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Problem 3.73
[Difficulty: 2]
Given:
Liquid concrete is poured into the form shown
R2
f
t
⋅
=
w1
5
f
t
⋅
=
SG
c
2.5
=
(Table A.1, App. A)
Find:
Magnitude and line of action of the vertical force on the form
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure  h is positive downwards)
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
(Vertical Hydrostatic Force)
x' F
v
⋅
F
v
x
⌠
⎮
⎮
⌡
d
=
(Moment of vertical force)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts at free surface of concrete
and
on outside of gate
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
A
ρ
g
⋅
h
⋅
sin
θ
()
⋅
⌠
⎮
⎮
⌡
d
=
where
dA
w R
⋅
d
θ
⋅
=
and
hRy
−
=
R
R sin
θ
⋅
−
=
Therefore,
F
v
0
π
2
θ
ρ
g
⋅
R
R sin
θ
⋅
−
⋅
w
⋅
R
⋅
sin
θ
⋅
⌠
⎮
⎮
⌡
d
=
ρ
g
⋅
w
⋅
R
2
⋅
0
π
2
θ
sin
θ
( )
sin
θ
2
−
⎣
⎦
⌠
⎮
⎮
⌡
d
⋅
=
Evaluating the integral:
F
v
ρ
g
⋅
w
⋅
R
2
⋅
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 Fall '07
 Lear
 Statics, Fluid Mechanics

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