PRINCIPLES OF NAVAL ENGINEERING 



en 

 Z. 



h 

 O 

 O 

 b 



Z 



u 



O 

 % 



o 



H 



Z 



o 



30,000 



20,000 



10.000 



10 20 30 



ANGLE OF HEEL IN DEGREES 



'40 50 60 70 



MAXIMUM RIGHTING MOMENT 

 - DEVELOPS AT 38' 



147.43 



Figure 3-27. — Curve of righting moments. 



The distance G3Z3 is the righting arm the 

 ship would have if the center of gravity had 

 risen from G to G3 and this virtual rise of G 

 may be computed from the formula: 



GG3= J- 

 V 



Referring to the formula, loss in GZ= — x 



sin . This formula is accurate for small angles 

 of heel only, due to the pocketing effect as the 

 angle increases. In case several compartments 

 or tanks have free surface, their surface mo- 

 ments of inertia are calculated individually 

 and their sum used in the correction for free 

 surface. The effect of a given area of loose 

 liquid at a given angle of heel is entirely in- 

 dependent of the depth of the liquid in the com- 

 partment, as is apparent in the formula, 



b"^l 



12 



where the only factors are the dimensions of 

 the surface and the displacement of the ship. 



The free surface effect is also independent of 

 the free surface location in the ship, whether 

 it is high or low, forward or aft, on the center- 

 line or off, as long as the boundaries remain 

 intact. 



The loss of metacentric height can obviously 

 can be reduced by reducing the breadth of the 

 free surface, as bythe installation of longitudinal 

 bulkheads. However, off-center flooding after 

 damage then becomes possible, causing the ship 

 to take on a permanent list and usually bringing 

 about a greater loss in stability than if the 

 bulkhead were not present. 



The loss of GZ due to free surface is always 

 lessened to some extent by pocketing. This is 

 the contact of the liquid with the top of the com- 

 partment or the exposure of the bottom surface 

 of the compartment, either of which takes place 

 at some definite angle and reduces the breadth 

 of the free surface area. To understand how 

 pocketing of the free surface reduces the free 

 surface effect, study figure 3-29. Part A shows 

 a compartment in which the free surface effect 

 is not influenced by the depth of the loose water. 

 The compartment shown in part B, however, con- 

 tains only a small amount of water; when the 

 ship heels sufficiently to reduce the waterline 



54 



