Pressure Distribution in a Fluid
2.6 Hydrostatic Forces on Curved Surfaces
The resultant pressure force on a curved surface is most easily computed by separating it into horizontal and vertical components. Consider the arbitrary curved surface sketched in Fig. 2.14a. The incremental pressure forces, being normal to the local area element, vary in direction along the surface and thus cannot be added numerically. We could sum the separate three components of these elemental pressure forces, but it turns out that we need not perform a laborious three-way integration.
Figure 2.14b shows a free-body diagram of the column of fluid contained in the vertical projection above the curved surface. The desired forces FH and FV are exerted by the surface on the fluid column. Other forces are shown due to fluid weight and horizontal pressure on the vertical sides of this column. The column of fluid must be in static equilibrium. On the upper part of the column bcde, the hor- izontal components F1 exactly balance and are not relevant to the discussion. On the lower, irregular portion of fluid abc adjoining the surface, summation of hori- zontal forces shows that the desired force FH due to the curved surface is exactly equal to the force FH on the vertical left side of the fluid column. This left-side force can be computed by the plane surface formula, Eq. (2.26), based on a verti- cal projection of the area of the curved surface. This is a general rule and simpli- fies the analysis:
The horizontal component of force on a curved surface equals the force on the plane area formed by the projection of the curved surface onto a vertical plane nor- mal to the component.
If there are two horizontal components, both can be computed by this scheme. Sum- mation of vertical forces on the fluid free body then shows that
(2.30) We can state this in words as our second general rule:
The vertical component of pressure force on a curved surface equals in magnitude and direction the weight of the entire column of fluid, both liquid and atmosphere, above the curved surface.
FVW1W2Wair
2.6 Hydrostatic Forces on
Thus the calculation of FV involves little more than finding centers of mass of a column of fluid—perhaps a little integration if the lower portion abc in Fig. 2.14b has a particularly vexing shape.
EXAMPLE 2.8
A dam has a parabolic shape z/z0(x/x0)2as shown in Fig. E2.8a, with x010 ft and z024 ft. The fluid is water, 62.4 lbf/ft3, and atmospheric pressure may be omitted.
Compute the forces FHand FVon the dam and their line of action. The width of the dam is 50 ft.
E2.8a Solution
• System sketch: Figure E2.8b shows the various dimensions. The dam width is b50 ft.
• Approach: Calculate FHand its line of action from Eqs. (2.26) and (2.29). Calculate FV and its line of action by finding the weight of fluid above the parabola and the centroid of this weight.
• Solution steps for the horizontal component: The vertical projection of the parabola lies along the z axis in Fig. E2.8b and is a rectangle 24 ft high and 50 ft wide. Its centroid is halfway down, or hCG24/212 ft. Its area is Aproj(24 ft)(50 ft) 1200 ft2. Then, from Eq. (2.26),
The line of action of FHis below the centroid of Aproj, as given by Eq. (2.29):
Thus FHis 12 4 16 ft, or two-thirds of the way down from the surface (8 ft up from the bottom).
• Comments: Note that you calculate FH and its line of action from the vertical projec- tionof the parabola, not from the parabola itself. Since this projection is vertical, its angle 90.
• Solution steps for the vertical component: The vertical force FV equals the weight of water above the parabola. Alas, a parabolic section is not in Fig. 2.13, so we had to look
yCP, proj Ixx sin hCGAproj
(1/12)(50 ft)(24 ft)3 sin 90 (12 ft)(1200 ft2) 4 ft FHhCGAproja62.4lbf
ft3b(12 ft)(1200 ft2)898,560 lbf899103 lbf z = z0
(
xx0(
2pa = 0 lbf/ft2 gage
FV FH z
x0
x z0
it up in another book. The area and centroid are shown in Fig. E2.8b. The weight of this parabolic amount of water is
E2.8b
This force acts downward, through the centroid of the parabolic section, or at a distance 3x0/8 3.75 ft over from the origin, as shown in Figs. E2.8b,c. The resultant hydrostatic force on the dam is
Ans.
This resultant is shown in Fig. E2.8c and passes through a point 8 ft up and 3.75 ft over from the origin. It strikes the dam at a point 5.43 ft over and 7.07 ft up, as shown.
• Comments: Note that entirely different formulas are used to calculate FH and FV. The concept of center of pressure CP is, in the writer’s opinion, stretched too far when applied to curved surfaces.
E2.8c
z
Resultant = 1028 × 103 lbf acts along z = 10.083–0.5555x
Parabola z = 0.24x2 29°
7.07 ft
5.43 ft 3.75 ft
x 8 ft
0
FV = 499 × 103 lbf
FH = 899 × 103 lbf
F(F2HFV2)1/2 3(899E3 lbf)2(499E3 lbf)241/21028103 lbf at 29 3z0
5
Area = 2x0z0 3 z0 = 24 ft
0 3x0
8
x0 = 10 ft Parabola FV FVAsectionba62.4lbf
ft3b c2
3 (24 ft)(10 ft)d(50 ft)499,200 lbf 499103 lbf
EXAMPLE 2.9
Find an algebraic formula for the net vertical force F on the submerged semicircular pro- jecting structure CDE in Fig. E2.9. The structure has uniform width b into the paper. The liquid has specific weight .
Solution
The net force is the difference between the upward force FLon the lower surface DE and the downward force FU on the upper surface CD, as shown in Fig. E2.9. The force FU equals times the volume ABDC above surface CD. The force FLequals times the vol- ume ABDEC above surface DE. The latter is clearly larger. The difference is times the volume of the structure itself. Thus the net upward fluid force on the semicylinder is
Ans.
This is the principle upon which the laws of buoyancy, Sec. 2.8, are founded. Note that the result is independent of the depth of the structure and depends upon the specific weight of the fluid, not the material within the structure.
The formulas for plane and curved surfaces in Secs. 2.5 and 2.6 are valid only for a fluid of uniform density. If the fluid is layered with different densities, as in Fig. 2.15, a single formula cannot solve the problem because the slope of the linear pressure