fluid_mechanics

2.10 Hydrostatic Force on a Curved Surface 67
A
B
F 1
A
CG
F 2
O
B
F H
(a)
F √(F H ) 2 + (F V ) 2
R =
O
B
C
C
C
(b) (c) (d)
F V
F I G U R E 2.23
force on a curved surface.
Hydrostatic
The resultant F R passes through the point O, which can be located by summing moments about an
appropriate axis. The resultant force of the fluid acting on the curved surface BC is equal and opposite
in direction to that obtained from the free-body diagram of Fig. 2.23c. The desired fluid
force is shown in Fig. 2.23d.
E XAMPLE 2.9
GIVEN A 6-ft-diameter drainage conduit of the type shown in
Fig. E2.9a is half full of water at rest, as shown in Fig. E2.9b.
Hydrostatic Pressure Force on a Curved Surface
FIND Determine the magnitude and line of action of the resultant
force that the water exerts on a 1-ft length of the curved section
BC of the conduit wall.
3 ft
A
B
F 1
1 ft
A
CG
B
F H
1.27 ft
A
F R = 523 lb
32.5°
O
1 ft
C
C
F V
(a)
(b)
(c)
(d)
F I G U R E E2.9
(Photograph courtesy of CONTECH Construction Products, Inc.)
SOLUTION
We first isolate a volume of fluid bounded by the curved section
BC, the horizontal surface AB, and the vertical surface AC, as
shown in Fig. E2.9c. The volume has a length of 1 ft. The forces
acting on the volume are the horizontal force, F 1 , which acts on
the vertical surface AC, the weight, w, of the fluid contained
within the volume, and the horizontal and vertical components of
the force of the conduit wall on the fluid, F H and F V , respectively.
The magnitude of F 1 is found from the equation
F 1 gh c A 162.4 lbft 3 2 1 3 2 ft2 13 ft 2 2 281 lb
and this force acts 1 ft above C as shown. The weight w gV ,
where V is the fluid volume, is
w g V 162.4 lbft 3 2 19p4 ft 2 2 11 ft2 441 lb
and acts through the center of gravity of the mass of fluid, which
according to Fig. 2.18 is located 1.27 ft to the right of AC as
shown. Therefore, to satisfy equilibrium
F H F 1 281 lb F V w 441 lb
and the magnitude of the resultant force is
F R 21F H 2 2 1F V 2 2
21281 lb2 2 1441 lb2 2 523 lb
(Ans)
The force the water exerts on the conduit wall is equal, but opposite
in direction, to the forces F H and F V shown in Fig. E2.9c.
Thus, the resultant force on the conduit wall is shown in
Fig. E2.9d. This force acts through the point O at the angle shown.