Chapter 3-STABILITY AND BUOYANCY 



when the draft is increased from 18 feet to 26 

 feet, when the ship is inclined at an angle of 20°. 

 At smaller angles up to 30°, certain hull types 

 show flat or slightly increasing righting arm 

 values with an increase in displacement. 



A reduction in the size of the righting arm 

 usually means a decrease in stability. When the 

 reduction in GZ is caused by increased dis- 

 placement, however, the total effect on stability 

 is more difficult to evaluate. Since the righting 

 moment is equal to W times GZ, the righting 

 moment will be increased by the gain in W at 

 the same time that it is decreased by the reduc- 

 tion in GZ. The gain in the righting moment, 

 caused by the gain in W, does not necessarily 

 compensate for the reduction in GZ. 



In brief, there are several ways in which an 

 increase in displacement affects the stability of 

 a ship. Although these effects occur at the same 

 time, it is best to consider them separately. 

 The effects of increased displacement are: 



1. Righting arms (GZ) are decreased as a 

 result of increased draft. 



2. Righting moments (foot-tons) are de- 

 creased as a result of decreased GZ (for a given 

 displacement). 



3. Righting moments may be increased as a 

 result of the increased displacement (W), if 

 (GZ X W) is increased. 



CROSS CURVES OF STABILITY 



To facilitate stability calculations, the design 

 activity inclines a lines drawing of the ship at a 

 given angle, and then lays off on it a series of 

 waterlines. These waterlines are chosen at 

 evenly spaced drafts throughout the probable 

 range of displacements. For each waterline the 

 value of the righting arm is calculated, using an 

 assumed center of gravity rather than the true 

 center of gravity. A series of such calculations 

 is made for various angles of heel— usually 10 , 

 20°, 30°, 40°, 50°, 60°, 7C°, 80°, and 90°- and the 

 results are plotted on a grid to forma series of 

 curves known as the cross curves of stability 

 (fig. 3-14). Note that, as draft and displacement 

 increase, the curves all slope downward, indi- 

 cating increasingly smaller righting arms. 



The cross curves are used in the preparation 

 of stability curves. To take a stability curve 

 from the cross curves, a vertical line (such as 

 line MN in fig. 3-14) is drawn on the cross curve 

 sheet at the displacement which corresponds to 

 the mean draft of the ship. At the intersection of 



this vertical line with each cross curve, the 

 corresponding value of the righting arm on the 

 vertical scale at the left can be read. Then this 

 value of the righting arm at the corresponding 

 angle of heel is plotted on the grid for the sta- 

 bility curve. When a series of such values of the 

 righting arms from 10 "through 90° of heel have 

 been plotted, a smooth line is drawn through 

 them and the uncorrected stability curve for the 

 ship at that particular displacement is obtained. 

 The curve is not corrected for the actual height 

 of the ship's center of gravity, since the cross 

 curves are based on an assumed height of G. 

 However, the stability curve does embody the 

 effect on the righting arm of the freeboard for 

 a given position of the center of gravity. 



Figure 3-15 shows an uncorrected stability 

 curve (A) for the ship operating at 11,500 tons 

 displacement, taken from the cross curves 

 shown in figure 3-14. This stability curve can- 

 not be used in its present form, since the cross 

 curves are made up on the basis of an assumed 

 center of gravity. In actual operation, the ship's 

 condition of loading will affect its displacement 

 and, therefore, the location of G. To use a curve 

 taken from the cross curves, therefore, it is 

 necessary to correct the curve for the actual 

 height of G above the keel (K)— that is, it is 

 necessary to use the distance KG. As far as the 

 new center of gravity is concerned, when a 

 weight is added to a system of weights, the 

 center of gravity can be found by taking moments 

 of the old system plus that of the new weight and 

 dividing this total moment by the total final 

 weight. Detailed information concerning changes 

 in the center of gravity of a ship can be obtained 

 from chapter 9880 of the Naval Ships Technical 

 Manual . 



Assume that the cross curves are made up 

 on the basis of an assumed KG of 20 feet, and the 

 actual KG, which includes the added effects of 

 Free Surface , for the particular condition of 

 loading, is 24 feet. This means that the true G 

 is 4 feet higher than the assumed G, and that the 

 righting arm (GZ) at each angle of inclination 

 will be smaller than the righting arm shown in 

 figure 3-15 (curve A) for the same angle. To 

 find the new value of GZ for each angle of 

 inclination, the increase in KG (4 feet) is multi- 

 plied by the sine of the angle of inclination, and 

 the product is subtracted from the value of GZ 

 shown on the cross curves or on the uncorrected 

 stability curve. In order to facilitate the correc- 

 tion of the stability curves, a table showing the 

 necessary sines of the angles of inclination is 



41 



