CHAPTER 7
TRANSVERSE STABILITY AT LARGE ANGLES OF HEEL
The essence of stability calculations is finding the force couple between buoyancy and weight. This is the moment of force which a stable ship develops to counteract the overturning moments arising from external forces. Reliance on the metacentric height as a measure of transverse stability is limited, as described in Chapter 6, to situations in which the ship heels to small angles from the upright, typically less than about 10 degrees. If the upsetting forces that act upon ships in service, such as those caused by wind, waves, cargo handling, and turning, could not produce inclinations larger than a few degrees, the study of metacentric, or initial transverse statical stability would be sufficient for both ship designer and operator. However, ships can and do heel and roll to larger angles under the influence of large heeling moments. To ensure proper design and safe operation we must know how a ship behaves when heeled to large angles.
7.1. Righting Arm and Righting Moment
Whatever the angle of heel, the proper measure of a ship’s ability to return to upright is the righting moment, equal to the product of the ship’s weight () and the righting arm (GZ), as shown in Figure 7.1. The difference between the small and large angles of heel is due to the fact that at large angles the buoyant force vector does not pass through the metacentre (M). The reason is that, as the angle of heel increases beyond a few degrees, the path of the centre of buoyancy (B) departs from a circular arc of radius BM. The consequence of this departure is that the righting arm is no longer related in any simple way to the metacentric height, that is, GZ is not equal to GM, as it is in the case of very small angles of heel. In fact, no exact formula is known that relates GM to the righting arms GZ for large angles, except for the very restrictive class of hull forms for which the centre of buoyancy traces a circular path when the vessel heels to any angle. This will be the case only for spheres, circular cylinders, or bodies of revolution floating with their axis of symmetry parallel to the water surface. For such forms, the transverse metacentre lies on the axis of symmetry and the righting arms for all angles of heel are equal to GM. The only practical hull forms satisfying these conditions are circular section pontoons and submarines whose hull forms are essentially bodies of revolution.
Once the righting arm, GZ, is determined for a given heel angle and loading condition the righting moment can be estimated as
where
: form stability
: weight stability
: residual stability
N is known as the prometacentre.
Figure 7.1. Transverse stability at large angles of heel
7.2. Cross Curves of Stability
The results of the righting arm calculations for a ship are plotted as a set of cross curves known as cross curves of stability. These curves are used to determine the length of the righting arm at any angle of inclination for a given displacement. A typical set of cross curves is shown in Figure 7.2. The range of displacements over which cross curves have been determined is from the light ship displacement at the lower end to a displacement usually well above the load displacement, so that stability can be assessed at deep draughts associated with potential flooding situations.
Since the centre of gravity is a function of loading condition the basis of the cross curves is taken as a fixed point, such as the keel (K). Then the righting arm is
In the preparation of cross curves of stability, certain assumptions have been made, as follows;
· The ship’s centre of gravity remains fixed at the pole point, or assumed centre of gravity, regardless of the angle of heel.
· The ship’s hull, consisting of the bottom, sides, and weather deck, is assumed to be perfectly watertight.
· Superstructures and deckhouses above the weather deck are normally assumed to be nonwatertight. Any actual watertightness of such structures, maintained by the proper closure of watertight doors, will provide a margin of safety of additional intact stability beyond that indicated by the cross curves at angles of heel that immerse the structures in question.
· Adjustments are made to account for the volumes and moments of immersed appendages suc as rudder, propellers, sonar domes, etc., and freely flooding spaces like large seachests.
Figure 7.2. Typical cross curves of stability
7.3. Statical Stability Curve
Cross curves of stability are a convenient form in which to store the information necessary to determine the large angle stability characteristics of a ship at any displacment, but at a single assumed position of the centre of gravity. For the ship operator as well as for the naval architect during the design process, what is needed is a determination of the righting arms or moments of a ship heeled to any angle while in a given loading condition at one displacement and with its centre of gravity at a position different from the one that was assumed in preparing the cross curves. The statical stability curve in which righting arms are plotted against angle of heel is the appropriate format.
By combining weight (both magnitude and CG) with the hydrostatic properties, righting arms are produced and the stability of the ship can be determined. Typically, righting arms are plotted for a range of heel angles where the heel is assumed to be induced by a moment about the longitudinal axis. A particular moment, such as that caused by the wind at a given speed, may also be imposed. Further analysis of the properties of this righting arm curve lead to a formal assessment of the vessel's stability. The ability to derive righting arms for inclinations in other directions is also useful for some types of ships.
The statical stability curve is determined by picking values of the righting arms from the cross curves at the appropriate displacement and correcting the values thus obtained to reflect the actuaş position of the centre of gravity. Two corrections may be required; one for the height of G above the keel (KG) and one for the distance of G off centreline (TCG). The vertical correction is almost always necessary, since it would be only by gratuitous coincidence that a ship’s KG in a given loading condition would be exactly at the pole point chosen arbitrarily when the cross curves were prepared. The lateral correction is, by contrast, rarely needed in routine loading conditions because good practice in loading a ship requires that the centre of gravity be on centreline so that the ship will float upright.
It is clear that all righting arms will be smaller than those plotted in the cross curves, because the actual centre of gravity is higher than the assumed centre of gravity, the keel, K. Thus stability at large angles decreases as G rises, just as initial stability measured by GM does.
The conventional plotting of a statical stability curve is shown in Figure 7.3. Since the centre of gravity is on centreline, the direction of assumed heel (staboard or port) is immaterial, because the centreline symmetry of the hull would cause the centre of buoyancy to assume symetrically corresponding positions on either side at any given angle of heel. Although the curve can be plotted for all angles of heel for which cross curves have been determined, it is typically terminated where the GZs become negative, that is, where righting arms change to capsizing arms.
To have a full understanding of intact ship stability, we must know not only how a statical stability curve is determined, but also why it is shaped as shown, and what significance is to be attached to its typical features.
The initial portion of the statical stability curve (the first 7-10 degrees) must be consistent with the measure of initial stability, that is, the metacentric height (GM).
As the heel angle approaches zero, . Thus for the small angles of heel we may write
Therefore the metacentric height (GM) is a measure of the slope of a staical stability curve at the origin and should always be used as an aid to plotting the curve, by running the curve in tangent to the straight line at the origin. At an angle of 1 radian (equal to 180/, or 57.3 degrees) the straight line passes through the value GZ=GM. Thus, as is shown in Figure 7.3 , if GM is laid out as an ordinate at 57.3 degrees and that point is connected to the origin by a straight line, the statical stability curve will approach that line asymptotically as it approaches the origin.
Figure 7.3. Typical static stability curve
For the great majority of ship hull forms, the statical stability curve from its initial path with increasing slope so that it rises above the tangent line as the angle of heel increases. Eventually, as the angle of heel increase, a point is reached at which deck edge immersion takes place. In fact, since a ship’s section shapes vary from bow to stern, deck edge immersion is not a sudden occurrence along the entire length of the ship, but take place gradually over a range of heel angles. But the general trend is accompanied by a much reduced growth in GZ, and thus an inflection point in the statical stability curve. This steadily decreasing slope of the statical stability curve beyond deck edge immersion leads to the peak of the curve, and ultimately to its rapid plunge beyond the peak.
This effect of deck edge immersion on large angle stability has important consequences for both the design naval architect and the ship operator. The design implications are that a ship designed to have a small freeboard may develop inadequate righting arms and moments at large angles because deck edge immersion and the peak of the GZ curve will occur at relatively small angles of heel.
To avoid this problem, low-freeboard ships must be designed with relatively large metacentric heights, because the large initial slope of the statical stability curve will tend to ensure that adequate righting moments will be achieved in spite of the small angle at which the curve reaches its peak.
The peak of a statical stability curve identifies two quantities that are important in evaluating the overall stability of a ship. They are the maximum righting arm and the angle of maximum stability. The importance of the maximum righting arm (GZmax) is that the product of the displacement and GZmax is the maximum steady heeling moment that the ship can experience without capsizing.
Beyond the angle of maximum stability, righting arms decrease, often more rapidly than they had increased up to that point. The rapid decrease ultimately leads to the point at which GZ becomes zero, and the curve recrosses the axis. The angle at which this occurs is the angle of vanishing stability, because thereafter the GZs are negative. That is, they are capsizing or upsetting arms, rather than righting arms. Any ship that inclines beyond its angle of maximum stability will capsize, regardless of the cause of the inclination or its duration. The typical statical stability curve like that in Figure 7.3 crosses the horizontal axis at two angles of inclination, each of which represents a condition of static equilibrium, since GZ equals zero. The first crossing (zero heel angle) is a stable equilibrium condition, because temporary inclinations to larger angles create righting moments that will restore the ship to the equilibrium angle when the cause of inclination is removed. The second crossing is at the angle of vanishing stability and represents a unstable equilibrium condition, because temporary inclinations to larger angles create upsetting moments that will cause the ship incline away from that equilibrium condition when the cause of inclination is removed. The range of heel angles between the two crossings is called the range of stability.
As a practical matter, one must be careful not to depend too much on any ship’s ability to recover from angles of inclination beyond its angle of maximum stability because the cross curves of stability were determined on the assumption of perfect watertight integrity of the weather deck. This assumption is incorrect for the great majority of ships. All the ships are provided in deck for cargo hatches, access to spaces below decks, ventillation, piping, etc. Properly designed deck penetrations are made weathertight, but few of them can be made truly watertight, and there is always the possibilty of human error, that is , leaving doors and hatches open that should be tightly closed again heavy weather. At angles of heel that immerse part of the deck, the possibility always exists that water will be shipped through such openings. This event is called downflooding, and the smallest angle of heel at which it can occur is known as the downflooding angle. If large quantities of water enter the ship, the statical stability curve is no longer correct; the ship’s loading has changed. Therefore at any angle of heel greater than the downflooding angle, a statical stability curve must be considered invalid. For large ships, downflooding is rarely considered to be a critical issue, but for smaller vessels, fishing boats in particular, the range of stability should be assumed to be terminated at the downflooding angle rather than the angle of vanishing stability.
7.4. Stability of a Body of Revolution
Although the body of revolution is not a realistic form for surface ships, it is instructive to examine the behaviour of such a vessel when it is heeled to large angles. Imagine a vessel, all of whose sections are circular, loaded such that it floats with its axis horizontal, but with its centre of gravity below its axis. It is immaterial whether all sections are of the same diameter (a circular cylinder) or not ( a body of revolution for eaxmple, shaped like a cigar or a football). Figure 7.4 depicts this body floating at various angles of heel, the upright condition (0) representing the position of stable equilibrium, in which G is directly below the centre of the circle. It is apparenr, as each of the positions shown are examined, that, at any angle of heel, the buoyant force acts throug the centre of the circle, which is thus the transverse metacentre M. The righting arms GZ are, for all angles of heel, equal to . Note that the righting arms increase to a maximum value equal to GM at 90 degrees, then decrease again to zero in the inverted position, or 180 degree heel. This upside-down position is therefore one of equilibrium (GZ=0), but thr slightest disturbance from it in either direction will cause the vessel to rotate until it returns to its original upright condition. Thus it is a position of unstable equilibrium.