4.9 Stability at Large Angles of Inclination

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4.9 Stability at Large Angles of Inclination
The transverse metacenter height is a measure
of the stability under initial stability (aka
small angle stability). When the angle of
inclination exceeds 5 degrees, the metacenter can
be no longer regarded as a fixed point relative
to the ship. Hence, the transverse metacenter
height (GM) is no longer a suitable criterion for
measuring the stability of the ship and it is
usual to use the value of the righting arm GZ for
this purpose.
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• The Derivation of Atwoods Formula

W.L. when the ship is at upright
position. W.L. when the ship is
inclined at an angle ?. If the ship section is
not vertically sided, the two W.L., underneath
which there must be the same volume, do not
intersect on the center line (as in the initial
stability) but at S.
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• GZ vs.

For each angle of ?, we compute GZ, the righting
arm. The ship is unstable beyond B. (even if the
upsetting moment is removed, the ship will not
return to its upright position). From 0 to B,
the range of angles represents the range of
stabilities.
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Ex. Righting arm of a ship vertically sided (A
special example to compute GZ at large angle
inclinations) Transverse moment of volume
shifted
Volume
arm
Transverse shift of C.B.
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Ex. Righting arm of a ship vertically sided (A
special example to compute GZ at large angle
inclinations) Similarly, vertical moment of
volume shifted
Volume
Arm
Vertical shift of C.B.
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Ex. Righting arm of a ship vertically sided (A
special example to compute GZ at large angle
inclinations)
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• Cross Curves of Stability

It is difficult to ascertain the exact W.L. at
which a ship would float in the large angle
inclined condition for the same displacement as
in the upright condition. The difficulty can be
avoided by obtaining the cross curves of
stability (see p44).

• How to Computing them
• Assume the position of C.G. (not known exactly)
• W.L. I - V should cover the range of various
  displacements which a ship may have.

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• Cross Curves of Stability

• Computation Procedures
• The transverse section area under waterline I,
  II, III, IV, V
• The moment about the vertical y-axis (passing
  through C.G)
• By longitudinal integration along the length, we
  obtain the displacement volume, the distances
  from the B.C. to y-axis (i.e. the righting arm
  GZ) under the every W.L.
• For every we obtain
• Plot the cross curves of stability.

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Cross Curves of Stability
These curves show that the righting arm (GZ)
changes with the change of displacement given the
inclination angle of the ship.
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For the sake of understanding cross curves of
stability clearly, here is a 3-D plot of cross
curves of stability.
The curved surface is
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• Curve of Static Stability

• Curve of static stability is a curve of
  righting arm GZ as a function of angle of
  inclination for a fixed displacement.
• Computing it based on cross curves of stability.
• How to determine a curve of statical stability
  from a 3-D of cross curves of stability.
  (C.C.S.), e.g., the curve of static stability is
  the intersection of the curved surface and the
  plane of a given displacement.
• Determining a C.S.S. from 2-D C.C.S. is to let
  displcement const., which intersects those
  cross curves at point A, B,, see the figure.

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GZ
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• Influences of movement of G.C on curve of static
  stability

1. Vertical movement (usually due to the correction
   of G.C position after inclining experiment.)

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• Influences of movement of G.C on curve of static
  stability

2. Transverse movement (due to the transverse
movement of some loose weight)
Weight moving from the left to the right
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• Features of A Curve of Static Stability

• Rises steadily from the origin and for the first
  few degrees is practically a straight line.
• Near the origin GZ ? slope slope
  ?, why?
• 2. Usually have a point of inflexion, concave
  upwards and concave downwards, then reaches
  maximum, and afterwards, declines and eventually
  crosses the base (horizontal axis).

1 radian
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The maximum righting arm the range of stability
are to a large extent a function of the
freeboard. (the definition of
freeboard) Larger freeboard Larger
GZmax the range of stability Using the
watertight superstructures Larger GZmax
the range of stability
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• 4.10 Dynamic Stability
• Static stability we only compute the righting
  arm (or moment) given the angle of inclination.
  A true measure of stability should considered
  dynamically.
• Dynamic Stability Calculating the amount of work
  done by the righting moment given the inclination
  of the ship.

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• Influence of Wind on Stability (p70-72)
• Upsetting moment due to beam wind

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When the ship is in upright position, the steady
beam wind starts to blow and the ship begins to
incline. At point A, the M(wind) M(righting),
do you think the ship will stop inclining at A?
Why?
The inclination will usually not stop at A.
Because the rolling velocity of a ship is not
equal to zero at A, the ship will continue to
incline. To understand this, lets review a
simple mechanical problem
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The external force F constant The work
done by it If at the work done by the spring
force R,
Hence, the block will continue to move to the
right. It will not stop until
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In a ship-rolling case Work done by the upright
moment Work done by the wind force
It will stop rolling (at E) In a static
stability curve or simply,
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• Consideration in Design (The most sever case
  concerning the ship stability)
• Suppose that the ship is inclining at angle
  and begins to roll back to its upright position.
  Meanwhile, the steady beam wind is flowing in the
  same direction as the ship is going to roll.

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• Standards of Stability ships can withstand
• winds up to 100 knots
• rolling caused by sever waves
• heel generated in a high speed turn
• lifting weights over one side (the C.G. of the
  weight is acting at the point of suspension)
• the crowding of passengers to one side.

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• 4.11 Flooding Damaged Stability
• So far we consider the stability of an intact
  ship. In the event of
• collision or grounding, water may enter the ship.
  If flooding is
• not restricted, the ship will eventually sink.
  To prevent this, the
• hull is divided into a number of watertight
  compartments by
• watertight bulkheads. (see the figure)
• Transverse (or longitudinal) watertight bulkheads
  can
• Minimize the loss of buoyancy
• Minimize the damage to the cargo
• Minimize the loss of stability

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• Too many watertight bulkheads will increase cost
  weight of the ship. It is attempted to use the
  fewest watertight bulkheads to obtain the largest
  possible safety (or to satisfy the requirement of
  rule).
• Forward peak bulkhead (0.05 L from the bow)
• After peak bulkhead
• Engine room double bottom
• Tanker (US Coast Guard) Double Hull (anti
  pollution)
• This section studies the effects of flooding on
  the
• hydrostatic properties
• and stability

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• Trim when a compartment is open to Sea

If W1L1 is higher at any point than the main
deck at which the bulkheads stop (the bulkhead
deck) it is usually considered that the ship will
be lost (sink) because the pressure of water in
the damaged compartments can force off the
hatches and unrestricted flooding will occur all
fore and aft.
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• (1) Lost buoyancy method

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Ex. p121-123 A vessel of constant rectangular
cross-section L 60 m, B 10 m, T 3 m. ZG
2.5 m l0 8 m.
2) Parallel sinkage
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3) Draft at midway between W0L0 W1L1
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Moment for Trim per meter
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if
Find trim. MTI ( at )
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• (2) Added Weight Method (considering the loss of
  buoyancy as added weight)
• also a Trial error (iterative) method
• 1) Find added weight v under W0L0. Total
  weight W v
• 2.) According to hydrostatic curve , determine
  W1L1 (or T) trim (moment caused by the added
  weight MTI).
• 3.) Since we have a larger T, and v will be
  larger, go back to step 1) re-compute v.
• The iterative computation continues until the
  difference
• between two added weights v obtained from the two
  
• consecutive computation is smaller than a
  prescribed error
• tolerance.

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• Stability in damaged condition

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• Asymmetric flooding
• If the inclination angle is large, then the
  captain should let the corresponding tank
  flooding. Then the flooding is symmetric.
• If the inclination angle is small,

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• Floodable length and its computation
• Floodable Length The F.L. at any point within
  the length of the ship is the maximum portion of
  the length, having its center at the point which
  can be symmetrically flooded at the prescribed
  permeability, without immersing the margin line.

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• Bulkhead deck The deck tops the watertight
  bulkhead
• Margin line is a line 75 mm (or 3) below the
  bulkhead at the side of a ship
• Without loss of the ship When the W.L. is
  tangent to the margin line.
• Floodable length (in short) The length of (part
  of) the ship could be flooded without loss of the
  ship.
• Determine Floodable length is essential to
  determine
• How many watertight compartments (bulkheads)
  needed
• Factor of subdivision (How many water
  compartments flooded without lost ship)

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8) Factor of Subdivision F Factor of
subdivision is the ratio of a permissible length
to the F.L. For example, if F is 0.5, the ship
will still float at a W.L. under the margin line
when any two adjacent compartments of the ship
are flooded. If F is 1.0, the ship will still
float at a W.L. under the margin line when any
one compartment of the ship is flooded. Rules
and regulations about the determination of F are
set by many different bureaus all over the world
(p126-127)