Determination Of The Horizontal (Radial) Coefficient Of Consolidation By From Oedometer Test With Horizontal Drainage Using Incremental Loading Method.pdf

ABSTRACT
The radial (horizontal) coefficient of consolidation (cr) is a key parameter which
impacts the total consolidation of the PVD-improved grounds. In practice, the cr value
can be interpreted from field tests and laboratory tests. A radial consolidation test
(RCT) might be conducted using incremental loading (IL) method with either a
central drain (CD) or a peripheral drain (PD). The key goals of the research are: (1)
to design and manufacture a multi-directional flow consolidometer (VCT, RCT-PD,
RCT-CD) using incremental loading method; (2) to make a comparative study on the
cr values obtained from the RCTIL using a PD and a CD; (3) to make a comparative
study on the cr values derived from RCT-based method and CPTu-based method.
A desk study is carried out to secure the following: (1) a literature review on
equipment used for the test and existing methods used to evaluate the cr value; (2)
graphical design of a multi-directional flow consolidation cell. Sampling and CPTu
dissipation tests are carried out at sites. Besides the basic physical lab tests, the RCT IL
with a CD and a PD will be performed using the designed consolidation cell under
same condition test.
Overall, the ratio of kr/kv is approximately equal the ratio of cr/cv. The cr, PD & cr,
CD

are double to triple higher than cv. The cr, CD values are about 1.5 times larger than

figures of cr, PD. Finally, the cr values determined from CPTu-based method are
doubled higher than cr values obtained from RCT-based method.
The reliability of the new device is confirmed. Moreover, the results of cr values
in the case of both PD and CD obtained by interpreting with traditional method like
square root time method is more reliable than non-graphical method.
The limitation of the study is that the amount of data is still limited so it is still not
enough to fully confirm the reliability of the multi-directional flow consolidation cell.
In the future, the author intends to perform more consolidation and permeability tests
to ensure that the new designated consolidation can be applied in routine
performance.

i


ACKNOWLEDGEMENTS
I would like to express my sincere appreciation for the lecturers of Master of
Infrastructure Engineering Program for their help during my undergraduate at
Vietnam Japan University (VJU).
My thesis supervisor Dr. Nguyen Tien Dung for his enthusiasm, patience, advice
and constant source of ideas. Dr. Dung has always been available to reply to my
questions. His support in professional matters has been priceless.
Special gratitude is given to LAS- XD 442 lab and the staff at Institute of
Foundation and Underground, Golden Earth Inc. for their kindly support for
performing the laboratory work.
And finally, I want to spent my thank to my parents and friends for their
unflinching support in the tough time. Their support, spoken or unspoken, has helped
me complete my master thesis.

ii


TABLE OF CONTENTS
ABSTRACT................................................................................................................ i
ACKNOWLEDGEMENTS ..................................................................................... ii
TABLE OF CONTENTS ........................................................................................iii
LIST OF FIGURES ................................................................................................ vii
LIST OF TABLES ................................................................................................... ix
LIST OF ABBREVIATIONS ................................................................................. xi
CHAPTER 1: INTRODUCTION ......................................................................... 1
1.1 Background .................................................................................................... 1
1.2 Consolidation ................................................................................................. 1

1.2.1 Settlement with Prefabricated Vertical Drains (PVD) .............................. 3
1.3 Problem statement .......................................................................................... 5
1.4 Objectives and scope of present study ............................................................ 6
CHAPTER 2: LITERATURE REVIEW ............................................................. 9
2.1 Fundamentals of One Dimensional Consolidation .......................................... 9
2.1.1 Consolidation Theory with Vertical Drainage ........................................ 10
2.1.2 Consolidation Theory with Horizontal Drainage .................................... 11
2.2 Consolidation Tests in Laboratory ................................................................ 15
2.2.1 Vertical oedometer consolidation test .................................................... 15
iii


2.2.2 Horizontal Consolidation Test................................................................ 16
2.3 Determination of Coefficient of Consolidation ............................................. 17
2.3.1 Analysis of Time-Compression Curve ................................................... 17
2.3.2 Graphical Method .................................................................................. 18
2.3.2 Non-graphical Method ........................................................................... 21
2.4 Falling Head Permeability Test .................................................................... 24
2.5 The piezocone penetration test (CPTu) ......................................................... 26
2.5.1 Introduction ........................................................................................... 26
2.5.2 Pore-water Dissipation Tests .................................................................. 27
2.5.3 Coefficient of Consolidation .................................................................. 28
CHAPTER 3: METHODOLOGY ...................................................................... 32
3.1 Introduction ................................................................................................. 32
3.2 Radial Consolidation Test ............................................................................ 33
3.2.1 Design of the equipment ........................................................................ 33
3.2.2 Manufacture of the equipment ............................................................... 35
3.2.3 Testing procedure .................................................................................. 37
3.2.4 Analysis procedure ................................................................................ 38
3.3 Vertical consolidation test ............................................................................ 38

3.3.1 Testing procedure .................................................................................. 38
3.3.2 Analysis of Time-Compression Curve ................................................... 38
3.4 Permeability test .......................................................................................... 39
3.4.1 Equipment of permeability test .............................................................. 39
3.4.2 Testing procedure .................................................................................. 39
iv


3.4.3 Analysis procedure ................................................................................ 40
3.5 CPTu dissipation test.................................................................................... 41
3.5.1 Equipment ............................................................................................. 41
3.5.2 Testing procedure .................................................................................. 42
3.5.3 Analysis procedure ................................................................................ 42
3.6 Results verification and comparison.......................................................... 42
CHAPTER 4: TEST RESULTS & DISCUSSIONS ......................................... 43
4.1 Introduction ................................................................................................. 43
4.2 Summary of test performed .......................................................................... 43
4.3 Comparison of cr,PD and cv ........................................................................... 46
4.3.1 Square root time method ........................................................................ 46
4.3.2 Non-graphical method ........................................................................... 47
4.3.3 Inflection point method .......................................................................... 48
4.4 Comparison cr,CD and cv................................................................................ 49
4.4.1 Square root time method ........................................................................ 49
4.4.2 Non-graphical method ........................................................................... 50
4.4.3 Inflection point method .......................................................................... 51
4.5 Comparison cr,PD and cr, CD ........................................................................... 52
4.5.1 Square root time method ........................................................................ 52
4.5.2 Non-graphical method ........................................................................... 53
4.5.3 Inflection point method .......................................................................... 54
4.6 Horizontal coefficient of consolidation (cr) from CPTu ................................ 55

4.6.1 Estimate cr value from monotonic dissipation curves ............................. 55
v


4.6.2 Estimate cr value from non-standard dissipation curves.......................... 59
4.7 Test verification ........................................................................................... 61
4.8 Comparison cr,PD, cr,CD vs cr, CPTu .................................................................. 62
CHAPTER 5: CONCLUSIONS & RECOMMENDATIONS ......................... 65
REFERENCES ....................................................................................................... 68

vi


LIST OF FIGURES
Figure 1.1: Soil phase diagram (Das, 2008) .................................................................. 1
Figure 1.2: Primary consolidation (Das, 2008).............................................................. 2
Figure 1.3: Typical oedometer settlement (Das, 2008) .................................................. 3
Figure 1.4: Settlement damage ..................................................................................... 4
Figure 1.5: Drainage with and without drains ............................................................... 5
Figure 2.1: Mechanism of consolidation ....................................................................... 9
Figure 2.2: Uv versus Tv relationship (Head, 1986) ..................................................... 11
Figure 2.3: Schematic diagram of an RCT with central drain and peripheral drain...... 11
Figure 2.4: (a) Scheme of arrangement of the consolidation test in the triaxial
apparatus, with drainage towards the cylindrical surface; (b) Cylindrical element
of the sample ....................................................................................................... 12
Figure 2.5: Distribution of pore pressures within the soil sample related to r and t...... 13
Figure 2.6: Schematic of oedometer test (Head, 1986) ................................................ 15
Figure 2.7: Schematic of the apparatus used for conducting radial consolidation test.. 16
Figure 2.8: Rowe cell test under equal strain loading, horizontal outward drainage .... 17
Figure 2.9: Shapes of consolidation curve gained from oedometer test ....................... 18

Figure 2.10: Theoretical curve linkage square-root time factor to degree of
consolidation for vertical drainage (Taylor, 1942) ............................................... 20
Figure 2.11: Consolidation curve relating square-root time factor to for drainage
radially outwards to periphery with equal strain loading (Head, 1986) ................ 21
Figure 2.12: (a) Theoretical Ur-log Tr curve for n = 5; (b) (dUr/d log Tr)-log Tr plot
showing the inflection point (Sridhar and Robinson, 2011) ................................. 23
Figure 2.13: Falling-head permeability test (Das, 2017) ............................................. 24
Figure 2.14: Principal sketch of horizontal and vertical trimming of samples from
determining vertical and horizontal coefficient of permeability ........................... 25
Figure 2.15: Overview of the cone penetration test per ASTM D 5778 procedures ..... 27
Figure 2.16: Strain path solution for CPTu1 dissipation tests (The and Houlsby,
1991) ................................................................................................................... 30
Figure 2.17: Strain path solution for CPTu2 dissipation tests (The and Houlsby,
1991) ................................................................................................................... 30
Figure 2.18: "Non-standard" dissipation curve ( Chai et al., 2012).............................. 30
Figure 3.1: Equipment for radial consolidation test with peripheral drainage .............. 33
vii


Figure 3.2: Flow chart of the study ............................................................................. 34
Figure 3.3: Equipment for radial consolidation test with central drainage ................... 35
Figure 3.4: Manufacture of the equipment for radial consolidation with PD ............... 36
Figure 3.5: Manufacture of equipment for radial consolidation test with CD .............. 36
Figure 3.6: Radial consolidation with peripheral drain and central drain setup............ 37
Figure 3.7: Equipment of falling head permeability test.............................................. 39
Figure 3.8: Falling head permeability test setup .......................................................... 40
Figure 3.9: The typical and complete electrical CPT system ....................................... 42
Figure 4.1: Comparison of cv and cr,PD obtained from square root time method at 400
kPa & 800 kPa .................................................................................................... 46
Figure 4.2: Comparison of cv and cr,PD obtained from non-graphical method at 400

kPa & 800 kPa .................................................................................................... 47
Figure 4.3: Comparison of cv and cr,PD obtained from inflection point method at 400
kPa & 800 kPa .................................................................................................... 48
Figure 4.4: Comparison of cv and cr,CD obtained from square root time method at 400
kPa & 800 kPa .................................................................................................... 49
Figure 4.5: Comparison of cv and cr,CD obtained from non-graphical method at 400
kPa & 800 kPa .................................................................................................... 50
Figure 4.6: Comparison of cv and cr,CD obtained from inflection point method at 400
kPa & 800 kPa .................................................................................................... 51
Figure 4.7: Comparison of cr,PD and cr,CD obtained from square root time method at 400
kPa & 800 kPa .................................................................................................... 52
Figure 4.8: Comparison of cr,PD and cr,CD obtained from non-graphical method at 400
kPa & 800 kPa .................................................................................................... 53
Figure 4.9: Comparison of cr,PD and cr,CD obtained from inflection point method at 400
kPa & 800 kPa .................................................................................................... 54
Figure 4.10: Strain path solution for monotonic dissipation tests at 11 & 17m ............ 57
Figure 4.11: Strain path solution for monotonic dissipation tests at 18.5 & 20.5m ...... 58
Figure 4.12: Dilatory dissipation curve at 8.5m .......................................................... 59
Figure 4.13: Dilatory dissipation curve at 9.5 & 11.3 m .............................................. 60
Figure 4.14: Results of test verification which compares the ratios between kr/kv with
cr/cv ..................................................................................................................... 61
Figure 4.15: Comparison between cr,CPTu with cr,PD obtained from square root time
method at 400 kPa ............................................................................................... 64

viii


Figure 5.1 Comparison between results obtained from square root time, non-graphical
and inflection point method at 400 kPa................................................................ 67
Figure 5.2: Comparison between results obtained from square root time, non-graphical

and inflection point method at 800 kPa................................................................ 67

ix


LIST OF TABLES
Table 4.1: Laboratory and in-situ tests done ............................................................... 44
Table 4.2: Soil profile of borehole BH08 .................................................................... 55
Table 4.3: Estimate cr value from modified time factor, T* ........................................ 56
Table 4.4: Estimate cr values from “non-standard” dissipation curves......................... 59
Table 4.5: Coefficient of permeability obtained from permeability tests ..................... 62
Table 4.6: Comparison cr,PD and cr,CD obtained by square root time method with
cr,CPTu ................................................................................................................... 63

x


LIST OF ABBREVIATIONS

ac

probe radius

Cc

compression index

ch

horizontal coefficient of consolidation

Ck

permeability change index

Cr

recompression index

cv

vertical coefficient of consolidation

de

diameter of influence area

dw

drain diameter

e

void ratio

f

source/sink term; function; cyclic load natural frequency

Gs

specific weight of soil solids

H

height of soil; drainage length

H0

initial height of soil

kh

horizontal coefficient of permeability

kv

vertical coefficient of permeability

M

summation term e.g. M 2m+1)π/2 in Terzaghi theory

mv

coefficient of volume compressibility

xi

n

ratio of influence radius to drain radius

OCR

over-consolidation ratio

r

radial coordinate

su

undrained shear strength

t

time

t50

time required to reach 50% consolidation

Th

time factor for horizontal consolidation

Tv

time factor for vertical consolidation

U

degree of consolidation

u

pore-water pressure

Δu

change in pore pressure

Λ

function

σ'vo

vertical effective stress

qc

cone tip resistance

qt

tip resistance

IR

rigidity index

Vs

shear wave velocity

xii


CHAPTER 1: INTRODUCTION
1.1 Background
Throughout the world, due to rapid urbanization and development, construction
projects are rapidly built on fine-grained soils. Natural soils in their original condition
may be inappropriate for short or long term structure activities and so must be enhanced
before use. In particular, many coastal areas contain deep multi-layers of compressible
clay initially deposited by sedimentation from lakes, rivers, and seas. These fine-grained
soils have poor bearing capacity and indicate excessive settlements under the load. One
of the most broadly and successfully used techniques to boost soft soils is preloading
with vertical drains to consolidate the soil and expedite strength growth. My thesis
mainly builds on the understanding of consolidation by both vertical and horizontal
drains developed in the last decades.
This chapter illustrates the perception of consolidation and how surcharge with
vertical drains can accelerate the water expulsion process. The progress of vertical and
horizontal drain concepts is discussed.
1.2 Consolidation

Figure 1.1: Soil phase diagram (Das, 2008)

1


Soil formed from two or three phase composition (see Figure 1.1). The space inside
the soil particles are replaced by water, air or a combination of both. Consolidation
associates the contraction of voids under load. It develops in three stages (see Figure
1.3). Immediate settlement happens instantly after loading with zero volume change, i.e.
shape change only. In saturated soil (i.e. no air) the expansion in pressure emerging from
the load is immediately carried out by the liquid which is incompressible. Such excess
pore-water pressure regularly disappears as water seeps out of the soil and the stress is
transferred to the soil skeleton. This is defined as primary consolidation (see Figure 1.2).
Primary consolidation may last year’s depending on soil permeability. When additional
pore-water pressure has expelled, the soil remains to consolidate continually as the soil
particles rearranges to fill into voids.

Figure 1.2: Primary consolidation (Das, 2008)

2


Figure 1.3: Typical oedometer settlement (Das, 2008)
Consolidation of soils can lead to serious problems for constructions like
embankments founded on them. If structures settle uniformly little damage is
experienced except perhaps to services feeding it. However, settlement is rarely uniform.
Varied loading and the nonhomogeneous characteristic of soil lead to differential
settlement. This produces added loads that often create cracking in the structure. If soils
have insufficient strength to withstand the applied loads it may be difficult to build such
structures in the first place. Soil density largely affects shear strength in soil. The
densification of soil due to consolidation thus results in considerable strength gain,
allowing increased loads to be subjected to the soil.

1.2.1 Settlement with Prefabricated Vertical Drains (PVD)
Pre-consolidation is a method adopted to reduce the consequence of consolidation on
structures and enhance the strength of the ground. Basically, a surcharge is applied to
the ground, usually in the form of an embankment, where a structure located on. This
embankment induces the foundation soil to consolidate. Once the required primary
consolidation is attained the pre-consolidation load is discharged and the structure built.
Thus after construction, the soil foundation experiences the slow gradual process of
3


secondary compression. Differential settlements are reduced so the structure is less likely
to crash or collapse.

Figure 1.4: Settlement damage
The rate at which preloading attains the required consolidation is accelerated by
increasing the magnitude of the surcharge. The magnitude of surcharge is restricted by
soil failure criteria. Thus preloading surcharges are raised in periods as the shear strength
of soil enhances and is able to prevent increased loads without failure. To hasten the
consolidation process so surcharges can be set up more quickly (or not built up as high
in the first place), one must speed the egress of water from the soil frame. The
establishment of vertical drains reduce the leakage path for water to seep out under the
additional pore-water pressure (see Figure 1.5). In particular, they equip both a radial
outflow path in addition to vertical outflow path. Clays have greater horizontal
permeability than in the vertical permeability. Usually, water only seeps out in the
vertical direction due to the large extent of the clay body. Vertical drains allow the
increased horizontal permeability to be exploited.

4

Figure 1.5: Drainage with and without drains
1.3 Problem statement
In practice, clay layers are often thick and drain spacing is small enough so that the
vertical drainage becomes insignificant and it might be ignorable (Chung et al., 2009).
The horizontal drainage, in which the radial (or horizontal) coefficient of consolidation
(cr) is an important factor, therefore plays a critical role in the total consolidation of the
PVD-improved grounds. The cr value can be interpreted from field tests (e.g., CPTU
dissipation test), from laboratory tests (e.g., radial consolidation tests (RCT)), or from
some empirical correlations, in which the RCT on undisturbed sample should provide
the most reliable value. Similar to methods for vertical consolidation test (VCT) (ASTM
D2435; ASTM D4186), an RCT might be conducted using either incremental loading
(IL) method (hereafter referred to as RCT IL) or constant rate of strain (CRS) method
(hereafter referred to as RCTCRS). However, both RCTIL and RCTCRS have not been
standardized in any formal standards. For the same soil material and same range of
applied pressure, the CRS method might be completed in a shorter time than the IL
method and it provides continuous profiles of consolidation parameters (e.g., cv or cr)
with applied pressure. However, key limitations of the CRS method are that it is very
complicated for routine performances and the consolidation parameters vary with varied
strain rate applied. The IL method is simple to carry out in practice, and in fact most
5


existing methods for determining the consolidation parameters were proposed for the IL
method. This research focuses on the CRSIL.
The RCTIL can be conducted using either a central drain (CD) (i.e., inward drainage)
or a peripheral drain (PD) (i.e., outward drainage). Existing methods for determining cr
value are mainly focused on the test with a CD rather than a PD. This is because the test
results can conveniently be interpreted using the well-known theoretical solutions of
Barron (Barron, Lane, Keene, & Kjellman, 2002). However, the test with a CD is often
associated with three typical problems: (1) the interpreted cr value is a function of n

=de/dw, where de is radius of soil specimen and dw is the radius of the central drain; (2)
soil disturbance resulted from preparation of the central drain affects the test results
significantly; and (3) the attainment of the test depends greatly on the central placement
of the central drain (if it is a porous stone), which is quite difficult in routine test. For a
given soil under the same magnitude of applied pressure, the RCT IL of both drainage
types should result in the same value of cr, but the problems associated with the test using
a CD can be avoided by using a PD. Since not much attention has been paid to explore
the advantages of the test with a PD, an in-depth study on this drainage condition is
therefore very necessary to take its advantages in routine performances.
1.4 Objectives and scope of present study
The key goals of the research are to investigate the effectiveness of the RCT IL using
a PD compared with the test using a CD and to propose a comprehensive method to
evaluate cr value of clays from the test so that the method can conveniently be applied
in routine performances. The particular objectives of the research are:
- To design and manufacture a multi-directional flow consolidation cell that should
be able to perform the RCTIL using either a CD or a PD.

6


- To highlight the advantages of the test using a PD over the test using a CD in testing
procedures.
- To make a comparative study on the cr values obtained from the RCTIL using a PD
and a CD on the same clay, equipment and procedures.
- To make a comparative study on the cr values obtained lab-based method and CPTu
based methods.
This research focuses on the RCTIL, not on the RCTCRS. To depict the advantages of
the test using a PD over that using a CD, both drainage types will be used for pairs of
specimens of the same depths. In order to fulfil the targets described above, the research
would cover field tests (sampling and CPTu) at a site in Dinh Vu Port, Hai Phong City,

Vietnam and laboratory tests on undisturbed samples obtained from the sites. Extensive
analytical analyses will be carried out to depict the test results. The following are main
activities of the research:
- At research site, boring and sampling was conducted for one borehole up to 23.0 m.
A total of 06 sampling tubes (at 06 sampling depths) as obtained. The sampling tubes
were brought to the laboratory and preserved for laboratory test.
- One CPTu sounding in association with CPTu dissipation test was conducted nearby
the sampling borehole location. The CPTu test results are used characterize the soil at
the site and the CPTU dissipation test results are used to determine the coefficient of
radial consolidation (cr) which will be compared with that with that obtained from radial
consolidation test in the lab. In total, 06 dissipation points at the centers of 06 sampling
depths, respectively, were conducted at the site. At each test depth the penetration was
halted to conduct the dissipation test until at least 50% of excess pore water pressure has
been dissipated.

7


- Designed and manufactured a consolidation cell that could be able to perform the
RCTIL using either a CD or a PD, and more importantly the cell could be able to function
using the standard loading frame and monitoring system of the conventional oedometer
test (ASTM D2435 / D2435M - 11, 2011).
- Perform all basic laboratory tests (e.g., Atterberg limits, water content, unit weight,
specific gravity, and conventional consolidation test) to characterize the soil, and RCT IL
using a CD and a PD.
- Analyze test data and compare the cr values obtained from laboratory and field tests.

8

CHAPTER 2: LITERATURE REVIEW
2.1 Fundamentals of One Dimensional Consolidation
A soil may be known to be a skeleton of solid particles enclosing voids which may
be filled with combination of gas and liquid. If a sample of soil subjected to sustained
pressure so that its volume is reduced in a drained manner.
As the compression happens, the pore water is seeped out based on Darcy’s law,
(Taylor 1948). At the same time, a slow expulsion of water accompanied by the reduction
in the volume of the soil mass, which results in settlement.

(a) Initial loading, water
takes load, soil (i.e.
spring) has no load

(b) Dissipation of excess
water pressure, water
seeping out and soil starts

(c) Final loading water
dissipated and soil has

to take load
Figure 2.1: Mechanism of consolidation

9

load


The consolidation of clay under a surcharge does not occur instantaneously; clays are
impermeable that the water is relatively trapped into the pores. When an increment of

load is subjected the pore water cannot seep out promptly. Since clay particles have a
tendency to approach one another and pressure increases in the pore water which is
known the excess pore pressure. The hydraulic gradients appear due to this excess
pressure lead the fluid to escape from the soil. As drainage continuous, the excess
pressures dissipate and later the externally constant applied stress is gradually transferred
to the soil frame. The part of pressure carried by soil frame is defined as effective stress.
Soil skeleton then changes under the rise in effective stresses. This is called
consolidation.
2.1.1 Consolidation Theory with Vertical Drainage
The soil property designated by cv is called the coefficient of consolidation:
cv

 2u u

z 2 t

(2.1)

In the consolidation understanding, the drainage path length is evaluated downward
from the surface of the clay sample. The thickness of the sample is nominated by 2H,
the distance H thus being the length of the longest drainage path.
A dimensionless time factor is illustrated as:
Tv 

cvt
H d2

(2.2)

and average degree of consolidation

U  1

8

2

e

 2Tv
4

10

(2.3)


The relation between Uavg and Tv can be observed from Figure 2.2.

Figure 2.2: Uv versus Tv relationship (Head, 1994)
2.1.2 Consolidation Theory with Horizontal Drainage
Case for Equal Vertical Strain – For the case where loading and compression are as
in the standard test, but where pore water flow is restricted to the radial direction only,
it has been shown in Eq. (2.4) that the differential equation of consolidation is:
  2u 1 u  u
cr  2 

r r  t
 r

(2.4)

The consolidation of the soil specimen in an RCT, as schematically shown in Figure
2.3, is an excellent case for represent the consolidation condition expressed by Eq. (2.4).

Figure 2.3: Schematic diagram of an RCT with central drain and peripheral drain
11


2.1.2.1 For the case of a peripheral drain (PD)
By isolating an element of the sample to a distance r from the axis (Figure 2.4, if we
call u=u(r,t) the pore pressure at a time t, then the difference between the volume of
water flowing into and out of the element will be:
V 

k  u  2u 
r  drdzd
 
 w  r r 2 

LOADING PLATE (IMPERVIOUS)



z

De
P

DRAINAGE

(2.5)

d
r

SOIL ELEMENT

dz

dr

SATURATED SOIL

POROUS
STONE

SOIL ELEMENT

STEEL PLATE (IMPERVIOUS)

(b)

(a)

Figure 2.4: (a) Scheme of arrangement of the consolidation test in the triaxial
apparatus, with drainage towards the cylindrical surface; (b) Cylindrical element of the
sample
If ε is the strain per unit length along the z axis, the change in volume may also be
expressed by
V   (1  2 ) rdrdzd 

Equating both values, we get

12

(2.6)


k
 
(1  2 ) w

  2u 1 u 
 r 2  r r 



(2.7)

Since the value of the vertica
l strain ε is independent of r and only a function of time (equal strain), we have

  2u 1 u 
k
 

 f (t )
(1  2 ) w  r 2 r r 

(2.8)

u  R(t ) exp( p2 t )

(2.9)

We obtain

At any given time, t after loading, diagram of u (hypothesis) changes as following:
u

u=f(r,t)

umax
uPD (t)= umax /2
LOADING PLATE (IMPERVIOUS)

De/2
0

Ht

SATURATED SOIL

DRAINAGE

r
DRAINAGE

STEEL PLATE (IMPERVIOUS)

De

Figure 2.5: Distribution of pore pressures within the soil sample related to r and t
Solving this equation is achieved by applying the following boundary conditions:
1) u = 0 at r = De/2 and t ≥ 0
2) u(r,t) = max,  u ( r , t ) / t  0
13