Consolidation
Now for each layer, sc = Uvsoed, which is proportional to Uv H. Hence if U is the overall degree of consolidation for the two layers combined:
4 5 1 5 6 0
4 5 0 825 1 5 0 97 6 0
1 2
. . .
. . . . .
U U U
U
+ =
(
´)
+(
´)
=i.e.
Hence U = 0.86 and the 3-year settlement is sc =0.86 182 157 mm´ =
of over 30 m. Prefabricated drains are now generally used, and tend to be more economic than backfilled drains for a given area of treatment. One type of drain (often referred to as a ‘sand‑
wick’) consists of a filter stocking, usually of woven polypropylene, filled with sand. Compressed air is used to ensure that the stocking is completely filled with sand. This type of drain, a typical diameter being 65 mm, is very flexible and is generally unaffected by lateral soil displacement, the possibility of necking being virtually eliminated. The drains are installed either by insertion into pre-bored holes or, more commonly, by placing them inside a mandrel or casing which is then driven or vibrated into the ground.
Another type of prefabricated drain is the band drain, consisting of a flat plastic core indented with drainage channels, surrounded by a layer of filter fabric. The fabric must have sufficient strength to prevent it from being squeezed into the channels, and the mesh size must be small enough to prevent the passage of soil particles which could clog the channels. Typical dimensions of a band drain are 100 × 4 mm, and in design the equivalent diameter is assumed to be the perim- eter divided by π. Band drains are installed by placing them inside a steel mandrel which is either pushed, driven or vibrated into the ground. An anchor is attached to the lower end of the drain to keep it in position as the mandrel is withdrawn. The anchor also prevents soil from entering the mandrel during installation.
Drains are normally installed in either a square or a triangular pattern. As the object is to reduce the length of the drainage path, the spacing of the drains is the most important design consideration. The spacing must obviously be less than the thickness of the clay layer for the consolidation rate to be improved; there is therefore no point in using vertical drains in relatively thin layers. It is essential for a successful design that the coefficients of consolidation in both the horizontal and the vertical directions (ch and cv, respectively) are known as accurately as pos- sible. In particular, the accuracy of ch is the most crucial factor in design, being more important than the effect of simplifying assumptions in the theory used. The ratio ch/cv is normally between 1 and 2; the higher the ratio, the more advantageous a drain installation will be. A design com- plication in the case of large-diameter sand drains is that the column of sand tends to act as a weak pile (see Chapter 10), reducing the vertical stress increment imposed on the clay layer by an unknown degree, resulting in lower excess pore water pressure and therefore reduced con- solidation settlement. This effect is minimal in the case of prefabricated drains because of their flexibility.
Vertical drains may not be effective in overconsolidated clays if the vertical stress after consol- idation remains less than the preconsolidation pressure. Indeed, disturbance of overconsolidated clay during drain installation might even result in increased final consolidation settlement. It should be realised that the rate of secondary compression cannot be controlled by vertical drains.
In polar coordinates, the three-dimensional form of the consolidation equation, with different soil properties in the horizontal and vertical directions, is
∂
∂ = ∂
∂ + ∂
∂
+ ∂
∂ u
t c u
r r u
r c u
z
e h e e
v e
2 1
2
2
2 (4.38)
The vertical prismatic blocks of soil which are drained by and surround each drain are replaced by cylindrical blocks, of radius R, having the same cross-sectional area (Figure 4.26). The solu- tion to Equation 4.38 can be written in two parts:
Uv =f T
( )
vConsolidation
and
Ur= f T
( )
rwhere Uv = average degree of consolidation due to vertical drainage only; Ur = average degree of consolidation due to horizontal (radial) drainage only;
T c t
v = dv2 (4.39)
= time factor for consolidation due to vertical drainage only T c t
r = Rh
4 2 (4.40)
= time factor for consolidation due to radial drainage only
The expression for Tr confirms the fact that the closer the spacing of the drains, the quicker consolidation due to radial drainage proceeds. The solution of the consolidation equation for radial drainage only may be found in Barron (1948); a simplified version which is appropriate for design is given by Hansbo (1981) as
U e
T r
r
= -1
8
m (4.41)
where
m = − − + +
≈ − n
n n
n n n
2
2 1 2 4
3 4
1 1 3
ln ln 4 (4.42)
In Equation 4.42, n = R/rd, R is the radius of the equivalent cylindrical block and rd is the radius of the drain. The consolidation curves given by Equation 4.41 for various values of n are plotted Figure 4.26 Cylindrical blocks.
in Figure 4.27. Some vertical drainage will continue to occur even if vertical drains have been installed and it can also be shown that
(1-U)=
(
1-Uv) (
1-Ur)
(4.43)where U is the average degree of consolidation under combined vertical and radial drainage.
Installation Effects
The values of the soil properties for the soil immediately surrounding the drains may be signifi- cantly reduced due to remoulding during installation, especially if boring is used, an effect known as smear. The smear effect can be taken into account either by assuming a reduced value of ch or by using a reduced drain diameter in Equations 4.41–4.43. Alternatively, if the extent and permeability (ks) of the smeared material are known or can be estimated, the expression for μ in Equation 4.42 can be modified after Hansbo (1979) as follows:
m»lnn+ ln - S
k k S
s
3
4 (4.44)
Example 4.7
An embankment is to be constructed over a layer of clay 10 m thick, with an impermeable lower boundary. Construction of the embankment will increase the total vertical stress in the clay layer by 65 kPa. For the clay, cv = 4.7 m2/year, ch = 7.9 m2/year and mv = 0.25 m2/ MN. The design requirement is that all but 25 mm of the settlement due to consolidation Figure 4.27 Relationships between average degree of consolidation and time factor
for radial drainage.
Consolidation
of the clay layer will have taken place after 6 months. Determine the spacing, in a square pattern, of 400-mm diameter sand drains to achieve the above requirement.
Solution
Final settlement
mm
= v ¢ = ´ ´
=
m DsH 0 25 65 10 162
.
For t = 6 months, U 162 25
= 162-
=0 85.
For vertical drainage only, the layer is half-closed, and therefore d = 10 m.
T c t
v = dv = ´
2 2 =
4 7 0 5
10 0 0235
. .
.
From curve 1 of Figure 4.15, or using the spreadsheet tool on the Companion Website Uv = 0.17.
For radial drainage the diameter of the sand drains is 0.4 m, i.e. rd = 0.2 m. The radius of the cylindrical block is:
R nr= d =0.2n and
T c t
R n n
r = h = ´
´ ´ =
4
7 9 0 5 4 0 2
24 7
2 2 2 2
. .
.
. i.e.
Ur= -e n n
´
( - )
é ë êê
ù û úú
1
8 24 7 2 0 75
. ln .
Now (1–U) = (1–Uv)(1–Ur), and therefore 0 15 0 83 1
0 82
. .
.
=
(
-)
=
U U
r
r
The value of n for Ur = 0.82 may then be found by evaluating Ur at different values of n using Equations 4.41 and 4.42 and interpolating the value of n at which Ur = 0.82. Alternatively a simple ‘Goal seek’ or optimisation routine in a standard spreadsheet may be used to solve the equations iteratively. For the first of these methods, Figure 4.28 plots the value of Ur against n, from which it may be seen that n = 9. It should be noted that this process is greatly shortened by the use of a spreadsheet to perform the calculations. Therefore
R=0 2 9 1 8. ´ = . m
The spacing of drains to achieve this, if placed in a square pattern is given by
S= R = =
0 564 1 8 0 564 3 2 .
.
. . m