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)

In applying this theory, GHS makes the assumption that the ground buoyancy vs. penetration function always has the form,

b= C * (d - d)_{0}^{2}ford>dEquation 1_{0}

whereb= 0 ford<d_{0}

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 =2C * (d - d) for_{0}d>dEquation 2_{0}

where D is the density of the water (weight per volume), anda*D = 0 ford<d_{0}

While this contribution to the effective waterplane area makes it possible to derive the usual weight-to-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 400 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

ADDThe /GR parameter indicates that a grounding point is being defined, rather than a fixed weight item. The point ("description" b, l,t,v/GR

The

The above form of the command assumes that the grounding point is in contact with the ground. If

It may seem that there is a difficulty with this method, since the value of

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: 2.0tells 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

(d-d) = 2.0_{0}

therefore C = 25.0.b= 100

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: -2.0As 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, 3.0In 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.

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 single-point 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.

Consider a case of single-point 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 followi ng series of commands could be used:

TRIM =This produces a series of status reports from 25 port heel to 30 starboard heel, each report showing the ground reaction and righting arm.t| HEEL =_{0}h| DEPTH =_{0}dADD_{0}"description" b, l,t,v/GR MACRO H HEEL = *+5 SOLVE TRIM STATUS / HEEL = 30P H (12)

Consider a two-point grounding case as illustrated in figure 3, and assume that longitudinal strength is to be checked. The following commands provide the answer:

TRIM =t| HEEL =_{0}h| DEPTH =_{0}dADD_{0}"fwd descr" b1, l1,t1,v1/GR ADD"aft descr" b2, l2,t2,v2/GR SOLVE TRIM LS

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 ce nterline; 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 =wheret| HEEL =_{0}hCRTPT (1)_{0}"description", l,t,vHEIGHT (1) =d

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

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