The Geometry of Ships

24 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
incidentally by the elastic and/or plastic deformation accompanying
stress and welding shrinkage that occurs
during forced assembly of the product. In any case, introduction
of in-plane strain is an expensive manufacturing
process, and it is highly desirable to minimize the
amount of it that is required. Also, it is very valuable to
predict the flat blank outlines accurately so each plate,
following forming, will fit “neat” to its neighbors, within
weld-seam tolerances.
There exist traditional manual lofting methods for
plate expansion, some of which have been “computerized”
as part of ship production CAD/CAM software.
Typically, these methods do not allow for the in-plane
strain and, consequently, they produce results of limited
utility for plates that are not nearly developable. A survey
by Lamb (1995) showed that expansions of a test
plate by four commercial software systems yielded
widely varying outlines.
Letcher (1993) derived a second-order partial differential
equation relating strain and Gaussian curvature
distributions, and showed methods for numerical solution
of this “strain equation” with appropriate boundary
conditions. In production methods where plates are subjected
to deliberate compound forming before assembly,
this method has produced very accurate results, even for
highly curved plates. When the forming is incidental to
stress applied during assembly, results are less certain,
as the details of the elastic stress field are not taken into
account, and the process depends to some degree on the
welding sequence (Fig. 20).
Fig. 20 Plate expansion by numerical solution of the “strain equation.”
(a) The plate is defined as a subsurface between snakes representing the
seams. (b) The required strain distribution is indicated by contours, which
are somewhat irregular on account of the discretization of the plate into
triangular finite elements.
Fig. 21 The “shell expansion” drawing is a 1:1 mapping of the hull
surface to a planar figure used for representing layout of structural
elements such as longitudinal stiffeners.
The “shell expansion” drawing (Fig. 21), used to plan
layout of frames and longitudinal stiffeners, is a quite different
mapping that produces a flat expansion of a curved
surface. The rule of correspondence is that each point on
the 3-D hull is mapped to a point on the same transverse
station, at a distance from the drawing base line that corresponds
to girth (arc-length) measured along the station
from the keel, chine, or a specified waterline.
4.15 Intersections of Surfaces. Finding intersections
between surfaces is in general a difficult problem, requiring
(in all but the simplest cases) iterative numerical
procedures with relatively large computational costs
and many numerical pitfalls. Intersection between two
parametric or two implicit surfaces is especially difficult
and expensive; one of each is a more tractable, but nevertheless
thorny, problem.
If we have two parametric surfaces X 1 (u, v) and X 2 (s,
t), the governing equations are:
X 1 (u, v) X 2 (s, t) (42)
i.e., three (usually nonlinear) equations in the four unknowns
u, v, s, t. The miscount between equations and
unknowns reflects the fact that the intersection is usually
a curve, i.e., a one-dimensional point set. Some of
the difficulties are as follows:
• The supposed intersection may not exist.
• The intersection may have varying dimensionality.
Two surfaces might intersect only at isolated points
(where they are tangent), in one or more closed or open
curves, or might have entire 2-D regions in common, or
a mixture of these.
• When the intersection is at a shallow angle, the equations
are ill-conditioned.
• Intersection curves can have cusps, branches, and
other singularities that make them hard to follow.
• It is difficult to get good starting locations. For example,
if the surfaces are approximated by meshes, it is
quite possible for the meshes to have no intersection
while the surfaces do.