1-25-00, Settlement and Consolidation
Ref:
Principles of Geotechnical Engineering, Braja M. Das, 1994
Coastal Engineering Handbook, J.B. Herbich, 1991
Topics:
Effective Stress
Compression and Consolidation Process
Settlement Categories
Preconsolidation Condition
Calculation of Primary Consolidation Settlement
Time Rate of Consolidation
--------------------------------------------------------------------------------------------------------------------Effective Stress
Total stress at point A below the soil surface is equal to (1) the stress carried by the water
in the continuous void spaces (i.e. pore water pressure, u) and (2) the stress carried by the
soil solids at their points of contact (i.e. the soil skeleton). Approximately the effective
stress (')
H
A
HA
  H w  H A  H  sat
W  Ww
where  sat    s
Vtotal
  'u1  a
where a is the fractional surface area of solids (usually a<<1 for all
practical conditions) so   'u
rearranging and subtracting the pore pressure ( u  H A  w , i.e. hydrostatic pressure)
  H w  H A  H  sat   H A  w  H A  H   , where      w
Compression of soil layers due to stress increase by construction of foundations or other loads.
Compression is caused by:
1. Deformation of soil particles
2. Relocation of soil particles
3. Expulsion of water or air from void spaces
Consolidation - Process the reduction of bulk soil volume under loading due to flow of pore
water. For saturated soils, any increment of loading (, called surcharge) will be initially taken
up by the pore pressure and result in consolidation until a new equilibrium is reached where the
soil solids (or skeleton) takes up the added load.
surcharge:   'u
For cohesive soils:
at t = 0,   u ;
t=,   '
For non-cohesive soils: water drains faster and the load is transferred immediately
Categories:
1. Immediate settlement - elastic deformation of dry soil and moist and saturated soils
without change to moisture content
a. due to high permeability, pore pressure in clays support the entire added load and no
immediate settlement occurs
b. generally, due to the construction process, immediate settlement is not important
2. Primary consolidation settlement - volume change in saturated cohesive soils because of
the expulsion of water from void spaces
a. high permeability of sandy, cohesionless soils result in near immediate drainage due
to the increase in pore water pressure and no primary consolidation settlement occurs
3. Secondary compression settlement - plastic adjustment of soil fabric in cohesive soils
Preconsolidation Condition
1. normally consolidated - present effective overburden pressure = maximum pressure the
soil has been subjected to in the past (pc)
2. overconsolidated - present effective overburden pressure < maximum pressure the soil
has been subjected to in the past (pc)
Normally consolidated
Maximum
past load
e
e
Overconsolidated
Preconsolidation pressure determination
(Casagrande, 1936)
1. Establish point a at which e-log p has
minimum radius of curvature
2. Draw horizontal line from a (line ab)
3. Draw tangent to curve at a (line ac)
4. Draw line ad to bisect angle bac
5. Project the straight-line portion of gh
back to intersect ad at f
6. Abscissa of point f is the
preconsolidation pressure, pc
Void ratio, e
log p
Non-linear
rebound when
load is removed
log p
pc
a
b
f
d
c
g
pc
log p
h
Calculation of Ultimate Primary Consolidation Settlement
Saturated clay soil layer of thickness H, cross-sectional area A, existing overburden
pressure po, increase in pressure p, and resulting ultimate primary consolidation
settlement S
Change in volume is V  SA , change in volume is equal to the change in volume of the
voids (definition of settlement) and by the definition of the void ratio:
V  SA  Vv  eVS
V
AH
Using the initial void ratio and total volume (eo and Vo) gives VS  o 
1  eo 1  eo
Combining and rearranging gives
e
AH
V  SA  eVS 
e  S  H
1  eo
1  eo
e1  e2
e
Compression index Cc = slope of the e-log p curve: Cc 

log  p2  log  po  p 
po 
 p1 

 p  p 
CH

S  c log  o
1  eo
p
o




For thick clay, more accurate to divide multiple layers
Consider depth to 2B for square foundation (BxB) or 4B for strip foundations
(BxL), B is the width
Compression vs. Swell Index
normally consolidated clays (pc = po + p)
Good
condition!
S 
 po i   pi  
Cc H i

log 


1  eo
p
o i 


overconsolidated clay (pc  po + p) use the Swell Index (Cs)
 p  p 
CH

NOTE: use of Cc vice Cs is conservative
S  s log  o
1  eo
p
o


most marine soils are overconsolidated - sedimentation increases the surcharge on
the soil, but subsequent erosion removes much of the load
underconsolidated clay (pc  po + p)
(RARE CONDITION)
p  CH
 p  p 
C H

(partially on both curves)
S  S log  c   c log  o
1  eo
p
1

e
p
o
c
 o



If the e-log p plot is known, can simply find e over the appropriate range of
e
pressures and use S  H
(works for all conditions)
1  eo
Compression index determination
1. Graphically from laboratory e-log p plot, use "virgin compression curve" (i.e.
straight line portion of the curve)
2.38
 1  eo 

2. Rendon-Herrero (1983) C c  0.141G 
 Gs 
 LL%  
Cc  0.2343
Gs
3. Nagaraj and Murty (1985)
 100 
1. 2
s
Swell Index… determined from lab tests, generally C S  101 CC to 15 CC
Time Rate of Consolidation
Derivation assumptions
1. Homogeneous clay-water system
2. Saturated
3. Water and soil grains are incompressible
4. Flow of water is unidirectional and in the direction of consolidation
5. Darcy's law assumed - v  ik , v = discharge velocity, k = coeff of
permeability, i = hydraulic gradient, i  h L
p
h = u/w
[vz + (dvz /dz)dz]dxdy
sand
dz
dy
2Hdr
dx
clay
z
vzdxdy
sand
[rate of water outflow] - [rate of water inflow] = [rate of change of volume]
restrict flow to vertical (z) direction (assumption 4)
v
V


 v z  z dz dxdy  v z dxdy 
z 
t

v z
V
dxdydz 
z
t
Darcy's law gives
v z  ki  k
combining gives

h
u
 k
since u   w h
z
z
k  2u
1 V

2
 w z
dxdydz t
during settlement
VS 
VS
V Vv  VS  eVS  dxdydz e
 0 and



, since
t
t
t
t
1  eo t
V
dxdydz

1  eo
1  eo


k  2u
1 e

2
 w z
1  eo t
assume that the decrease in void ratio is proportional to the increase in effective stress (or
the decrease in pore pressure)
e  av u , av = coeff. of compressibility
define the coeff. of volume compressibility mv 
av
1  eo
k
k  2u
u
 2u
u
c

,
define
coeff.
of
consolidation

,



m
 cv
v
v
2
2
 w mv
 w z
t
t
z
solving gives a time factor Tv 
cv t
H dr2
estimate mv from e-log p plot at appropriate pressures, mv 
e e
1 e
, eav  1 2
2
1  eav p
Variation of Degree of Consolidation with Time Factor
degree of consolidation, U(%)/100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.001
0.01
0.1
1
2
Time Factor, Tv=Cvt/H
Sivaram & Swamee (1977) empirical relationship for U (degree of settlement) from 0-100%
2
   U % 
4Tv
 

U%
4  100 


Tv 
and

0.357
2.8 0.179
100 
  U %  5.6 

 4Tv 
 
 
1  
1  
  100  
    