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‡oating structure is said to be wall-sided if, for the angles of heel to be considered, those portions of the hull covered or uncovered by the changing water plane are vertical when the structure is ‡oating upright; see …gure 2.10. The wedges in the cross sections are bounded by vertical lines - as is the case for the barge - so that the emerged and immersed wedges are right angle symmetrical triangles and the position of the metacenter NÁ can be calculated easily in such a special case. It can be found that - regardless the under-water geometry of a structure - equation 2.30 is valid for all wall-sided structures:¯¯¯¯ BNÁ = IT r ¢ (1 + 1 2 tan2 Á) ¯¯¯¯ Scribanti Formula (2.33) 2-16 CHAPTER 2. HYDROSTATICS in which IT is the transverse moment of inertia (second moment of areas) of the not heeled water plane about the axis of inclination for both half water planes. This is a fairly good approximation for ships that have vertical sides over a great part of the length and which are not so far heeled that the deck enters the water or the bottom comes above the water level. At small angles of heel (up to about 10 degrees), the e¤ect of the vertical shift of the center of buoyancy, B 0ÁBÁ = MNÁ, can be ignored. This can be seen in the Scribanti formula too, because in that case 1 2 tan2 Á is very small compared to 1.0. The work line of the vertical buoyancy force, ½gr, intersects the work line of the vertical buoyancy force for Á = 0± at the initial metacenter, M, which location is de…ned by equation 2.26, useful for Á < §10±. Up to angles of heel of about 10 degrees, this is a fairly good approximation of the metacenter of the ships that are almost wall-sided at the ”zone between water and wind” over a relatively large part of the length. Sailing yachts for instance, do not ful…l this requirement. At larger angles of heel, the e¤ect of the vertical shift of the center of buoyancy must be taken into account. The work line of the vertical buoyancy force, ½gr, intersects the work line of the vertical buoyancy force for Á = 0± at the metacenter,NÁ, which location is de…ned by equation 2.32, useful for Á > §10±. Use of this formula yields su¢ciently accurate results for ”non-Scribanti” ships until heel angles of heel of about 20 to 25 degrees. Keep in mind that this formula is not longer valid when the angle of heel gets so large that the water plane changes rapidly. This is the case when the bilge or chine (the ”corner” where the sides of a ship meet its bottom) comes out of the water or the deck enters the water, for example. This possibility always has to be checked, when carrying out stability calculations! It has been shown above that the metacenter is de…ned by the intersection of lines through the vertical buoyant forces at a heel angle Á and a heel angle Á + ¢Á. Depending on the magnitude of Á and the increment ¢Á, two di¤erent metacenter de…nitions have been distinguished here, as tabled below. metacenter point symbol Á ¢Á initial metacenter M 0 very small metacenter NÁ 0 larger Note that - for symmetrical under water forms like ships - both the metacenters M and NÁ are situated in the middle line plane (Á = 0). The stability lever armGZ = GNÁ ¢sinÁ is determined by the hydrostatic properties of the submerged structure (form and heel dependent) and the position of the center of gravity of this structure (mass distribution dependent). This is the reason why the following expression for GNÁ has been introduced:¯¯ GNÁ =KB + BNÁ ¡KG ¯¯ (2.34) Here, K is the keel point of the structure; see …gure 2.10. The magnitude of KB follows from the under water geometry of the structure and the magnitude of KG follows from the mass distribution of the structure. 2.3. STATIC FLOATING STABILITY 2-17 If - within the range of considered angles of heel - the shape of the water plane does not change very much (wall-sided structure), a substitution of equation 2.32 in equation 2.34 results in: GNÁ = KB + BM ¢ µ 1 + 1 2 tan2 Á ¶ ¡KG = GM + 1 2 BM ¢ tan2 Á (2.35) For small angles of heel, the stability lever arm becomesGZ = GM ¢sinÁ andGM 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 ¢ sinÁ = ½gr ¢ ©GM +MNÁª ¢ sinÁ (2.38) In these relations:¯¯ GZ = GNÁ ¢ sinÁ = © GM +MNÁ ª ¢ sinÁ¯¯ (stability lever arm) (2.39) 2-18 CHAPTER 2. HYDROSTATICS 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 2.3. STATIC FLOATING STABILITY 2-19 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 Á1 is 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. 2-20 CHAPTER 2. HYDROSTATICS 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: