Problem 3.68
[Difficulty: 4]
Given:
Various dam crosssections
Find:
Which requires the least concrete; plot crosssection area
A
as a function of
α
Solution:
For each case, the dam width
b
has to be large enough so that the weight of the dam exerts enough moment to balance the
moment due to fluid hydrostatic force(s).
By doing a moment balance this value of
b
can be found
a) Rectangular dam
Straightforward application of the computing equations of Section 35 yields
b
D
F
H
y
mg
O
F
H
p
c
A
⋅
=
ρ
g
⋅
D
2
⋅
w
⋅
D
⋅
=
1
2
ρ
⋅
g
⋅
D
2
⋅
w
⋅
=
y'
y
c
I
xx
A y
c
⋅
+
=
D
2
w D
3
⋅
12 w
⋅
D
⋅
D
2
⋅
+
=
2
3
D
⋅
=
so
y
D
y'
−
=
D
3
=
Also
m
ρ
cement
g
⋅
b
⋅
D
⋅
w
⋅
=
SG
ρ
⋅
g
⋅
b
⋅
D
⋅
w
⋅
=
Taking moments about
O
M
0.
∑
0
=
F
H
−
y
⋅
b
2
m
⋅
g
⋅
+
=
so
1
2
ρ
⋅
g
⋅
D
2
⋅
w
⋅
⎛
⎜
⎝
⎞
⎠
D
3
⋅
b
2
SG
ρ
⋅
g
⋅
b
⋅
D
⋅
w
⋅
(
)
⋅
=
Solving for
b
b
D
3 SG
⋅
=
The minimum rectangular crosssection area is
A
b D
⋅
=
D
2
3 SG
⋅
=
For concrete, from Table A.1, SG = 2.4, so
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 Fall '07
 Lear
 Fluid Mechanics

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