intersecting the 45 degrees line in D and a vertical line is drawn through D. On this vertical line, the distance DM represents the height of the centre of metacentre above the waterplane ans DB represents the depth of the centre of buoyancy below the waterplane. This process is repeated a 6.7
sufficient number of times to define the loci of metacentre and centre of buoyancy. These are termed the metacentre, M, and buoyancy, B, curves. A table may be constructed to the left of the diagram, in which are listed the displacements and T 1 values for each a number of draughts corresponding to typical ship loading conditions. Since KB is approximately proportional to draught over the normal operating range, the B curve is usually nearly staright for conventional ship forms. The M curve, on the other hand, usually falls steeply with increasing draught at shallow draught than levels out and may even begin to rise at very deep draught.
M
S N O T DRAUGHT : 6.3 m N I M M DISPLACEMENT : 12700 t T C N : 12.7 t/cm N R I T1 E E T M P H E S G C N U A O A L T R P D S I D
.
KM
G KB
B
K
Figure 6.8. Metacentric Diagram
6.2.2.1. Metacentric Diagrams for Simple Geometrical Forms 6.8
a) Rectangular section
Consider a barge of length L and breadth B with constant rectangular cross sections floating at draught T. The volume of the barge is
∇ = LBT The height of the centre of buoyancy is
KB =
T 2
The metacentric radius is LB3 BM =
I
∇
=
12 LBT
B2 = 12T
Then the height of metacentre above the keel is
KM = KB + BM =
T 2
B2 + 12T
As can be seen the height of metacentre depends upon the beam and draught but not the length. At zero draught KM would be infinite, and the second term predominates for small draught values. The draught at which KM is minimum can be found by diffrentiating the equation for KM with respect to T and equating to zero. Then the draught at which KM is minimum can be found as follows, dKM dT
1
B2
2
12T 2
= −
=0
⇒
T
=
B 6
b) Triangular section
Consider a barge of length L and waterline breadth B with constant triangular cross sections floating apex down at draught T. The volume of the barge is
1
∇ = LBT 2
The height of the centre of buoyancy is
KB =
2T 3
6.9
The metacentric radius is
I
BM =
=
∇
LB3 12 1 LBT 2
=
B2 6T
Then the height of metacentre above the keel is
KM = KB + BM =
2T 3
+
B2 6T
The draught at which KM is minimum can be found by differentiating the equation for KM with respect to T and equating to zero. Then the draught at which KM is minimum can be found as follows,
dKM dT
2 3
= −
B2 6T 2
=0
⇒
T=
B 2
c) Circular section
Consider a circular cylinder of radius r and centre of section O, floating with its axis horizontal. For any waterline, above or below the centre, and for any inclination, the buoyancy force always acts through the centre. That is, KM is independent of draught and equal to r.
6.3. Measurement of Initial Stability The difference between the centre of gravity and the metacentre is defined as the metacentric height (GM) and this distance can be used the initial stability of a ship at smalll angles of heel, from 0 0 to 70-100. For small angles of heel the metacentric height (GM) is calculated by subtracting the height of the ship’s centre of gravity above the keel (KG) from the height of metacentre above the keel (KM), i.e GM = KM − KG The height of metacentre above the keel is calculated by the summation of the height of centre of buoyancy above the keel (KB) and the metacentric radius (BM) as follows KM
= KB + BM
where the metacentric radius is the distance between the centre of buoyancy and the metacentre. Therefore GM = KB + BM − KG
It is evident that GM is the key indicator of initial transverse stability. Whilst it should obviously be positive, too high values should be avoided. GM is a measure of the ship’s stiffness in roll motion and largely governs the period of roll motion. Too high a value of GM leads to a very short roll period. 6.10
The actual value of GM for a ship may be found by an inclining experiment, which is discussed in later sections. At the initial stages of design, to ensure that the ship has sufficient initial stability, the metacentric height can be calculated by using approximate formulae.
Normand
⎛ C WP ⎞ ⎟⎟ + C C WP ⎠ ⎝ B C ⎞ 5T T ⎛ ∇ − = ⎜⎜ 2.5 − B ⎟⎟ KB = 6 3A WP 3 ⎝ C WP ⎠ KB = T (0.9 − 0.36C M )
Schneekluth
KB = T(0.9 − 0.3C M
KB = T⎜⎜
Posdunine Morrish
− 0.1C B ) ⎛ C ⎞ KB = T⎜⎜ 0.828 − 0.343 B ⎟⎟ C WP ⎠ ⎝ KB = T (1.1 − 0.6C M )
Bauer Henschke
BM =
Robb
⎞ B 2 ⎛ 1 ⎜⎜ ⎟ T ⎝ 16 − 7C B ⎠⎟
⎛ 0.0067 ⎞ ⎜⎜ 0.09 − ⎟ T ⎝ C B ⎠⎟ B 2 ⎛ C WP (0.727C WP + 0.0106 − 0.003) ⎞ ⎜ ⎟⎟ BM = T ⎜⎝ CB ⎠ B 2 ⎛ 0.1216C WP − 0.0410 ⎞ ⎜ ⎟⎟ BM = T ⎜⎝ CB ⎠ B 2 ⎛ 0.09377C P − 0.0122 ⎞ ⎜ ⎟⎟ BM = T ⎜⎝ CB ⎠ B 2 ⎛ 0.0861C P − 0.008 ⎞ ⎜ ⎟⎟ BM = T ⎜⎝ CB ⎠ 2 B 2 ⎛ C WP ⎞ ⎜ ⎟ BM = T ⎜⎝ 12C B ⎠⎟ B 2 ⎛ C WP (C WP + 0.04) ⎞ ⎜ ⎟⎟ BM = T ⎜⎝ 12C B ⎠ 2 B ⎛ 57C WP − 22 ⎞ ⎜ ⎟ BM = T ⎜⎝ 420C B ⎠⎟ BM =
Riddlesworth Eames D’arcengelo Kiss Brown Mc Cloghrie Posdunine Rauert
B2
KG=aD Ship type
a
6.11
Tanker Bulk carrier Dry Cargo Passenger
0.69 0.68 0.72 0.75
6.4. Wall sided formula For small angles of heel, we found that the righting lever GZ ≅ GMϕ . A more accurate formula for GZ at angles around 100 is available for wall sided ships, i.e. those having vertical sides in the region of the waterline. The vessel can have a turn of bilge provided it is not exposed by the inclination of the ship. Nor must the deck edge be immersed. Because the vessel is wall sided the emerged and immersed wedges will have sections which are right angled triangels of equal area. Consider the wall sided section, shown in Figure 6.9. The transverse moment of volume shift can be expressed as y 4y 2 2 = y 3 tan ϕdx = tan ϕ y 3dx = I tan ϕ MT = y tan ϕdx × 2 3 3 3
∫
∫
∫
where y is the half ordinate an I is the second moment of area of the waterplane about the centreline. Therefore M I tan ϕ = BM tan ϕ BB1 = T =
∇
∇
Similarly the vertical moment
MV
=∫
1 2 2 y tan ϕ × y tan ϕdx = 2 3
∫
y3 1 tan 2 ϕdx = I tan 2 ϕ 3 2
Therefore B1 R =
MV
∇
=
1 I tan 2 ϕ 2
∇
1
= BM tan 2 ϕ 2
From Figure 6.9, BR = BB1 cos ϕ + B1R sin ϕ BR = BM tan ϕ cos ϕ + BR = sin ϕ(BM +
1 BM tan 2 ϕ sin ϕ 2
1 BM tan 2 ϕ) 2
Therefore, the righting arm, GZ, is
6.12
GZ = BR − BG sin ϕ 1 ⎛ ⎞ BM tan 2 ϕ − BG ⎟ 2 ⎝ ⎠ 1 ⎛ ⎞ GZ = sin ϕ⎜ GM + BM tan 2 ϕ ⎟ 2 ⎝ ⎠ GZ = sin ϕ⎜ BM +
At small angles, this degenerates to GZ = GM sin ϕ . The formula is invalid at large heel angles.
M
W
ϕ h1 g1 W1
y
G
Z
ϕ
L1 g B
R B1
h
L
B2 K
Figure 6.9.
6.5. Angle of Loll In certain conditions of loading it is possible for a ship to have negative GM when upright. The GZ curve will have a negative slope at the origin, as shown in Figure 6.10. As the ship heels to larger angles, GZ increase to become positive at an angle, known as the angle of loll. The ship is in unstable equilibrium when upright and will flop over rapidly to the angle of loll. The wall sided formula can be used to estimate the angle of loll. If the ship has a positive GM it will be in equilibrium when GZ is zero, that is
⎛ ⎝
GZ = sin ϕ⎜ GM +
1 2
⎞ ⎠
BM tan 2 ϕ ⎟ = 0
6.13
This condition is satisfied by two values of upright. The second value is given by
GM +
1 2
ϕ . The first is ϕ = 0 , which is the case with the ship
BM tan 2 ϕ = 0
⇒
tan 2 ϕ = −
2GM BM
With both GM and BM positive there is no solution to this meaning that the upright position is the only one of equilibrium. This also applies to the case of GM=0, which means that in the upright position the ship is stable, not neutral. When, however, the ship has a negative GM there are two possible solutions for ϕ in addition to that of zero, which in this case would be a position of unstable equilibrium. These other solutions are at ϕ either side of the upright position being given by, tan ϕ =
2GM BM
Such an angle is known as the angle of loll. The ship would show no preference for one side or the other.
GZ (m)
angle of loll Heel angle
GM
Figure 6.10. Angle of loll
If ϕ1 is the angle of loll, the value of GM for small inclinations about the loll position, will be given by the slope of the GZ curve at that point, i.e.
6.14
⎛ ⎝
GZ = sin ϕ⎜ GM +
1 2
⎞ ⎠
BM tan 2 ϕ ⎟
1 ⎞ = cos ϕ⎛ ⎜ GM + BM tan 2 ϕ ⎟ + sin ϕBM tan 2 ϕ sec 2 ϕ = 0 d ϕ 2 ⎝ ⎠
dGZ
It is important to recognize the loll condition as it is potentially dangerous. It cannot, unlike heel, be corrected by applying a counter moment to the ship. This would merely cause the ship to flop over rapidly to a larger angle on the other side.
6.6. Statical Stability Curve A typical curve of GZ variation with heel angle is shown in Figure 6.11. This is known as a curve of static stability or GZ curve. Important features of this curve are as follows;
• At small angles GZ ≅ GMϕ . The slope of the GZ curve near the origin is given by dGZ d (GMϕ) = = GM . Thus, if the slope of the curve at the origin is extrapolated to a value of d ϕ d ϕ ϕ = 1 radian the ordinate has a value equal to GM, the metacentric height. • As ϕ increases beyond small values, the slope of the GZ curve usually increases above the • •
initial slope. For a wall sided ship, this is predicted by the wall sided formula for GZ (BM is always positive). For a round bilge vessel, this increase in slope may be small or even negative. At point B, there is a point of inflexion in the GZ curve and the slope begins to decrease. This change is associated with immersion of the deck edge or emergence of the turn or bilge. At this point, the waterplane width ceases to increase and begins to reduce. At point C, GZ reaches its maximum value GZmax. A steady overturning moment applied to the ship of value greater than ΔGZ max would cause it to capsize.
• At point D, GZ becomes zero. This point is called the point of vanishing stability and distance •
OD is called the range of stability. The area under the GZ curve is a measure of the work done in steady conditions to heel the ship. It may also be called the dynamical stability, as it is related to the energy absorbing capability of the ship in roll.
GZ curves are normally prodyced for a range of loading conditions. GZ curve is normally corrected for any expected free surface effects in the condition concerned. There are some deficiencies in GZ curves. For example C
• The GZ curve makes no allowance for changes in centre of gravity at very large heel angles, GZ (m) • •
due to cargo shift, for example B It makes no allowance for water flooding into the ship at large angles, such as through deck openings or engine intakes The GZ curve assumes quasi-static conditions which are certainly not present in a heavy sea
GMsinϕ+1/2BMtan2ϕsinϕ P
P
GMϕ
6.15
GMsinϕ
A range of stability
D
Heel angle
Figure 6.11. Statical stability curve
6.7. Cross Curves of Stability These curves show the relationship between the righting levers KZ and various displacements for a ship inclined at a constant angle. The displacement cover the range from lightship to maximum possible loading condition.
Figure 6.12. Cross curves of stability
6.8. Free Surface Effect We have so far assumed that the contents of the ship do not move as the ship heels over. In practice, ships contain tanks of liquids such as oil and water. If these tanks are pressed full then the contained fluid will behave as though it were solid. If, however, the tank is only partly full, the contained liquid will move towards the downward side as the ship heels. In Figure 6.13 the local centre of gravity of the liquid of density ρ1 moves to g´. This has the effect of shifting the ship’s centre of gravity G to G´, closer to the line of action of the buoyancy force, and thus reduces the righting moment. For small angles of heel, it is shown in Figure 6.13, that the new righting arm is given by
G ' Z' ≅ GZ − GG' ≅ GZ − GG' ' ϕ ≅ G' ' Mϕ ≅ (GM − GG' ' )ϕ There is an apparent loss of metacentric height, of value GG´´. Taking the moments about G,
ΔGG' = w 1gg ' where w1 is the weight of the liquid in the tank. We may assume that gg' ≅ bb' . Thus
6.16
gg' ≅ bb' = bmϕ =
i ϕ v
where i is the area moment of inertia of the free surface of the liquid in the tank and v is the volume of the liquid. From this we can deduce that GG ' =
ρ1 ϕ ∇ ρS i
and GG ' ' =
ρ1 ∇ ρS i
This is the apparent loss in GM due to the free surface of the liquid in the tank. If there are a number of tanks, then the total loss of GM due to the free surface effect is the sum of the individual effects, i.e. i ρ1 GG ' ' =
∑
∑∇ ρ
S
There are several points to note about the free surface effect:
• The free surface effect is always destabilizing • It is independent of the volumes of the liquids. Even a small amount of liquid can have a large • •
effect if the free surface is large The free surface effect is independent of tank position, either vertically or horizontally It is very dependent on size of the free surface, particularly on width, in the same way as ship beam has a large effect on GM
The best way to reduce the free surface effect is to subdivide the tanks so as to prevent the liquids in bulk. The subdivision must form completely separate tanks; subdivision of the free surface alone has no effect as the liquid as a whole can still move in bulk beneath the surface. The free surface effect is particularly dangerous for ships with large deck areas close to waterline level, such as a Ro-Ro ferry. The deck of a car ferry often has an area and second moment of area equal to or greater than that of the ship’s load waterplane. In the case of an accident when the deck is covered by even a shallow depth of sea water, the loss of GM is
i
∇
≅
I
∇
≅ BM
M
This loss is more than the original value of GM and the ship will therefore have negative upright L´ stability. This will cause the ship heel over rapidly and possibly capsize.
W
G´´
Z´´ L Z
G W´
B’
B
6.17 Z´
Figure 6.13. Free surface effect
g
g´
6.9. Solid Weight Shifts, Additions and Removals 6.9.1. Vertical Weight Shifts
w
M
W
G1 G
h
L
B
w
K
Figure 6.14.
If a weight of w tons already on board is raised through some distance h m, the centre of gravity will be raised to a new position such that
GG1
=
w×h
Δ
Since there is no change in the displacement or draught of the ship, there will be no change in the position of the metacentre, and therefore the metacentric height will be reduced by the amount GG 1 meters. 6.9.2. Horizontal Weight Shifts 6.18
Shifting weight horizontally, no matter where onboard it is, will always cause the ship’s centre of gravity to move in the same direction as the weight shift. Thus, horizontal weight shifts will always reduce the stability of the ship. The resulting heel angle can be calculated by
tan ϕ =
w × d Δ × GM
where w is the amount of weight shifted and d the horizontal distance the weight is shifted.
d
w M
w W G W1
ϕ
G1 L1
B
B1
L
K
Figure 6.15. 6.9.3. Weight Additions/Removals
This is a more complicated problem, compared with vertical weight shifts, because the addition or removal of weights from a ship changes the position of the centre of gravity of the ship and it also changes the draught and hence the position of the metacentre;
6.19
d w
M
W1 W
G1 G
p
h
ϕ
L1 L
B1 B
K
Figure 6.16.
Parallel sinkage
: p =
w T1
Metacentric height
ΔKG + wh Δ+w p ⎞ ΔKB + w ⎛ ⎜T + ⎟ ⎝ 2 ⎠ : KB1 = Δ+w ∇ : BM1 = BM ∇1 : GM1 = KB1 + BM1 − KG 1
Heel angle
: tan ϕ =
Height of centre of gravity
Height of centre of buoyancy Metacentric radius
: KG1
=
w × d Δ1 × GM1
6.10. Inclining Experiment The purpose of the inclining experiment is to find the KG and GM of a ship in a known condition. Owing to the importance of GM and the difficulty of estimating KG accurately, an inclining experiment is usually carried out on all new vessels. It is repeated after significant modifications to the ship during its service life. The position of the centre of gravity of the lightship could be determined by considering all the weights such as steel, outfit and fittings, and machinery, and ascertaining their individual centres of 6.20
gravity, from which could be calculated the position of the centre of gravity for the complete ship. For a particular condition of loading the weights of all the items which the ship carries would then have to be added on at their appropriate centres of gravity. In this way the position of the centre of gravity of the loaded ship could be determined. Whilst it is customary to calculate the volumes and centres of gravity of all the hold, fuel, fresh water and ballast spaces, etc., from the drawings the calculation of the lightship weight and the position of its centre of gravity would be a long and tedious process and the accuarcy of the result would be in some doubt. For this reason the lightship weight and the position of its centre of gravity are determined experimentally. The inclining experiment is based on applying a known heeling moment and then measuring the resulting angle of heel. The heel can be measured by means of a long pendulum with length l , suspended, preferably into the hold, and free to move. The heeling angle, ϕ (in radians), is
ϕ=
a l
where a is the deviation of the pendulum. It is advisable to suspend the small weights at the end of the pendulum into oil or some other kind of liquid to dampen oscillations. Consider a known weight of amount w moved transversely across the ship a distance d. The heeling moment wd must be balanced by the restoring moment ΔGMϕ , where ϕ is the small angle of heel involved. Thus wd GM = Δ tan ϕ All the quantities on the right hand side of the equation are measured as accurately as possible. The ship’s displacement is found from the hydrostatic curves for the draughts measured during the experiment.
6.21
Figure 6.17. Inclining experiment 6.10.1. Practical Execution of the Inclining Experiment
The inclining experiment must be carried out in calm water in still conditions. The GM as measured will include any free surface effects due to liquids on board. This virtual GM must be adjusted to a solid GM or a standard liquid condition. The information for this adjustment is obtained by dipping all tanks on board at the time of the experiment and calculating the free surface effect in each case. Draughts and/or freeboards should be read immediately before or immediately after the test. All persons required to be on board for the test should be in location during these readings. Draughts should be read forward and aft and draughts or freeboards lifted amidships to determine hog or sag. At the same time the density of the water should be recorded at various locations along the vessel. The weights used should be sufficient to give a total inclination of 1 1/2° to 3 ° on each side. It may be necessary to have a larger inclination in small vessels in order to get sufficient deflection of the pendulum. However, care should be taken to ensure: (a) that the angle of heel does not exceed the angle at which GZ no longer equals GM sin α . (b) that the deflection of the pendulum for each shift is sufficiently large to give meaningful readings, and (c) that changes in the waterplane area during the shifts are kept as small as possible. In this regard vessels with appreciable flare at the waterline should be carefully considered and the angle of inclination should not exceed 1° . The weights to be used in the experiment should, preferrably consist of 4w tons, i.e. 2w tons on each side of the ship. The inclinations must be recorded with the following weight movements, each through distance d meters
• • • • • •
w from port to starboard 2w from port to starboard 2w from starborad to port to chack initial zero w from starboard to port 2w from starboard to port 2w from port to starboard to check initial zero
Thus, the average inclination in degrees may be obtained for a movement of w tons through d meters. As w is moved across the deck the centre of gravity of the ship moves from G to G 1 and the resulting angle of heel can be found as
tan ϕ = In the inclining experiment the values of w, d, calculated.
wd
ΔGM Δ and ϕ are all known, and thus GM may be
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The weights to be used must produce a reasonable heel angle. They can be determined by estimating GM, the moment arm, and the angle of heel. The estimation of GM can be made by preliminary weight and KG calculations. Then the required weight is
w
=
ΔϕGM d
6.10.2. The Accuracy of the Inclining Experiment
The accuracy of the inclining experiment depends on many factors and measures must be taken to reduce the possibility of errors. These measures may include
• A calm day with no wind should be chosen. If possible the test should not be conducted in wind • • • • • • • • • •
• • •
• • •
conditions heavier than a light breeze. The ship should ideally be inclined in a dry dock in order to avoid any disturbances caused by passing ships. Failing this, it should be moored at a buoy with head to wind. The depth of water under the hull should be sufficient to allow the vessel to move freely Excessive accumulations of rain, snow or ice should be removed before the test All moorings should be slack and gangways removed All loose items such as derricks, boats, etc. should be secured The ship should be as nearly complete as possible, and an accurate list should be made of items to go on to complete, items to be taken off and items already on board which have to be shifted As few men as possible should remain on board, and those that do should remain on the centreline All tanks should be empty or pressed full. However, it is emphasized that empty tanks are preferred for an accurate test The vessel should be upright or within 1/2° from upright Heavy trim should be avoided. The vessel should be as close to the design trim as possible and should not differ from this by more than 0.01 LBP. Unless the vessel is at the designed trim, soundings of tanks will not give true readings and the inclining stability calculation will require to be calculated for the trimmed waterline All measuerements should be taken with care, and be checked as the experiment proceeds Special emphasis should be placed on draught readings and control of heeling weights At least two pendulums should be used, one at each end of the vessel. The length of each pendulum should be as long as possible and they should be located in an area protected from the wind. Pendulum weights should be suspended into a liquid such as oil to dampen excessive movement. The natural period for the pendulum and the roll period of the vessel should not be close to each other At least 4 readings of moment versus heeling angle should be taken The maximum heeling angle should be in the range 5-100
6.10.3. Free Surface Corrections
After the value of GM has been calculated for the ship in the inclined condition there may be a correction to be made for the presence of free surfaces in double bottom, or other tanks. The presence of free surfaces reduces the actual value of GM by an amount given by
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δGM = ∑
ρ1 ∇ ρS i
where i is the transverse moment of inertia of each free surface about its own centreline, ∇ the displacement volume, ρ1 the density of liquid with free surface, and ρ S the density of water in which the ship floats.
6.11. Determination of Ship’s Metacentric Height (GM) by Means of the Rolling Period Test This method of approximating a ship’s initial metacentric height (GM) may be used for small vessels, where it is not practicable to carry out an inclining experiment, or as a supplement to an inclining experiment. The method depends upon the relationship between the metacentric height and the rolling period in terms of the extreme breadth of the vessel. It should be noted that a roll test is not acceptable as a basis for determining a ship’s stability characteristics, where, the vessel is of hard chine construction (i.e. of knuckled hull form) or is fitted with bilge keels, or reasonable doubt exists as to the adequacy of the intact stability characteristics of the ship, over its complete range of operating conditions. The rolling period required is the time for one complete oscillation of the vessel. To ensure the most accurate results in obtaining this value the following precautions should be observed:
• The test should be conducted with the vessel in harbour, in smooth water with the minimum •
•
•
interference from wind and waves. Starting with the vessel at the extreme end of a roll to one side (say port) and the vessel about to move towards the upright, one complete oscillation will have been made when the vessel has moved right across to the other extreme side (i.e. starboard) and returned to the original starting point and is about to commence the next roll. By means of a stop watch, the time should be taken for about five (5) of these complete oscillations; the counting of these oscillations should begin when the vessel is at the extreme end of a roll. After allowing the roll to completely fade away, this operation should be repeated at least twice more. If possible, in every case the same number of complete oscillations should be timed to establish that the readings are consistent, i.e. repeating themselves within reasonable limits. Knowing the total time for the total number of oscillations made, the time for one complete oscillation can be calculated. The vessel can be made to roll by rhythmically lifting up and putting down a weight as far off middle-line as possible; by pulling on the mast with a rope; by people running athwartships in unison; or by any other means. However, and this is most important, as soon as this forced rolling has commenced the means by which it has been induced must be stopped and the vessel allowed to roll freely and naturally. If rolling has been induced by lowering or raising a weight it 6.24
• •
• •
is preferable that the weight is moved by a dockside crane. If the ship’s own derrick is used, the weight should be placed on the deck, at the middle-line, as soon as the rolling is established. The timing and counting of the oscillations should only begin when it is judged that the vessel is rolling freely and naturally, and only as much as is necessary to accurately count these oscillations. The mooring should be slack and the vessel "breasted off" to avoid making any contact during its rolling. To check this, and also to get some idea of the number of complete oscillations that can be reasonably counted and timed, a preliminary rolling test should be made before starting to record actual times. Care should be taken to ensure that there is a reasonable clearance of water under the keel and at the sides of the vessel. Weights of reasonable size which are liable to swing (e.g. a lifeboat), or liable to move (e.g. drum), should be secured against such movement. The free surface effects of slack tanks should be kept as small as is practicable during the test.
Having calculated the period for one complete oscillation, say T seconds, the metacentric height GM can be calculated from the following formula:
⎛ fB ⎞ GM = ⎜⎜ ⎟⎟ ⎝ Tr ⎠
2
Where: f = factor for the rolling period B = breadth of the ship in metric units, T = time for a full rolling period in seconds (i.e. for one oscillation ‘to and fro’ port-starboard-port, or vice versa). The factor ‘f’ is of the greatest importance and data from a number of tests have been used in determining the following typical values: (A) For unloaded fishing boats (but with fuel, stores, and equipment). f = 0.761 (B) For vessels of normal size (excluding tankers): (a) empty ship or ship carrying ballast f = 0.88 (b) ship fully loaded and with liquids in tanks comprising the following percentage of the total load on board (i.e. cargo, liquids, stores, etc.) 1. 20 percent of total load f = 0.78 2. 10 percent of total load f = 0.75 3. 5 percent of total load f = 0.73
The stated values are mean values. Generally, f - values are within ± 0.05 of those given above. It must be noted that the greater the distance of masses from the rolling axis, the greater the rolling coefficient will be. Therefore, it can be expected that:
• the rolling coefficient for an unloaded ship, i.e. for a hollow body, will be higher than that for a loaded ship;
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• the rolling coefficient for a ship carrying a great amount of bunkers and ballast - both groups are usually located in the double bottom, i.e. far away from the rolling axis - will be higher than that of the same ship having an empty double bottom. The above recommended rolling coefficients were determined by tests with vessels in port and with their consumable liquids at normal working levels; thus, the influences exerted by the vicinity of the quay, the limited depth of water and the free surface of liquids in service tanks are covered. Experiments have shown that the results of the rolling test method get increasingly less reliable the nearer they approach GM-values of 0.20 metres and below. For the following reasons, it is not generally recommended that results be obtained from rolling oscillations taken in a seaway:
• Exact coefficients for tests in open waters are not available. • The rolling periods observed may not be free oscillations but forced oscillations due to seaway. • Frequently, oscillations are either irregular or only regular for too short an interval of time to •
allow accurate measurements to be observed. Specialized recording equipment is necessary.
However, sometimes it may be desirable to use the vessel’s period of roll as a means of approximately judging the stability at sea. If this is done, care should be taken to discard readings which depart appreciably from the majority of other observations. For oscillations corresponding to the sea period and differing from the natural period at which the vessel seems to move should be disregarded. In order to obtain satisfactory results, it may be necessary to discard a considerable number of observations. In view of the foregoing circumstances, it needs to be recognized that the determination of the stability by means of the rolling test in disturbed waters should only be regarded as a very approximate estimation.
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