Handbook of Civil Engineering Calculations

wedge of emersion (or immersion). Then

"-^-

The displacement volume and the draft
when the boat is upright are W = 182
(2240) = 407.700 Ib (1813 N); V=WIw =
407.700/62.4 = 6530 ft^3 (184.93 m^3 ); d =
6530/[64(2O)] = 5.10 ft (155.448 cm).

2. Find h, using Eq. 6
   Since <f) is relatively small, apply this ap- FIGURE 5. Location of resultant forces on
   proximation: h' = 2b/3 — 2(20)/3 = 13.33 ft inclined vessel.
   (406.298 cm), h =^1 X 2 (IO)(IO tan 6°) x
   (13.33)/[5.10(20)] = 0.687 ft (20.940 cm).


3. Compute the horizontal distance a (Fig. 5)
   Thus, BG = 8.6 -
   1
   X 2 (S.10) = 6.05 ft (184.404 cm); a = 6.05 sin 6° = 0.632 ft (19.263 cm).


4. Compute the moment of the vertical forces
   Thus, M= W(h-a) = 407,700(0.055) = 22,400 ft-lb (30,374.4 N-m). Since h>a, the mo-
   ment is righting. This constitutes the solution to part a. The remainder of this procedure is
   concerned with part b.
   In Fig. 5, let M denote the point of intersection of the vertical line through B' and the
   line BG prolonged. Then Mis termed the metacenter associated with this position, and the
   distance GM is called the metacentric height. Also BG is positive if G is above B, and GM
   is positive if Mis above G. Thus, the moment of vertical forces is righting or upsetting de-
   pending on whether the metacentric height is positive or negative, respectively.


5. Find the lever arm of the vertical forces
   Use the relation for metacentric height:

J-WT

GM= ,-BG (7)

Fcos 0

where IWL = moment of inertia of original waterline section about axis through O. Or,
IWL = C/i2)(64)(20)^3 = 42,670 ft^4 (368.3 m^4 ); GM = 42,670/6530 cos 6° - 6.05 = 0.52 ft
(15.850 cm); h-a = 0.52 sin 6° = 0.054 ft (1.646 cm), which agrees closely with the pre-
vious result.

Mechanics of Incompressible Fluids

The notational system is a = acceleration; A = area of stream cross section; C = discharge
coefficient; D = diameter of pipe or depth of liquid in open channel; F = force; g = gravi-
tational acceleration; H = total head, or total specific energy; hF = loss of head between
two sections caused by friction; hL = total loss of head between two sections; hv= differ-
ence in velocity heads at two sections if no losses occur; L = length of stream between
two sections; M = mass of body; NR = Reynolds number; p = pressure; Q = volumetric
rate of flow, or discharge; s = hydraulic gradient = -dH/dL; T = torque; V = velocity;
w = specific weight; z = elevation above datum plane; p = density (mass per unit volume);