The Geometry of Ships

42 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
M y
M z
ydxdydz m y
(x) dx
zdxdydz m z
(x) dx
(78)
(79)
The 3-D centroid (CB, center of buoyancy) is the point
with coordinates M x /V, M y /V, M z /V.
For complex bodies such as offshore structures, the
cutting of sections is difficult, and the sections themselves
may be quite complex. Also, section properties can
change rapidly in a short distance, often requiring a large
number of closely spaced sections to achieve an adequate
representation. A fully 3-D approach to hydrostatic
calculations is then advantageous. A 3-D discretization of
the body, analogous to approximation of sections by
polygons in 2-D, then becomes preferable. We assume the
discretization is in the form of a triangle mesh having all
watertight junctions (i.e., no gaps between adjacent triangles).
Applying Archimedes’ principle, the hydrostatic
force and moment can be calculated from the volume and
centroid of the solid enclosed by the triangle mesh surface
including the waterplane area. We calculate this
solid as the sum of a set of triangular prismatic elements,
each formed by taking one triangular panel, projecting it
onto the plane z 0, and connecting the panel to its projection
with three vertical trapezoidal faces (Fig. 34). The
volume of the prism will be positive if the panel faces
downward, i.e., if its outward normal has a negative z
component; otherwise, the prism volume is negative.
The corner points of the panel are x 1 , x 2 , x 3 , numbered
in counterclockwise order as viewed from the
water. (It is essential that each triangle have this consistent
orientation.) Half the cross-product of two sides
a (x 2 x 1 ) (x 3 x 1 )/2 (80)
is a vector normal to the panel (pointing outward, into
the water), with magnitude equal to the panel area.
Points on the panel are parameterized with parameters
u, v as follows:
x(u, v) x 1 (x 2 x 1 )u (x 3 x 1 )v (81)
where the range of v is 0 to 1 and the range of u is 0 to
1 v. In particular, over the panel surface,
z z 1 (z 2 z 1 )u (z 3 z 1 )v (82)
We perform integrations over the triangle that is the
vertical projection of the panel onto the z 0 plane (the
top of the prism), e.g.,
Q(x, y) dx dy 2 A
(83)
where Q is any function of x and y, and A a 3 , the
(signed) area of the waterplane triangle. (A is positive
for a panel whose normal has a downward component.)
The volume and moments of volume of the prism are
evaluated as follows:
V A (z 1 z 2 z 3 ) / 3 (84)
M x A [(x 1 x 2 x 3 )(z 1 z 2 z 3 )
+ x 1 z 1 + x 2 z 2 + x 3 z 3 ]/12
M y A [(y 1 y 2 y 3 )(z 1 z 2 z 3 )
+ y 1 z 1 + y 2 z 2 + y 3 z 3 ]/12
(85)
(86)
M z A [(z 1 z 2 z 3 ) 2 z 2 1 z 2 2 z 2 3] / 24 (87)
The waterplane area of this prism is A, with its
centroid at {(x 1 x 2 x 3 )/3, (y 1 y 2 y 3 )/3, 0}.
Its contributions to the waterplane moments of
inertia are:
I xx A [y 2 1 y 2 2 y 2 3 y 1 y 2 y 2 y 3 y 3 y 1 ] / 6 (88)
1
0
1v
0
Q(x, y) du dv
I xy A [(x 1 x 2 x 3 )(y 1 y 2 y 3 )
x 1 y 1 x 2 y 2 x 3 y 3 ]/12
(89)
Fig. 34
Triangular prismatic element used for hydrostatic calculations
with a triangular-paneled discretization.
I yy A [x 2 1 x 2 2 x 2 3 x 1 x 2 x 2 x 3 x 3 x 1 ] / 6 (90)
9.5 Weight Estimates, Weight Schedule. Archimedes’
principle states the conditions for a body to float in
equilibrium:
• its weight must be equal to that of the displaced fluid,
and
• its center of mass must be on the same vertical line as
the center of buoyancy.
The intended equilibrium will only be obtained if the
vessel is actually built, and loaded, with the correct
weight and weight distribution. Preparation of a reasonably
accurate weight estimate is therefore a critical step
in the design of essentially any vessel, regardless of size.
Enormous expense and disappointment await the designer
who shortcuts this element of design.