Theory
The method
assumes that forces due to grounding are buoyancies applied at the
points of grounding. These buoyancies are derived by treating the
ground as a very dense liquid.
Consider a
point on the vessel where there is a present or potential contact
between the vessel and the ground. This will be called a "grounding
point". The surface of the ground near such a point is set
to a certain level relative to the surface of the waterplane. This
represents the depth of the water at the grounding point. When the
vessel tries to sink into the ground, it receives a large buoyant
force at the grounding point which prevents it from penetrating
very far into the ground. Conversely, if the vessel is raised up
(relative to the ground) so that the grounding point is actually
above the ground surface, the buoyant force due to the ground becomes
zero.
Just as the
buoyant parts of the vessel receive upward, buoyant forces from
the water which depend on the depth of immersion, so the grounding
points receive upward buoyant forces which depend on their depths
of immersion into the ground.
In the case
of the water buoyancy, the displaced volume of water (which also
depends on the exact shape of the vessel) determines the magnitude
of the force. In the case of the grounding buoyancy, the mechanics
may be quite different. However, if the "density" of the
ground is very great relative to the density of the water, then
for a given increment of weight on the vessel there will be little
penetration into the ground relative to the sinkage which would
take place due to the same weight increment if the ground were not
present. In other words, this model effectively "stops"
the vessel from sinking much further when it hits the ground (which
is all it needs to do) even if the exact mechanics of the "grounding
buoyancy" are not precisely modeled. When a weight is added
to the vessel after it is aground, nearly all of the opposing buoyancy
required to support the weight comes from the ground. In order to
achieve this, the only requirement is that the ground buoyancy vs.
penetration function be steep relative to the water buoyancy vs.
sinkage function.
Implementation
In applying
this theory, GHS makes the assumption that the ground buoyancy vs.
penetration function always has the form,
b = C * (d
 d0)2 for d > d0 Equation 1
b = 0 for d d0
where b is the
buoyant force at the grounding point, d is the depth of the grounding
point below the waterplane, and d0 is the depth of the ground surface
below the waterplane near that point.
Figure 1 depicts
this function.
The constant
C is chosen such that the buoyant force becomes sufficiently large
to "stop" the vessel so that it sinks not very far into
the ground.
GHS assumes
that "not very far" means a small distance relative to
the size of the vessel. If the user allows GHS to select the value
of C automatically, it selects a value which would produce a buoyant
force equal to the entire weight of the vessel if the ground/vessel
penetration were 0.2% of the length of the vessel. The user is free
to provide a different value, as will be shown below.
Note that this
is one of the simplest ways of meeting the requirement that the
grounding buoyancy respond sharply to the penetration. It is not
intended to model any other characteristic of the ground/vessel
interaction. Unlike real ground, this model springs back when the
penetration is reduced: the same buoyancy appears at the same penetration
regardless of whether it is approached from a lesser or a greater
penetration.
Equation 1
also allows an effective "waterplane area" to be derived
at a given penetration by taking the first derivative:
a*D = ½
C * (d  d0) for d > d0 Equation 2
a*D = 0 for
d d0
where D is
the density of the water (weight per volume), and a is the effective
waterplane area. In this manner the total effective waterplane area,
center and moments of inertia can be calculated (assuming, for inertia
purposes, that the grounding point is truly a point).
While this
contribution to the effective waterplane area makes it possible
to derive the usual weightto immerse, center of flotation and
GM values in a grounded condition, these values have large changes
for small changes in the vessel's draft, trim and heel due to the
nonlinearity of equation 1.
The number
of grounding points required to model a particular grounding will,
of course, depend on the details of the situation. GHS allows as
many as 20 grounding points. Note that a single grounding point
may be sufficient to represent a large area of contact with the
ground. The location of the point should be close to the center
of such an area.
Grounding points
are defined through an extension of the ADD command in a manner
analogous to the definition of weight items; viz. description, magnitude
of force and location are given. The simplest form is
ADD "description"
b, l,t,v /GR
The /GR parameter
indicates that a grounding point is being defined, rather than a
fixed weight item. The point (l,t,v) is the location of the grounding
point (usually a point on the bottom of the hull) where the coordinates
are in the ship's coordinate system (grounding points are tied to
the vessel).
The b parameter
is the magnitude of the buoyancy due to the grounding point at the
present water depth. Therefore, the waterplane must be set up to
represent the grounded condition before the ADD command is issued.
The above form
of the command assumes that the grounding point is in contact with
the ground. If b = 0, then the depth to the ground is assumed to
be exactly the same as the depth of the grounding point. If b >
0, the depth of the grounding point is taken to be greater than
the depth of the ground; ie. enough greater to cause the force b
to result from the penetration according to equation 1. Values of
b < 0 are not allowed.
It may seem
that there is a difficulty with this method, since the value of
b will seldom be accurately unknown. However, due to firmness of
the ground, an error in the initial specification of b will usually
not have significant consequences. Note that the b value given with
the ADD command is only at the present depth. As soon as the depth
changes, the value of b automatically changes also. Therefore, if
the given value is too small, the vessel will settle down to a slightly
deeper draft. If the given value is too great it will rise up a
bit until the ground reaction makes up whatever buoyancy is missing
from the displacement of water. But the change in draft required
will often be relatively small compared with the accuracy required
in the analysis. If a more precise result is required, a twostep
process may be used wherein the initial estimate of b is refined
after the equilibrium value is found.
As with other
adjustments to the state of the vessel, GHS does not adjust the
waterplane (draft, trim and heel) until instructed to do so. Thus
the procedure is to issue one or more ADD /GR commands followed
by a SOLVE command to find equilibrium. After that, a STATUS command
may be given to examine the magnitude of the ground reaction at
each grounding point.
If the firmness
of the ground which GHS automatically assigns "by default"
is inappropriate, an additional parameter may be given. For example,
ADD "Sand
Bar", 100, 75F, 0, 0 /GR: 2.0
tells the computer
that an estimated 100 weight units of ground reaction is located
at the center keel 75 length units forward, and that the penetration
into the ground at that point is 2.0 length units. Referring to
equation 1, this means that
(dd0) = 2.0
b = 100
therefore C = 25.0.
Thus the ADD
command has had the effect of specifying the value of C for that
grounding point. (Other grounding points may use different values
of C.)
It is possible
to define a grounding point which is, at the time of definition,
not in contact with the ground. For example,
ADD "Port
Bilge", 0, 12A, 35P, 1 /GR: 2.0
As before,
the "2.0" is the penetration into the ground, but being
negative it indicates a distance above the ground surface. In other
words, this grounding point will have to go 2.0 length units deeper
before it contacts the ground. Obviously, the b value must be zero
when the penetration parameter is negative.
The firmness
of the ground in this example is at its "default" value.
If it is necessary to specify the firmness when the grounding point
is above the ground, an additional parameter may be given. For example,
ADD "Port
Bilge", 0, 12A, 35P, 1 /GR: 2.0, 3.0
In this case,
the 3.0 is the "maximum" penetration, which is defined
as the penetration which would occur if the entire weight of the
vessel (except tank loads) were pressing at that point. When making
such an estimation, it should be remembered that a ground point
represents the middle of an area of contact, and the size of the
area may increase with the penetration.
It should be
clear that unless the ground is unusually soft, great accuracy in
the setting of the ground firmness is not ordinarily necessary.
Comparison
with the Negative Weight Method
A traditional
method of representing a ground reaction is to locate a "negative
weight" at the point of grounding (GHS is still able to apply
this method also).
While the same
ground reactions can be simulated with either method, the "negative
weight" method has limited usefulness for ascertaining stability
when grounded. This is due to the inability of the negative weights
to respond to changes in the trim or heel of the vessel.
While the GM
calculated with a negative weight is valid for singlepoint grounding
where the point of ground contact happens to be under the center
of flotation, it is less valid in other cases. On the other hand,
the positive buoyancy method, by contributing to the waterplane
properties, results in a GM which is theoretically valid, though
it may change very rapidly due the inherent nonlinearity of the
ground forces.
Another advantage
of the new method is that the distribution of the ground reaction
among several grounding points is automatic.
Example
1  SinglePoint Grounding
Consider a case
of singlepoint grounding where the grounding point is to one side
of the vessel (see figure 2). For some range of heel, the vessel
would have to pivot on the point of grounding, after which it would
float free. The angle at which it would float free would depend
on the direction of heel. This is exactly the behavior which GHS
would simulate with a single grounding point. The following series
of commands could be used:
TRIM = t0 
HEEL = h0  DEPTH = d0 ADD "description" b, l,t,v /GR
MACRO H
HEEL = *+5
SOLVE
TRIM
STATUS
/
HEEL =
30P
H (12)
This produces
a series of status reports from 25 port heel to 30 starboard heel,
each report showing the ground reaction and righting arm.
Example
2  TwoPoint Grounding
Consider a twopoint grounding case as illustrated in figure 3,
and assume that longitudinal strength is to be checked. The following
commands provide the answer:
TRIM = t0  HEEL = h0  DEPTH = d0 ADD "fwd descr" b1,
l1,t1,v1 /GR
ADD "aft
descr" b2, l2,t2,v2 /GR
SOLVE TRIM
LS
Setting Depth
and Changing Tide
In a salvage
situation, when the stranded vessel is not on an even keel, it may
be necessary to set the vessel's depth by means of a measurement
taken at some point on the hull which is not near the centerline;
hence it would inconvenient to use the DEPTH or DRAFT command. In
such a case, the new HEIGHT command is useful.
The HEIGHT
command takes the height of a Critical Point relative to the water
measured perpendicular to the waterplane. Hence, any convenient
point on the vessel may be used to establish its depth or draft.
For example,
TRIM = t0 
HEEL = h0
CRTPT (1) "description",
l,t,v
HEIGHT (1) =
d
where d is
the distance from critical point #1 to the water (positive if above
the water, negative if below the surface).
It was mentioned
above that the depth of the ground at each grounding point is fixed
at the time that the ADD command is given. If a DEPTH or HEIGHT
command is given after the grounding points are defined, all of
the depths at the grounding points are changed by the same amount
that the vessel's draft is changed. This is a convenient way of
simulating a change of tide level.
