The Geometry of Ships

THE GEOMETRY OF SHIPS 49
Molded displacement for a metal vessel is the volume
of the molded form, i.e., inside of shell or outside of
frames — the reference or control surface of the hull, exclusive
of shell plating and other appendages. (Yes, the
shell plating is considered an “appendage”!) Total or
gross displacement includes the volume of shell plating
and other appendages such as rudder, propeller, shaft
bossings, sonar domes, bilge keels, etc. A thruster tunnel,
moon pool, or other flooded space removed from
the displacement of the molded form should be treated
as a negative appendage.
In a single-screw cargo vessel, the volume of shell
plating is typically less than 1 percent of the molded
volume (as little as 0.5 percent for the largest ships),
and volume of other appendages is only about 0.1 to
0.2 percent.
11.1.2 Longitudinal Center of Buoyancy (LCB).
x B is found by dividing the x-moment of displaced volume
by the displaced volume, equation (77). The longitudinal
coordinate of the vessel’s center of mass must be
at x B in order for the vessel to float without trim at this
displacement.
If S(x) is the section area curve at a particular draft,
(118)
Alternatively, the integration can be performed vertically.
If A wp (z) is the area of the waterplane at height z
above base, and x w (z) is the x-position of its centroid, then
x B
A wp
(z) x W
(z) dz
A wp
(z) dz
(119)
The LCB is commonly expressed as a percentage of
waterline length, from bow to stern; or may be in units of
length, usually measured forward or aft of the midship
section. It is usually in the range from 1 percent LWL forward
to 5 percent LWL aft of midships. There is fairly
consistent tank-test evidence that minimum resistance
for displacement vessels is obtained with LCB at 51 to 52
percent of waterline length (referring to the molded
form).
11.1.3 Vertical Center of Buoyancy (VCB). z B is
found by dividing the z-moment of displaced volume by
the displaced volume, equation (77). VCB has an important
effect on initial stability, equation (106).
If S(x) is the section area curve at a particular draft,
and z s (x) is the height of the centroid of the transverse
section, then
z B
x B
xS(x) dx
S(x) dx
S(x) z S
(x) dx
S(x) dx
A wp
(z) x W
(z) dz
(120)
Alternatively, the integration can be performed vertically.
If A wp (z) is the area of the waterplane at height z,
xS(x) dx
S(x) z S
(x) dx
then
z B
(121)
VCB is expressed in length units above the base plane.
11.1.4 Waterplane Area and Incremental
Displacement. The waterplane area A wp has units of
length squared. Its use is primarily to furnish a ready calculation
of the incremental displacement due to a small
additional immersion. The volume dV added by a change
dz in draft is A wp dz, therefore dV/dz A wp . In SI units,
this is usually expressed in tonnes per cm immersion, for
salt water TPC 1.025A wp /100 0.01025A wp , with A wp
in square meters.
11.1.5 Longitudinal Center of Flotation (LCF). x F
(center of flotation, CF) is the centroid of waterplane
area; this is effectively the pivot point for small changes
of trim or heel. If b(x) is the breadth of waterplane as a
function of x, the LCF is calculated as:
x F
A wp
(z) zdz
A wp
(z) dz
b(x) xdx
(122)
b(x) dx A wp
Like LCB, LCF is usually expressed as a percentage of
waterline length, or a distance forward or aft of midships.
There is a general experience that LCF 2 to 4 percent
aft of LCB is advantageous in providing a favorable
coupling between heave and pitch motions, resulting in
reduction of pitching motions and of added resistance in
head seas.
11.1.6 Transverse Metacenter. In Section 9.6, equation
(106) was given relating transverse initial stability to
geometric properties of the displaced volume and waterplane
area (and to vertical center of gravity z G ):
dL/d pg (z Mt z G ) (123)
where z Mt z B I xx /.
z Mt z G is called transverse metacentric height, not
to be confused with height of metacenter, which means
z Mt alone. The term I xx / is called transverse metacentric
radius, and is denoted BM T .
The curves of form need to reflect geometric attributes,
which are fixed in the vessel geometry, as opposed
to variable attributes such as mass distribution. KM T ,
KB, and BM T are the candidates from the above list. KM T
is generally chosen over BM T because it is one step
closer to the initial stability, which is the real quantity of
interest.
It is generally desirable, of course, for a vessel to have
positive initial stability. However, too large an initial stability
(unless combined somehow with large mass moment
of inertia about the longitudinal axis, or large roll
damping) produces a quick rolling response (short period,
high natural frequency) which is uncomfortable
and an impediment to many shipboard operations.
Consequently, most cargo and passenger vessels operate
with GM T in the range 0.5 to 1.5 m.
A wp
(z) zdz
b(x) dx