Consolidation of soils II · 2018-01-04 · Consolidation involves movement of water through soil...
Transcript of Consolidation of soils II · 2018-01-04 · Consolidation involves movement of water through soil...
Consolidation involves movement
of water through soil skeleton:
It takes time!
This usually matters only in clays Water
drainage
Terzaghi’s consolidation theory (テルツァギの圧密理論)
• Soil is saturated (土は飽和している)
• Darcy’s law holds (ダルシーの法則が成り立つ)
• Principle of effective stress holds (有効応力の法則が成り立つ)
• Compression expressed with mv (mvによる圧縮の表現)
• 1-D Consolidation: No lateral deformation (横方向には変形しない)
• No compression of soil particles and water (土粒子と水は圧縮しない)
■ Consolidation process (圧密過程) 1
Apply pressure p instantly
Water is far more compressible than
soil skeleton
At the first moment, all the applied
pressure is taken by water
As time passes, water is drained and
gradually p falls on soil
Before
loading
Moment of
loading
After long enough
time
Total stress s s0 (z) s0 (z) + p s0 (z) + p
Pore water pressure, u u0 (z) u0 (z) + p u0 (z)
Effective stress, s' s0'(z) s0' (z) s0' (z) + p
z
= Excess pore water pressure (過剰間隙水圧)Du
■ Consolidation process (圧密過程) 2
① Soil skeleton compression
zD
xD
yD
z : Compression
Volumetric strain – effective stress
relationship (体積ひずみ-有効応力関係)
s DD vv m
The volume reduction is
vzyxV DDDDD
■ Deriving consolidation equation (圧密方程式の導出) 4
② Net water flux
zD
xD
yD
ii D
i
Hydraulic gradient
Net water outflow (正味の流出体積) per unit time:
: Coefficient of permeability
(透水係数)
Water flux, q = v × Area
k
z
■ Deriving consolidation equation (圧密方程式の導出) 5
(Darcy’s law)
yxki DD
h = u /gw+ z
yxikQ DDD
Piezometric head = Pressure head + Potential head (ピエゾ水頭) (圧力水頭) (位置水頭)
■ Deriving consolidation equation (圧密方程式の導出) 6
zD
xD
yD
ii D
i
Hydraulic gradient
z
Soil skeleton compression = Net water outflow
zyxumV v DDDDD styx
z
uktQ
w
DDD
DD 1
1
g
tyxz
ukzyxum
w
v DDD
DDDDD 1
1
gs
1
1
z
u
zk
t
um
w
vg
sWhen
s is constant
0
t
s
■ Deriving consolidation equation (圧密方程式の導出) 7
Pore water pressure = Hydrostatic pressure + Excess pore water pressure (間隙水圧) (静水圧) (過剰間隙水圧)
2
2
z
u
m
k
t
u
wv
g
2
2
z
uc
t
uv
Terzaghi’s (one-dimenssional) consolidation equation (テルツァギの(一次元)圧密方程式)
zzu wg)(0 uDu
2
2
z
uzc
t
uz wv
w
D
D gg
2
2
z
uc
t
uv
D
DValid for both u and Du
■ Deriving consolidation equation (圧密方程式の導出) 8
• Can be solve with the method of separation of variables (変数分離法)
• 1 initial condition (初期条件), 2 boundary conditions (境界条件) necessary
Example
• Boundary condition 1: Du = 0 at z = 0
• Boundary condition 2: Du = 0 at z = 2d (or, at z = d )
• Initial condition: Du = p at any z
z
p
Du = p
Apply pressure p instantly
■ Solving consolidation equation (圧密方程式を解く) 9
0/ zu
0
2d
Obtained in form of Fourier series (フーリエ級数)
D
4
12exp
2
12sin
12
14 22
0
v
n
Tn
H
zn
npu
tH
cT v
v 2
Function of z
(location)
Function of t
(time)
0
* In textbook, u is used as “excess pore water pressure” in place of Du.
∞ t
■ Solution (解) 10
2d
z
Initial condition: Du=p at Tv=0
Tv =
As Tv increases, the excess pore water pressure Du dissipates (間隙水圧が消散する)
Du/p
■ Solution (解) 11
D
4
12exp
2
12sin
12
14 22
0
v
n
Tn
H
zn
npu
2d
k ×2
Tv
H ×2
Tv
tH
cT v
v 2
20mm thick specimen
in laboratory:
1 hour for consolidation
20m thick real ground:
114 years for consolidation
Impervious (不透水性)
■ Meaning of time factor, Tv 12
Equivalent to having half thickness.
Consolidation takes 1/4 of the time for
single-side drainage case.
Du/p Single-side drainage (片面排水)
Double-side drainage (両面排水)
■ Influence of boundary conditions 13
20m
10m
10m
Impervious (不透水性)
Pervious (透水性): Sand, gravel
20mm thick specimen
in laboratory:
1 hour for consolidation
20m thick real ground:
114 years for consolidation
20m thick real ground,
But with 10m drainage length:
29 years for consolidation
■ Example 14
In (b) ~ (e), how much time does consolidation take
when compared to (a)?
■ Further example 15
Impervious
10m
Consol. layer
(k = 1.0
×10-8 m/s)
Permeable
10m
Consol. layer
(k = 1.0
×10-8 m/s)
Permeable
10m
Consol. layer
(k = 2.0
×10-8 m/s)
Permeable
5m
Consol. layer
(k = 2.0
×10-8 m/s)Impervious
5m
Consol. layer
(k = 1.0
×10-8 m/s)
(a) (b) (c) (d) (e)
Degree of consolidation (圧密度), Uz :
Uz = 0:
Uz = 1:
21
1),(ee
eeTzU vz
Final reduction of void ratio
Or,
Current reduction of void ratio
■ Degree of consolidation (圧密度) 16
00
0 1),(u
u
u
uuTzU vz
D
D
D
DD
Du/p
Uz depends on depth
Take the average over layer Depth
0
d
Uz(Tv=0.6)=90% 0.2H
0.8H Uz(Tv=0.6)=73%
dzTzUH
TUH
vzvz 0
),(1
)(
)( vz TU
1
2
22 2
12exp
)12(
181
n
vTn
■ Degree of consolidation (圧密度) 17
d
d
Initial Du
distribution