CSE30307 Soil Mechanics (Consolidation)

CSE30307 Soil Mechanics

for Civil Engineering
Dr. Andy YF Leung

Room ZS938
Tel: 2766 6064
Email: yfleung@polyu.edu.hk
http://www.cee.polyu.edu.hk/~leung_yf/
 CSE30307 Soil Mechanics (Consolidation)

• By the end of lecture, you will be able to

– estimate the amount of consolidation settlement
– estimate the degree of consolidation with time using
Terzaghi’s 1D consolidation theory
– understand the use of vertical drains to reduce time
of consolidation.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Reclamation projects in Hong Kong

 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Reclamation projects in Hong Kong

(Source: Airport Authority; CHEC)

 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Reclamation projects in Hong Kong

• Reclamation in Central
• The government vision on
“Lantau tomorrow”

The key issues:

• Amount of total consolidation settlements
• Time taken for the settlements to take place
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Oedometer test
• Oedometer test commonly used to obtain the characteristics of
clayey soils in one-dimensional consolidation or swelling
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Phase diagram in oedometer test

• Recall the phase relationships in Chapter 1, and consider saturated
clayey soils (S = 1).
• By definition, if volume of solid (Vs) = 1, volume of void = volume of
water = e (e0 – before compression; e1 – after compression).
• Change in volume of soil is due to change in void ratio, i.e. De

De 1  e0

DH H0
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Compressibility characteristics
• Compressibility of clayey soils usually represented in one of two
different plots: e – s’ or e – log s’

Slope proportional
to mv
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Compressibility characteristics (mv)

• Slope of e – s’ plot is proportional to the Coefficient of volume
compressibility, mv (unit: m2/MN)

1 dV
mv 
V ds '
1  H 0  H1 
mv    (Area is constant in Slope proportional
H 0  s '1 s '0 
to mv
oedometer test)

1  e0  e1 
mv   
1  e0  s '1 s '0 

• mv is not constant and changes with

stress (stress-dependent)
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Compressibility characteristics (Cc and Ce)

• The plot on e – log s’ usually shows two portions with different
slopes: virgin compression and recompression (expansion)
• The two slopes are represented by
the Compression index, Cc, and
Expansion (Recompression) index, Ce
• Overconsolidated soil (OCR>1)
follows the recompression curve, Cc 
e0  e1
log10 (s '1 / s '0 )
while normally consolidated soil
(OCR=1) follows the virgin
compression curve.
• The two curves are separated by
preconsolidation pressure (sc’)
(sv’,max in Chapter 6) sc’
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Overconsolidation ratio
• The overconsolidation ratio (OCR) describes the stress history of the
soil, which also affects its behaviour, such as in lateral pressure and
consolidation.
Ground surface in the past
• OCR is defined as the maximum value Ground surface today (e.g.
of effective vertical stress in the past due to erosion)

divided by the present value.

𝜎′𝑣, 𝑚𝑎𝑥
𝑂𝐶𝑅 =
𝜎′𝑣0 s’v, 0
s’v, max

• If the present effective stress is the maximum that the soil has ever
experienced, then the soil is ‘normally consolidated’ (OCR = 1).
• If the effective stress in the soil in the past has been greater than
the present value, then the soil is ‘overconsolidated’ (OCR > 1).
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Typical values of mv and Cc for Hong Kong soils

(by Prof. Peter Lumb)

 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Preconsolidation pressure
• Preconsolidation pressure is an
important parameter in determining
settlements.
• Due to disturbance during
retrieval/transportation/handling of
soil samples, the results from lab
oedometer test may deviate from in-
situ (field) behaviour.
• The Casagrande procedure is
commonly used to determine the in-
situ preconsolidation pressure.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Casagrande method for preconsolidation stress

1. Produce back the straight line part (BC)
of the curve
2. Determine the point D of the
maximum curvature on the re-
compression part of AB of the curve
3. Draw the tangent to the curve at D
and bisect the angle between the
tangent and the horizontal through D
4. The vertical through the point of
intersection of the bisector and CB
produced gives the approximate value
of the preconsolidation pressure.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

In-situ e – logs’ curve

1. In-situ value of e0 taken as the
same as the start of oedometer
G
test.
E
2. In-situ recompression curve (GE)
approximated to be parallel to
the mean slope of laboratory
recompression curve
3. The in-situ and laboratory virgin
compression lines intersect at
0.42 times of e0.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – oedometer test

• The following compression readings were obtained in an oedometer test
on a specimen of saturated clay (Gs = 2.73):

Pressure (kPa) 0 54 107 214 429 858 1716 3432 0

Dial gauge
5.000 4.747 4.493 4.108 3.449 2.608 1.676 0.737 1.480
after 24h (mm)

• The initial thickness of the specimen was 19.0 mm and at the end of the
test the water content was 19.8%. Plot the e-logs’ curve and determine
the preconsolidation pressure. Determine the values of mv for the stress
increments 100-200 kPa and 1000-1500 kPa. What is the value of Cc for
the latter increment?
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – oedometer test

• Hints:
1. At the end of test:
𝑒1 = 𝑤𝐺𝑠
2. De during test:
∆𝑒 ∆𝐻
=
1 + 𝑒0 𝐻0
3. Plot e against log s’
4. Determine sc’ by the
Casagrande method
• mv = 0.2 m2/MN
(100-200 kPa)
• mv = 0.067 m2/MN
• Cc = 0.31
(1000-1500 kPa)
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Consolidation settlements (using Cc and Ce)

Cc : Compression index
• Changes in void ratio under Void ratio, e
Ce : Recompression index
changes in stress measured
Ce 1
in lab
• Different behaviours during Cc =
-(e1-e0)
log(s’1/s’0)
virgin compression and
Cc
recompression, separated by
1
the Preconsolidation
pressure (s’c) Ce 1

s’c : maximum effective stress the

soil has experienced before
s’c log s'
Current s’ = s’c  normally consolidated
Current s’ < s’c  overconsolidated
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Consolidation settlements (using Cc and Ce)

• For normally consolidated clays (OCR=1): e Ce
1
𝐻 𝐻 𝜎0′ + ∆𝜎′
𝑠𝑐 = ∆𝑒 = 𝐶 log
1 + 𝑒0 1 + 𝑒0 𝑐 𝜎0 ′
Cc
• For overconsolidated clays (OCR>1): 1
for 𝜎0′ + ∆𝜎′ < 𝜎𝑐 ′ : Ce 1
𝐻 𝐻 𝜎0′ + ∆𝜎′
𝑠𝑐 = ∆𝑒 = 𝐶 log
1 + 𝑒0 1 + 𝑒0 𝑒 𝜎0 ′

for 𝜎0′ + ∆𝜎′ > 𝜎𝑐 ′ : s’c log s'

𝐻 𝐻 𝜎𝑐′ 𝜎0′ + ∆𝜎′

𝑠𝑐 = ∆𝑒 = 𝐶 log + 𝐶𝑐 log
1 + 𝑒0 1 + 𝑒0 𝑒 𝜎0 ′ 𝜎𝑐 ′

Recompression Virgin compression

phase phase
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Consolidation settlements (using mv)

1 𝑑𝑉
• Recall the definition of mv: 𝑚𝑣 =
𝑉 𝑑𝜎′
• For one-dimensional case:
1 𝐻0 − 𝐻1 1 𝑒0 − 𝑒1
𝑚𝑣 = =
𝐻0 𝜎′1 − 𝜎′0 1 + 𝑒0 𝜎′1 − 𝜎′0

• Consider the consolidation settlement, sc:

𝐻
𝑒0 − 𝑒1
𝑑𝑠𝑐 = 𝑑𝑧 = 𝑚𝑣 ∆𝜎 ′ 𝑑𝑧 𝑠𝑐 = න 𝑚𝑣 ∆𝜎 ′ 𝑑𝑧 = 𝑚𝑣 ∆𝜎 ′ 𝐻
1 + 𝑒0
0

Assuming mv is constant
throughout the layer
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – consolidation settlements (using mv)

A long embankment 30 m wide is to be built on layered ground as shown. The net
vertical pressure applied by the embankment (uniformly distributed) is 90 kPa. The
value of mv for the upper clay is 0.35 m2/MN, and for the lower clay mv = 0.13m2/MN.
Determine the final settlement under the centre of the embankment due to
consolidation. (For this example, assume that Dsv = 90kPa for the two clay layers).
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – consolidation settlements (using Cc and Ce)

(Example 4.6 of textbook)
An 8 m depth of sand overlies a 6 m layer of clay, below which is an impermeable
stratum. The water table is 2 m below the surface of the sand. Over a period of one
year, a 3-m depth of fill (unit weight 20 kN/m3) is to be placed on the surface over an
extensive area. The saturated unit weight of the sand is 19 kN/m3 and that of the clay
is 20 kN/m3. above the water table the unit weight of the sand is 17 kN/m3. For the
clay, the relationship between void ratio and effective stress (units kPa) can be
represented by the equation:
𝜎′
𝑒 = 0.88 − 0.32 log
100
Calculate the final settlement of the area due to consolidation
of the clay.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Degree of consolidation
• Degree of consolidation, U, represents
the progress of the consolidation
process, and is defined by:
𝑒0 − 𝑒
𝑈=
𝑒0 − 𝑒1
• Assuming linear e-s’ for the stress range:
𝜎′ − 𝜎′0 𝑢𝑒
𝑈= =1−
𝜎′1 − 𝜎′0 𝑢𝑖

• Degree of consolidation, in general, is a

function of depth, therefore, the average
degree of consolidation, can be
represented by:
2𝑑
‫׬‬0 𝑢𝑒 𝑡 𝑑𝑧
𝑈 =1− 2𝑑
‫׬‬0 𝑢𝑖 𝑡 = 0 𝑑𝑧
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Stress transfer during consolidation

Dsv
Du Ds'v

7 6 5 4 3 2 6 7
2 3 4 5
1 1

Dsv Isochrones: curves representing excess

pore pressure profile within the sample

1. Initially, applied stress taken by water (excess pore water pressure)

2-6. As water drains out, compressive stress is transferred to the soil, and
deformation takes place
7. At the end of consolidation, all the applied stress is taken by soil skeleton
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Terzaghi’s theory of 1-D consolidation

• The soil is homogeneous.
• The soil is fully saturated.
• The solid particles and water are incompressible.
• Compression and flow are one-dimensional (vertical).
• Strain are small.
• Darcy’s law is valid at all hydraulic gradients.
• Permeability remains constant.
• Unique relationship between void ratio and effective stress
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Terzaghi’s theory of 1-D consolidation

(Volume change with

time related to mv)
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

(Partial differential equations will be covered in other subjects of the curriculum)

Problem to be solved: ue  2ue

 cv , 0  z  2d , t  0
t z 2

Boundary conditions: ue (0, t )  0,

ue (2d , t )  0, t 0

Initial conditions: ue ( z, 0)  ui ( z ),
0  z  2d

d = maximum flow distance

 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

(Partial differential equations will be covered in other subjects of the curriculum)

cv ue, zz  ue,t 0  x  2d , t  0
ue (0, t )  0, ue (2d , t )  0, t 0
ue ( x, 0)  ui , 0  x  2d

Separation of variables
ue ( z, t )  Z ( z )T (t )
A function of time cannot be a
cv Z  T  Z T  function of position  =constant
Z  1 T 
 Z    Z  0 Key: 1 PDE becomes 2 ODE
Z cv T 
T   cv  T  0
Z  1 T 
  
Z cv T ue (0, t )  0, ue (2d , t )  0, t 0
Boundary condition (BC) ue (0, t )  Z (0)T (t )  0 Z (0)  0, Z (2d )  0

Z    Z  0, Z (0)  Z (2d )  0 Z  A cos  z  B sin  z

Have solution only for certain 
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

(Partial differential equations will be covered in other subjects of the curriculum)

Z (0)  A  0 Z (2d )  B sin  2d  0 sin  2d  0   2d  n

n  n2 2 / (2d )2 , n  1, 2,3,
boundary value problem are the eigenfunctions
Zn ( x)  sin  n z / (2d )  , n  1, 2,3,
associated with the eigenvalues
n  n2 2 / L2 , n  1, 2, 3,
Time-dependent function
dT
T   cv  T  0  cv  dt
T cv t
ln T  cv n2 2t / (2d )2  C Tv 
d2
 cv ( n /2 d )2 t [( n )2 /4]Tv
Tn  kne  kn e , kn constant.
Fundamental solutions

un ( x, t )  e  ( n /2)2 Tv
sin  n z / (2d )  , n  1, 2,3, ,
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

(Partial differential equations will be covered in other subjects of the curriculum)

Fourier Series Expansion in x
u( x,0)  f ( x), 0  x  L
 
ue ( x, t )   cnun ( x, t )   cne ( n /2) sin  n z / (2d ) 
2
Tv

n 1 n 1

Initial condition (IC)


u ( x, 0)  ui   cn sin  n z / (2d ) 
n 1
4u
1 2d 2ui  i n  odd
cn   ui sin  n z / 2d  dz  ( )[cos n  1] n
d 0 n
0 n  even
m 
 2ui  Mz    M 2Tv
ue ( z , t )    sin   e
m 0  M  d 

M  2m  1 Tv 
cv t
2 d2
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

One-dimensional consolidation theory

• Exact solution for excess pore water pressure and degree of
consolidation:
𝑚=∞
2𝑢𝑖 𝑀𝑧
𝑢𝑒 = ෍ sin exp −𝑀2 𝑇𝑣
𝑀 𝑑
𝑚=0

𝜋 Time factor Tv:

𝑀 = (2𝑚 + 1)
2 𝑐𝑣 𝑡
𝑚=∞ 𝑇𝑣 = 2 (no unit)
2 𝑑
𝑈 = 1 − ෍ 2 exp −𝑀2 𝑇𝑣
𝑀
𝑚=0

• Approximate solutions:
𝜋 2
𝑈 𝑈 < 0.6
𝑇𝑣 = ቐ 4
−0.933 log 1 − 𝑈 − 0.085 𝑈 > 0.6
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Average degree of consolidation versus time factor

 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Average degree of consolidation versus time factor

Degree of 𝜋 2
Time factor, Tv 𝑈 𝑈 < 0.6
consolidation, U 𝑇𝑣 = ቐ 4
−0.933 log 1 − 𝑈 − 0.085 𝑈 > 0.6
0 0
0.1 0.008
0.2 0.031 𝑈 ⇒ 𝑇𝑣 ⇒ 𝑐𝑣
0.3 0.071
0.4 0.126 𝑐𝑣 𝑡
𝑇𝑣 = 2
0.5 0.196 𝑑
0.6 0.287
0.7 0.403 Used in log time method to
determine cv
0.8 0.567
0.9 0.848 Used in root time method to
1.0 ∞ determine cv
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Determining cv by root time method

• Proposed by D.W. Taylor to obtain
cv from oedometer test results
• The theoretical curve is linear up
to ~60% consolidation
• At 90% consolidation, AC is 1.15
times of AB of the extrapolation of
the linear part
• The characteristic is used to
determine t90, time for 90%
consolidation
𝑐𝑣 𝑡90
0.848 =
𝑑2
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Determining cv by log time method

• Proposed by A. Casagrande to obtain cv from oedometer
test results
• Settlement at 0% consolidation (as) determined by the
initial parabolic portion.

𝑐𝑣 𝑡50
0.196 =
𝑑2
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Determining cv by log time method

• Settlement at 100% consolidation (a100) determined by
intersection of the linear portions of primary
consolidation and secondary compression.
• Settlement at 50% consolidation (a50) is midway
between as and a100. The corresponding time is t50.

𝑐𝑣 𝑡50
0.196 =
𝑑2
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Secondary compression (creep)

• Experiments show that compression does not stop when ue=0, but
continues at a decreasing rate under constant s’.
• The gradual readjustment of fine-grained particles (creep) causes
the secondary compression, with the rate defined by slope (Ca) of
the final part of compression-log time curve.
• The ‘conventional’ way to estimate secondary compression
settlements (tp is time at the ‘end of primary consolidation’):
𝐶𝛼 𝑡
𝑠𝑐𝑟𝑒𝑒𝑝 = 𝐻 log
1 + 𝑒0 𝑡𝑝
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – determining cv
The following compression readings were taken during an oedometer test on a
saturated clay specimen (Gs = 2.73) when the applied pressure was increased from
214 to 429 kPa:

After 1440 min the thickness of the specimen was 13.60 mm and the water content
was 35.9 %. Determine the coefficient of consolidation from both the log time and
the root time plots. Determine also the value of the coefficient of permeability.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – determining cv
Solution:

(previous plot)

(previous plot)
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Correction for construction time

• In practice, structural loads are not
applied instantaneously.
• An empirical method was proposed
by Terzaghi (1943) to correct the
settlement curve for such effects.
• Assuming the construction period is
represented by tc,
• When t < tc (e.g., at time t1),
– The corrected settlement is
offset by 0.5 t1, and reduced in
proportion by t1/tc.
• When t > tc,
– The corrected settlement curve is
offset by 0.5 tc.
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Example – consolidation settlements

(Example 4.6 of textbook)
Returning to Example 4.6 discussed earlier, the coefficient of consolidation is 1.26 m2
/year for the clay. The 3-m fill is to be placed over a period of 1 year.
(a) Calculate the settlements after a period of three years from the start of fill
placement.
(b) If a very thin layer of sand, freely draining, existed 1.5 m above the bottom of the
clay layer, what would be the values of the final and three-year settlements?
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Vertical drains
• Vertical drains speed up consolidation
process because:
– Distance of drainage path is
shortened
– For many clayey soils, permeability
in horizontal direction (kx) usually
higher than that in vertical direction
(kz)
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Vertical drains
 Consolidation One-dimensional
Oedometer test Vertical drains
settlements consolidation theory

Design of vertical drains

𝜕𝑢𝑒 𝜕 2 𝑢𝑒 1 𝜕𝑢𝑒 𝜕 2 𝑢𝑒
= 𝑐ℎ + + 𝑐𝑣 1 − 𝑈 = (1 − 𝑈𝑣 )(1 − 𝑈𝑟 )
𝜕𝑡 𝜕𝑟 2 𝑟 𝜕𝑟 𝜕𝑧 2
(Horizontal flow) (Vertical flow) 𝑐𝑣 𝑡 𝑐ℎ 𝑡
𝑇𝑣 = 2 𝑇𝑟 = 2
𝑑 4𝑅
𝑈𝑟 = 𝑓(𝑇𝑟 ) 𝑈𝑣 = 𝑓(𝑇𝑣 )