Soil Mechanics A

1
Flow Nets
2
Flow through a Dam
Phreatic line
Unsaturated Soil
Drainage blanket
Flow of water
z
x
3
Graphical representation of solution
1. Equipotentials Lines of constant head,
h(x,z)
Equipotential (EP)
4
Graphical representation of solution
2. Flow lines Paths followed by water
particles - tangential to flow
Phreatic line
Flow line (FL)
Equipotential (EP)
5
Properties of Equipotentials
Flow line (FL)
Equipotential (EP)
h(x,z) constant
(1a)
6
Properties of Equipotentials
Flow line (FL)
Equipotential (EP)
h(x,z) constant
(1a)
(1b)
Thus
7
Properties of Equipotentials
Flow line (FL)
Equipotential (EP)
h(x,z) constant
(1a)
(1b)
Thus
(1c)
Equipotenial slope
8
Properties of Flow Lines
Geometry
Flow line (FL)
vz
vx
Equipotential (EP)
Kinematics
From the geometry

(2b)
9
Properties of Flow Lines
Geometry
Flow line (FL)
vz
vx
Equipotential (EP)
Kinematics
From the geometry

(2b) Now from Darcys law
10
Properties of Flow Lines
Geometry
Flow line (FL)
vz
vx
Equipotential (EP)
Kinematics
From the geometry

(2b) Now from Darcys law Hence

(2c)
11
Orthogonality of flow and equipotential lines
Flow line (FL)
Equipotential (EP)
On an equipotential
On a flow line
12
Orthogonality of flow and equipotential lines
Flow line (FL)
Equipotential (EP)
On an equipotential
On a flow line
(3)
Hence
13
Geometric properties of flow nets
h?h
Y
h
Z
FL
h2?h
T
EP
t
X
y
FL
X
z
14
Geometric properties of flow nets
From the definition of flow
(4a)
h?h
Y
h
Z
FL
h2?h
T
EP
t
X
y
FL
X
z
15
Geometric properties of flow nets
From the definition of flow
(4a)
h?h
Y
h
From Darcys law
Z
FL
h2?h
(4b)
T
EP
t
X
y
FL
X
z
16
Geometric properties of flow nets
From the definition of flow
(4a)
h?h
Y
h
From Darcys law
Z
FL
h2?h
(4b)
T
EP
t
Combining (4a)(4b)
X
y
FL
X
(4c)
z
17
Geometric properties of flow nets
From the definition of flow
(4a)
h?h
Y
h
From Darcys law
Z
FL
h2?h
(4b)
T
EP
t
Combining (4a)(4b)
X
y
FL
X
(4c)
z
Similarly
(4d)
18
Geometric properties of flow nets
From the definition of flow
(4a)
h?h
Y
h
From Darcys law
Z
FL
h2?h
(4b)
T
EP
t
Combining (4a)(4b)
X
y
FL
X
(4c)
z
Similarly
(4d)
19
Geometric properties of flow nets
B
EP( h )
C
D
FL
A
d
EP ( h ?h )
b
a
c
20
Geometric properties of flow nets
From the definition of flow
B
(6a)
EP( h )
C
D
FL
A
d
EP ( h ?h )
b
a
c
21
Geometric properties of flow nets
From the definition of flow
B
(6a)
EP( h )
From Darcys law
C
(6b)
D
FL
A
d
EP ( h ?h )
b
a
c
22
Geometric properties of flow nets
From the definition of flow
B
(6a)
EP( h )
From Darcys law
C
(6b)
D
FL
A
d
Combining (6a)(6b)
EP ( h ?h )
b
a
(6c)
c
Similarly
(6d)
23
Geometric properties of flow nets
From the definition of flow
B
(6a)
EP( h )
From Darcys law
C
(6b)
D
FL
A
d
Combining (6a)(6b)
EP ( h ?h )
b
a
(6c)
c
Conclusion
Similarly
(6d)
24
Geometric properties of flow nets

• When drawing flow nets by hand it is most
  convenient to draw them so that
• Each flow tube carries the same flow ?Q
• The head drop between adjacent EPs, ?h, is the
  same
• Then the flow net is comprised of SQUARES

25
Geometric properties of flow nets
Demonstration of square rectangles with
inscribed circles
26
Drawing Flow Nets
To calculate the flow and pore pressures in the
ground a flow net must be drawn. The flow net
must be comprised of a family of orthogonal lines
(preferably defining a square mesh) that also
satisfy the boundary conditions.
27
Common boundary conditions
a. Submerged soil boundary - Equipotential
Water
H-z
H
u
w
h
z
?
?
?
w
z
(7)
Datum
28
Common boundary conditions
a. Submerged soil boundary - Equipotential
Water
H-z
H
u
w
h
z
?
?
?
w
z
now
u
H
z
?
?
?
(
)
(7)
w
w
Datum
29
Common boundary conditions
a. Submerged soil boundary - Equipotential
Water
H-z
H
u
w
h
z
?
?
?
w
z
now
u
H
z
?
?
?
(
)
(7)
w
w
Datum
so
H
z
?
?
(
)
w
h
z
H
?
?
?
?
w
30
Common boundary conditions
b. Impermeable soil boundary - Flow Line
Permeable Soil
vn0
Flow Line
vt
Impermeable Material
31
Common boundary conditions
c. Line of constant pore pressure - eg. phreatic
surface
Head is given by
32
Common boundary conditions
c. Line of constant pore pressure - eg. phreatic
surface
Head is given by and thus
33
Common boundary conditions
c. Line of constant pore pressure - eg. phreatic
surface
Head is given by and thus now if pore pressure
is constant
34
Common boundary conditions
c. Line of constant pore pressure - eg. phreatic
surface
Head is given by and thus now if pore pressure
is constant and hence
(8)
35
Common boundary conditions
c. Line of constant pore pressure - eg. phreatic
surface
36
Procedure for drawing flow nets

• Mark all boundary conditions
• Draw a coarse net which is consistent with the
  boundary conditions and which has orthogonal
  equipotentials and flow lines. (It is usually
  easier to visualise the pattern of flow so start
  by drawing the flow lines).
• Modify the mesh so that it meets the conditions
  outlined above and so that rectangles between
  adjacent flow lines and equipotentials are
  square.
• Refine the flow net by repeating the previous
  step.

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Example Suction Caisson Installation
38
Value of head on equipotentials
(9)
Phreatic line
15 m
Datum
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
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Calculation of flow
Phreatic line
15 m
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
For a single Flow tube of width 1m ?Q k
?h (10a)
40
Calculation of flow
Phreatic line
15 m
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
For a single Flow tube of width 1m ?Q k
?h (10a)
For k 10-5 m/s and a width of 1m ?Q
10-5 x 3 m3/sec/m (10b)
41
Calculation of flow
Phreatic line
15 m
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
For a single Flow tube of width 1m ?Q k
?h (10a)
For k 10-5 m/s and a width of 1m ?Q
10-5 x 3 m3/sec/m (10b)
For 5 such flow tubes Q
5 x 10-5 x 3 m3/sec/m (10c)
42
Calculation of flow
Phreatic line
15 m
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
For a single Flow tube of width 1m ?Q k
?h (10a)
For k 10-5 m/s and a width of 1m ?Q
10-5 x 3 m3/sec/m (10b)
For 5 such flow tubes Q
5 x 10-5 x 3 m3/sec/m (10c)
For a 25m wide dam Q
25 x 5 x 10-5 x 3 m3/sec (10d)
43
Calculation of flow
Phreatic line
15 m
h 15m
h 0
h 3m
h 12m
h 6m
h 9m
For a single Flow tube of width 1m ?Q k
?h (10a)
For k 10-5 m/s and a width of 1m ?Q
10-5 x 3 m3/sec/m (10b)
For 5 such flow tubes Q
5 x 10-5 x 3 m3/sec/m (10c)
For a 25m wide dam Q
25 x 5 x 10-5 x 3 m3/sec (10d)
(10e)
Note that per metre width
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Calculation of pore pressure
Phreatic line
15 m
h 15m
h 0
5m
P
h 3m
h 12m
h 6m
h 9m
5m
P
(11a)
Pore pressure from
45
Calculation of pore pressure
Phreatic line
15 m
h 15m
h 0
5m
P
h 3m
h 12m
h 6m
h 9m
5m
P
(11a)
Pore pressure from
At P, using dam base as datum
(11b)
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Example Calculating Pore Pressures
20 m
10 m
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Step 1 Choose a convenient datum. In this
example the sea floor has been chosen Then H1
40 m H2 1 m. The increment of head, ?h
39/9 4.333 m
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A
B
C
D
E
Step 2 Calculate the head at points along the
base of the vessel. For convenience these are
chosen to be where the EPs meet the vessel (B
to E) and at the vessel centerline (A). Hence
calculate the pore water pressures. At B
Head H1 - 5 ?h H2 4 ?h 18.33 m Pore
pressure at B
18.33 ?w 179.8 kPa
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Step 3 Calculate the upthrust (Force/m) due to
pore pressures
3218 kN/m Without pumping Upthrust 20 ? 1
? 9.81 196 kN/m Upthrust due to Pumping
3218 196 3022 kN/m
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