Principles of naval engineering - Historic Naval Ships Association

Chapter 3-STABILITY AND BUOYANCYwhen the draft is increased from 18 feet to 26feet, when the ship is inclined at an angle of 20°.At smaller angles up to 30°, certain hull typesshow flat or slightly increasing righting armvalues with an increase in displacement.A reduction in the size of the righting armusually means a decrease in stability. When thereduction in GZ is caused by increased displacement,however, the total effect on stabilityis more difficult to evaluate. Since the rightingmoment is equal to W times GZ, the rightingmoment will be increased by the gain in W atthe same time that it is decreased by the reductionin GZ. The gain in the righting moment,caused by the gain in W, does not necessarilycompensate for the reduction in GZ.In brief, there are several ways in which anincrease in displacement affects the stability ofa ship. Although these effects occur at the sametime, it is best to consider them separately.The effects of increased displacement are:1. Righting arms (GZ) are decreased as aresult of increased draft.2. Righting moments (foot-tons) are decreasedas a result of decreased GZ (for a givendisplacement).3. Righting moments may be increased as aresult of the increased displacement (W), if(GZ X W) is increased.CROSS CURVES OF STABILITYTo facilitate stability calculations, the designactivity inclines a lines drawing of the ship at agiven angle, and then lays off on it a series ofwaterlines. These waterlines are chosen atevenly spaced drafts throughout the probablerange of displacements. For each waterline thevalue of the righting arm is calculated, using anassumed center of gravity rather than the truecenter of gravity. A series of such calculationsis made for various angles of heel—usually 10 ,20°, 30°, 40°, 50°, 60°, 7C°, 80°, and 90°- and theresults are plotted on a grid to forma series ofcurves known as the cross curves of stability(fig. 3-14). Note that, as draft and displacementincrease, the curves all slope downward, indicatingincreasingly smaller righting arms.The cross curves are used in the preparationof stability curves. To take a stability curvefrom the cross curves, a vertical line (such asline MN in fig. 3-14) is drawn on the cross curvesheet at the displacement which corresponds tothe mean draft of the ship. At the intersection ofthis vertical line with each cross curve, thecorresponding value of the righting arm on thevertical scale at the left can be read. Then thisvalue of the righting arm at the correspondingangle of heel is plotted on the grid for the stabilitycurve. When a series of such values of therighting arms from 10 "through 90° of heel havebeen plotted, a smooth line is drawn throughthem and the uncorrected stability curve for theship at that particular displacement is obtained.The curve is not corrected for the actual heightof the ship's center of gravity, since the crosscurves are based on an assumed height of G.However, the stability curve does embody theeffect on the righting arm of the freeboard fora given position of the center of gravity.Figure 3-15 shows an uncorrected stabilitycurve (A) for the ship operating at 11,500 tonsdisplacement, taken from the cross curvesshown in figure 3-14. This stability curve cannotbe used in its present form, since the crosscurves are made up on the basis of an assumedcenter of gravity. In actual operation, the ship'scondition of loading will affect its displacementand, therefore, the location of G. To use a curvetaken from the cross curves, therefore, it isnecessary to correct the curve for the actualheight of G above the keel (K)— that is, it isnecessary to use the distance KG. As far as thenew center of gravity is concerned, when aweight is added to a system of weights, thecenter of gravity can be found by taking momentsof the old system plus that of the new weight anddividing this total moment by the total finalweight. Detailed information concerning changesin the center of gravity of a ship can be obtainedfrom chapter 9880 of the Naval Ships TechnicalManual .Assume that the cross curves are made upon the basis of an assumed KG of 20 feet, and theactual KG, which includes the added effects ofFree Surface , for the particular condition ofloading, is 24 feet. This means that the true Gis 4 feet higher than the assumed G, and that therighting arm (GZ) at each angle of inclinationwill be smaller than the righting arm shown infigure 3-15 (curve A) for the same angle. Tofind the new value of GZ for each angle ofinclination, the increase in KG (4 feet) is multipliedby the sine of the angle of inclination, andthe product is subtracted from the value of GZshown on the cross curves or on the uncorrectedstability curve. In order to facilitate the correctionof the stability curves, a table showing thenecessary sines of the angles of inclination is41