Dave Gerr - Boat Mechanical Systems Handbook-How to Design, Install, and Recognize Proper Systems in Boats

Formula 6-5b. Moment of Inertia of a
Rectangular Plane Area (Metric)

I =bh^3 ÷12

Where (for evaluating tank free surface)

b =the length of the tank fore-and-aft, m

h =the width of the tank athwartships, m

In our case, the diesel tanks are 2.743 m
long by 2.073 m wide athwartships. Thus the
moment of inertia of each tank is

I =2.743 m× (2.073 m)^3 ÷ 12 =2.036 m^4

Then

Multiply by 2 for both diesel tanks =

0.0552 m GMred (diesel)

The water tanks are 0.762 m long by 1.981
m wide athwartships. Thus the moment of
inertia for each tank is

I =0.762 m× (1.981 m)^3 ÷ 12 =0.494 m^4

Then

Multiply by 2 for both water tanks =

0.01574 m GMred (water)

Now

and

This is the loss in stability due to free-

surface effect and can be translated into

reduced GZ or RM using the preceding rela-

tionship between GZ and GM.

In our case, stability after free-surface

effect is

GZ 1°=1.02 m GM× sin 1°=0.0178 m

RM 1°=0.0178 m GZ 1°× 62,588 kg

Disp. =1,114 kgm

This is a reduction in righting moment of

6% due to free-surface effect.

Free-Surface Effect and

Loading Conditions

Keep in mind that the free-surface reduc-

tion changes for various conditions of load-

ing (for different displacements). Using

boat displacement to the DWL (or to the

expected flotation waterline at^2 / 3 load) is

adequate for most work on small craft; for

commercial projects, however, carefully

tabulated GMs for different loading condi-

tions will be required. Performing stability

calculations for all reasonable conditions of

loading and flooding are detailed and time-

consuming. Usually, these should be con-

ducted using a software package, such as

GHS, intended for this purpose; however,

you need to understand the concept of free-

surface effect to interpret the results

correctly.

GMeff=−1.091 m GMs 0.071 m GMredT=1.02 m

GM 0.0552 m GM

0.01574 m G

redT= red (diesel)

+ MMred (water)=0.071 m

GM

0.494 m

61.05 m

1.000

1.028

red

4

= 3

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟⎟

=0.00787 m

(for each water tank)

GM

2.036 m

61.05 m

0.852

1.028

red

4

= 3

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞⎞

⎠⎟

=0.0276 m

(for each diesel tank)

Chapter 6:Tank Capacity and Range

Formula 6-5b.