2.3 Static Floating Stability
2.3.7 Stability Curve
¡ KG
= GM +1
2BM ¢ tan2Á (2.35)
For small angles of heel, the stability lever arm becomes GZ = GM ¢sin Á and GM becomes:
¯¯GM = KB + BM ¡ KG¯¯ (2.36)
Submerged Structures
Figure 2.11: Submerged Floating Structure
Fully submerged structures, such as tunnel segments during installation or submarines (see
…gure 2.11), have no water plane. The de…nitions of BM and BNÁ show that these values are zero, which means that the metacenter coincides with the center of buoyancy. In this case the previous equations reduce to:
¯¯GNÁ = GM = KB ¡ KG¯
¯ (for fully submerged bodies only) (2.37)
2.3.7 Stability Curve
If a ‡oating structure is brought under a certain angle of heel Á, see …gure 2.12, then the righting stability moment is given by:
MS = ½gr ¢ GZ
= ½gr ¢ GNÁ¢ sin Á
= ½gr ¢©
GM + M NÁ
ª¢ sin Á (2.38)
In these relations:
¯¯GZ = GNÁ¢ sin Á =©
GM + M NÁª
¢ sin Á¯¯ (stability lever arm) (2.39)
Figure 2.12: Stability Lever Arm The value GZ determines the magnitude of the stability moment.
For practical applications it is very convenient to present the stability in the form of righting moments or lever arms about the center of gravity G, while the ‡oating structure is heeled at a certain displacement,Á. This is then expressed as a function of Á. Such a function will generally look something like …gure 2.13 and is known as the static stability curve or the GZ-curve.
Figure 2.13: Static Stability Curves
Because the stability lever arm is strongly dependent on the angle of heel, Á, a graph of GZ, as given in …gure 2.13 is very suitable for judging the stability. For an arbitrarily (non symmetric) ‡oating structure form, this curve will not be symmetrical with respect to Á = 0, by the way.
For symmetric forms like ships however, the curve of static stability will be symmetric with respect to Á = 0. In that case, only the right half of this curve will be presented as in
…gure 2.14.
The heel angle at point A in this …gure, at which the second derivative of the curve changes sign, is roughly the angle at which the increase of stability due to side wall e¤ects (Scribanti
Figure 2.14: Ship Static Stability Curve
formula) starts to be counteracted by the fact that the deck enters the water or the bilge comes above the water.
Figure 2.15 shows the static stability curve when the initial metacentric height, GM , is negative while GZ becomes positive at some reasonable angle of heel Á1, the so-called angle of loll.
Figure 2.15: Static Stability Curve with a Negative GM
If the ‡oating structure is momentarily at some angle of heel less than Á1, the moment acting on the structure due to GZ tends to increase the heel. If the angle is greater than Á1, the moment tends to reduce the heel. Thus the angle Á1is a position of stable equilibrium.
Unfortunately, since the GZ curve is symmetrical about the origin, as Á1 is decreased, the
‡oating structure eventually passes through the upright condition and will then suddenly lurch over towards the angle ¡Á1 on the opposite side and overshoot this value (because of dynamic e¤ects) before reaching a steady state. This causes an unpleasant rolling motion, which is often the only direct indication that the heel to one side is due to a negative GM rather than a positive heeling moment acting on the structure.
Characteristics of Stability Curve
Some important characteristics of the static stability curve can be summarized here:
1. Slope at The Origin
For small angles of heel, the righting lever arm is proportional to the curve slope and the metacenter is e¤ectively a …xed point. It follows, that the tangent to the GZ curve at the origin represents the metacentric height GM . This can be shown easily for the case of a wall-sided structure:
d This derivative becomes GM for zero heel. This means that the initial metacentric height GM can be of great importance for the further form of the curve, especially at smaller angles of heel, and for the area under the curve (see item 5 below).
2. Maximum GZ Value
The maximum GZ value is rules the largest steady heeling moment that the ‡oating structure can resist without capsizing. Its value and the angle at which it occurs are both important. The shape of the above-water part of the ‡oating structure is of great importance for the maximum attainable value of the stability lever arm.
3. Range of Stability
At some angle of heel (sometimes even greater than 90 degrees) the GZ value de-creases again to zero and even becomes negative for larger inclinations. This angle is known as the angle of vanishing stability. The range of angles for which GZ is positive is known as the range of stability. This range is of great importance for the maximum attainable area under the stability curve and thereby also on the maximum potential energy that the structure can absorb via a roll motion. The shape of the above-water part has a large in‡uence on the angle of vanishing stabil-ity; compare curves I and II in …gures 2.12 and 2.13. For angles within the range of stability, a ‡oating structure will return to the upright state when the heeling moment is removed.
4. Angle of Deck Edge Immersion
For most ‡oating structures, there is a point of in‡ection in the stability curve, corresponding roughly to the angle at which the deck edge becomes immersed. This point is not so much of interest in its own right as in the fact that it provides guidance to the designer upon the possible e¤ect of certain design changes on stability. The shape of the above water part of the structure can have a large in‡uence on its static stability. More or less the same statement can be made when the bilge or the bottom chine emerges, because of the decrease of the breadth of the water line. Keep in mind that for wall-sided structures, when the deck enters the water or the bottom chine comes above the water level, the immersed and emerged wedges are no longer nice triangles; calculations become much more cumbersome!
5. Area Under The Static Stability Curve
An important parameter when judging the stability properties of a structure ‡oating upright is the work that has to be done to reach a chosen heel angle, Á¤:
PÁ¤ =
Á¤
Z
0
MS¢ dÁ
= ½gr ¢
Á¤
Z
0
GNÁ¢ sin Á ¢ dÁ (2.41)
This means that the area under the static stability curve is an important quantity for the evaluation of the stability properties. It represents the ability of the ‡oating structure to absorb roll energy imparted to it by winds, waves or any other external e¤ect.