Problem 3.84
[Difficulty: 3]
Given:
Curved surface, in shape of quarter cylinder, with given radius R and width w; liquid concrete stands to depth H.
R1
f
t
⋅
=
w4
f
t
⋅
=
F
vmax
350 lbf
⋅
=
SG
2.50
=
From Table A.1, App A
Find:
(a) Maximum depth of concrete to avoid cracking
(b) Line of action on the form.
(c)
Plot the vertical force and line of action over H ranging from 0 to R.
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure  h is positive downwards from
free surface)
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
(Vertical Hydrostatic Force)
d
h
θ
1
F
V
x
x’
y
θ
x' F
v
⋅
F
v
x
⌠
⎮
⎮
⌡
d
=
(Moment of vertical force)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts on free surface of the concrete
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
From the geometry:
y
R sin
θ
()
⋅
=
x
R cos
θ
⋅
=
hyd
−
=
dRH
−
=
dA
w R
⋅
d
θ
⋅
=
Therefore the vertical component of the hydrostatic force is:
F
v
A
y
p
⌠
⎮
⎮
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 Fall '07
 Lear
 Fluid Mechanics

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