Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

18.6 Calculus 621

Example 18.14 As a civil engineer, you may be assigned the task of determining the force exerted by water

that is stored behind a dam. We discussed the concept of hydrostatic pressure in Chapter 10

and stated that, for fluid at rest, the pressure increases with the depth of fluid as shown in

Figure 18.20 and according to

(18.34)

where

Pfluid pressure at a point located a distanceybelow the

water surface (Pa or lb/ft

2

)

rdensity of the fluid (kg /m

3

or slugs /ft

3

)

gacceleration due to gravity (g9.81 m/s

2

or

g32.2 ft /s

2

)

ydistance of the point below the fluid surface ( m or ft)

Since the force due to the water pressure varies with

depth, we need to add the pressure exerted on areas at various

depths to obtain the net force. Consider the force acting at

depthyover a small areadA, as shown in Figure 18.21.

The procedure for computing the total force is demonstrated using the following steps. Note

these steps make use of integrals.

step 1 step 2 step 3 step 4 step 5 step 6

Step 1: The net force is equal to the sum (integral) of all the little forces acting at different depths.

Step 2: We substitute fordFpdA(recall, force is equal to pressure times area).

Net Force

H

0

dF

H

0

pdA

H

0

rgydArg (^) 
H
0
ydArgw (^) 
H
0
y dy
1
2
rgwH
2
V u u u u u
Prg y
h
w
dx
y dA
y
x
■Figure 18.19
Differential element used to calculate the second moment of area.
y
■Figure 18.20 The variation of pressure with depth.
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