15 Fluid Statics
20. INTACT STABILITY: STABILITY OF FLOATING OBJECTS
The gas densities are
air¼ p
¼ 0:07682 lbm=ft3
air¼ g
¼ 0:01061 lbm=ft3
helium¼ 0:01061 lbf=ft3
The total weight of the balloon, payload, and helium is W¼ 500 lbf þ 0:01061 lbf
ft3
VHe
The buoyant force is the weight of the displaced air.
Neglecting the payload volume, the displaced air volume is the same as the helium volume.
Fb¼ 0:07682 lbf ft3
VHe
At lift-off, the weight of the balloon is just equal to the buoyant force.
W¼ Fb 500 lbfþ 0:01061 lbf
ft3
19. BUOYANCY OF SUBMERGED PIPELINES Whenever possible, submerged pipelines for river crossings should be completely buried at a level below river scour. This will reduce or eliminate loads and movement due to flutter, scour and fill, drag, colli-sions, and buoyancy. Submerged pipelines should cross at right angles to the river. For maximum flexibility and ductility, pipelines should be made of thick-walled mild steel.
Submerged pipelines should be weighted to achieve a minimum of 20% negative buoyancy (i.e., an average density of 1.2 times the environment, approximately 72 lbm/ft3or 1200 kg/m3). Metal or concrete clamps can be used for this purpose, as well as concrete coat-ings. Thick steel clamps have the advantage of a smaller lateral exposed area (resulting in less drag from river flow), while brittle concrete coatings are sensitive to pipeline flutter and temperature fluctuations.
Due to the critical nature of many pipelines and the difficulty in accessing submerged portions for repair, it is common to provide a parallel auxiliary line. The auxiliary and main lines are provided with crossover and mainline valves, respectively, on high ground at both sides of the river to permit either or both lines to be used.
20. INTACT STABILITY: STABILITY OF FLOATING OBJECTS
A stationary object is said to be in static equilibrium.
However, an object in static equilibrium is not necessar-ily stable. For example, a coin balanced on edge is in static equilibrium, but it will not return to the balanced position if it is disturbed. An object is said to be stable (i.e., in stable equilibrium) if it tends to return to the equilibrium position when slightly displaced.
Stability of floating and submerged objects is known as intact stability.22 There are two forces acting on a sta-tionary floating object: the buoyant force and the object’s weight. The buoyant force acts upward through the centroid of the displaced volume. This centroid is known as the center of buoyancy. The gravitational force on the object (i.e., the object’s weight) acts down-ward through the object’s center of gravity.
For a totally submerged object (as in the balloon and submarine shown in Fig. 15.24) to be stable, the center of buoyancy must be above the center of gravity. The object will be stable because a righting moment will be created if the object tips over, since the center of buoy-ancy will move outward from the center of gravity.
The stability criterion is different for partially submerged objects (e.g., surface ships). If the vessel shown in
22The subject of intact stability, being a part of naval architecture curriculum, is not covered extensively in most fluids books. However, it is covered extensively in basic ship design and naval architecture books.
WaterResources
Fig. 15.25 heels (i.e., lists or rolls), the location of the center of gravity of the object does not change.23 How-ever, the center of buoyancy shifts to the centroid of the new submerged section 123. The centers of buoyancy and gravity are no longer in line.
This righting couple exists when the extension of the buoyant force, Fb, intersects line O–O above the center of gravity at M, the metacenter. For partially sub-merged objects to be stable, the metacenter must be above the center of gravity. If M lies below the center
of gravity, an overturning couple will exist. The dis-tance between the center of gravity and the metacenter is called the metacentric height, and it is reasonably constant for heel angles less than 10. Also, for angles less than 10, the center of buoyancy follows a locus for which the metacenter is the instantaneous center.
The metacentric height is one of the most important and basic parameters in ship design. It determines the ship’s ability to remain upright as well as the ship’s roll and pitch characteristics.
“Acceptable” minimum values of the metacentric height have been established from experience, and these depend on the ship type and class. For example, many submarines are required to have a metacentric height of 1 ft (0.3 m) when surfaced. This will increase to approx-imately 3.5 ft (1.2 m) for some of the largest surface ships. If an acceptable metacentric height is not achieved initially, the center of gravity must be lowered or the keel depth increased. The beam width can also be increased slightly to increase the waterplane moment of inertia.
For a surface vessel rolling through an angle less than approximately 10, the distance between the vertical center of gravity and the metacenter can be found from Eq. 15.62. Variable I is the centroidal area moment of inertia of the original waterline (free surface) cross sec-tion about a longitudinal (fore and aft) waterline axis;
V is the displaced volume.
If the distance, ybg, separating the centers of buoyancy and gravity is known, Eq. 15.62 can be solved for the metacentric height. ybg is positive when the center of gravity is above the center of buoyancy. This is the normal case. Otherwise, ybgis negative.
ybgþ hm¼ I
V 15:62
The righting moment (also known as the restoring moment) is the stabilizing moment exerted when the ship rolls. Values of the righting moment are typically specified with units of foot-tons (MNm).
Mrighting¼ hmwVdisplacedsin 15:63
The transverse (roll) and longitudinal (pitch) periods also depend on the metacentric height. The roll charac-teristics are found from the differential equation formed by equating the righting moment to the product of the ship’s transverse mass moment of inertia and the angu-lar acceleration. Larger metacentric heights result in lower roll periods. If k is the radius of gyration about the roll axis, the roll period is
Troll¼ 2ffiffiffiffiffiffiffiffiffipk ghm
p 15:64
The roll and pitch periods must be adjusted for the appropriate level of crew and passenger comfort. A
“beamy” ice-breaking ship will have a metacentric
23The verbs roll, list, and heel are synonymous.
Figure 15.25 Stability of a Partially Submerged Floating Object
A A
A A
1
1
2
2 3
3O O
CG CG
CB CB
hm
Fb ybg
W M
heel angle, θ metacenter solid force Figure 15.24 Stability of a Submerged Object
Fb
W
Fb
W
Fb
W Fb
W
balloon
submarine
F L U I D S T A T I C S
15-17
WaterResources
height much larger than normally required for intact stability, resulting in a short, nauseating roll period.
The designer of a passenger ship, however, would have to decrease the intact stability (i.e., decrease the meta-centric height) in order to achieve an acceptable ride characteristic. This requires a moderate metacentric height that is less than approximately 6% of the beam length.
Example 15.9
A 600,000 lbm (280 000 kg) rectangular barge has exter-nal dimensions of 24 ft width, 98 ft length, and 12 ft height (7 m 30 m 3.6 m). It floats in seawater (w = 64.0 lbf/ft3; w = 1024 kg/m3). The center of gravity is 7.8 ft (2.4 m) from the top of the barge as loaded. Find (a) the location of the center of buoyancy when the barge is floating on an even keel, and (b) the approximate location of the metacenter when the barge experiences a 5 heel.
SI Solution
(a) Refer to the following diagram. Let dimension y represent the depth of the submerged barge.
3.6 m
7 m
y ⫽ 1.3 m end view
CB 0.65 m
From Archimedes’ principle, the buoyant force equals the weight of the barge. This, in turn, equals the weight of the displaced seawater.
Fb¼ W ¼ V wg ð280 000 kgÞ 9:81 m
s2
¼ y
ð7 mÞð30 mÞ
1024 kg m3
9:81 m s2
y¼ 1:3 m
The center of buoyancy is located at the centroid of the submerged cross section. When floating on an even keel, the submerged cross section is rectangular with a height of 1.3 m. The height of the center of buoyancy above the keel is
1:3 m
2 ¼ 0:65 m
(b) While the location of the new center of buoyancy can be determined, the location of the metacenter does not change significantly for small angles of heel. For approximate calculations, the angle of heel is not significant.
The area moment of inertia of the longitudinal waterline cross section is
I¼Lw3
12 ¼ð30 mÞð7 mÞ3
12 ¼ 858 m4 The submerged volume is
V¼ ð1:3 mÞð7 mÞð30 mÞ ¼ 273 m3
The distance between the center of gravity and the center of buoyancy is
ybg¼ 3:6 m 2:4 m 0:65 m ¼ 0:55 m
The metacentric height measured above the center of gravity is
hm¼ I
V ybg¼858 m4
273 m3 0:55 m ¼ 2:6 m Customary U.S. Solution
(a) Refer to the following diagram. Let dimension y represent the depth of the submerged barge.
12 ft
24 ft
CB 2 ft y ⫽ 4 ft
end view
From Archimedes’ principle, the buoyant force equals the weight of the barge. This, in turn, equals the weight of the displaced seawater.
Fb¼ W ¼ V w
600;000 lbf ¼ y
ð24 ftÞð98 ftÞ 64 lbf
ft3
y¼ 4 ft
The center of buoyancy is located at the centroid of the submerged cross section. When floating on an even keel, the submerged cross section is rectangular with a height of 4 ft. The height of the center of buoyancy above the keel is
4 ft 2 ¼ 2 ft
(b) While the location of the new center of buoyancy can be determined, the location of the metacenter does not change significantly for small angles of heel. There-fore, for approximate calculations, the angle of heel is not significant.
WaterResources
...
The area moment of inertia of the longitudinal waterline cross section is
I¼Lw3
12 ¼ð98 ftÞð24 ftÞ3
12 ¼ 112;900 ft4 The submerged volume is
V¼ ð4 ftÞð24 ftÞð98 ftÞ ¼ 9408 ft3
The distance between the center of gravity and the center of buoyancy is
ybg¼ 12 ft 7:8 ft 2 ft ¼ 2:2 ft
The metacentric height measured above the center of gravity is
hm¼ I
V ybg¼112;900 ft4
9408 ft3 2:2 ft
¼ 9:8 ft
21. FLUID MASSES UNDER EXTERNAL