NUMERICAL MODELING OF CONSTANT RATE OF STRAIN
CONSOLIDATION TESTS
by
Yew Choong Patrick, Lee
Bachelor of Engineering in Civil Engineering, Second Upper Honors
National University of Singapore, Singapore, June 2000
Submitted to the Department of Civil and Environmental Engineering in Partial
Fulfillment of the Requirements for the Degree of
MASTER OF ENGINEERING
in Civil and Environmental Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
0 2003Yew Choong Patrick, Lee. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distributepubliclypaper and electronic
copies of the thesis document in whole or in part.
Signature of Author:
Department of Civil and E /ironmentat Engineering
May 09, 2003
Certified by:
Dr John T. Germaine
Thesis Supervisor
Accepted by:
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t
bOral
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Numerical Modeling of Constant Rate of Strain Consolidation Tests
NUMERICAL MODELING OF CONSTANT RATE OF STRAIN
CONSOLIDATION TESTS
by
Yew Choong Patrick, Lee
Submitted to the Department of Civil and Environmental Engineering on May 09, 2003
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Civil Engineering
ABSTRACT
Constant Rate of Strain Consolidation Testing (CRSC) is an effective testing method to
determine cohesive soil properties including stress history, compressibility, hydraulic
conductivity and coefficient of consolidation. However, testing at varying strain rates
have resulted in inconsistent results with respect to expected soil behavior. This
theoretical study develops at numerical model to simulate the important details of a
CRSC test using a finite difference approach in a spreadsheet in Microsoft Excel. The
model provides a tool which can be used to study the test interpolation errors.
The model based on void ratio versus log of effective vertical stress and versus log of
hydraulic conductivity relationship of soil and includes the pore pressure compressibility
of base measuring system. Parametric studies are preformed on Resedimented Boston
Blue Clay normally consolidated behavior to investigate the effects of apparatus
compressibility and strain rate. Finally, the predicted base pore pressure from the model
simulation are used with the conventional interpretation equations to quantify the errors
in soil parameters due to apparatus compressibility.
It was found that bottom drainage of the specimen into the measuring system in a pore
pressure distribution such that the base excess pore water pressure is not equal to the
maximum excess pore water pressure. The magnitude of this difference increases with
strain rate and neutral plane location. When using the conventional interpretation, this
causes a shift in the compression curve to higher effective vertical stress. The error is
small when the pore water pressure ratio is less than 5%. The error in hydraulic
conductivity is much more significant and causes an overestimate of the hydraulic
conductivity.
This research forms the basis for which future experimental validation, normally
consolidated to overly consolidated and steady state to transient state analysis. It can form
the basis of new theories for the interpretation of constant rate of strain consolidation
tests.
Thesis Advisor: John T. Germaine, Dr.
Title: Principal Research Associate in Civil and Environmental Engineering
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Acknowledgments
Special thanks go to Dr John Germaine, for being my advisor and supervisor throughout
the year and being always there when I need him, being always encouraging and for
making my time in MIT an enriching and rewarding experience.
I would also like to thank the following people:
Dr Lucy Jen for giving me advice with regards to class work, thesis, family issues, for
making my transition from work back to studying a smooth and enjoyable one; for being
so approachable and kind; and for being a great teacher and a good friend.
To the Geotechnical Engineering M. Eng/M. S/Ph.D Group, especially Isabella Batista,
Jedediah Greenwood, Kartal Toker, Maria Ai-katerini Nikolinakou, Brain Tan, Louis
Ngai Yuen Wong, Michael Edward Paonessa, Jean-Louie S. Locsin, Frangois and Aw
Eng Sew - for being so helpful to a person new in a foreign country; for giving advice
and encouragement when I need the most.
3
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Table of Contents
Chapter 1 ..............................................................................
......... ........ -----.................
Introduction.....................................................................................................................
13
13
1.1 Background and Problem Statem ent.......................................................................
13
1.2 Research Objective ..............................................................................................
15
1.3 Thesis Organization ............................................................................................
15
Chapter 2 .........................................................................................................................
17
Existing Testing M ethods and Theories......................................................................
17
2.1 Consolidation Theory..........................................................................................
17
2.1.1 One-Dim ensional Consolidation........................................................................
17
2.2 Constant Rate of Strain Consolidation Test........................................................
18
2.2.1 Constant rate of strain testing apparatus ............................................................
19
2.3 Constant Rate of Strain Theory...........................................................................
20
2.3.1 Sm ith and W ahl's Linear Theory......................................................................
20
2.4 W issa's Theories.................................................................................................
23
2.4.1 Transient State Conditions..............................................................................
25
2.4.2.1 M odified Linear Theory.................................................................................
26
2.4.2.2 W issa's Non-linear Theory............................................................................
27
2.4.3 Comparison of Wissa's Linear and Nonlinear Theory......................................
28
4
Numerical Modeling of Constant Rate of Strain Consolidation Tests
2.5 Previous Research on 1-D Consolidation at Constant Rate of Strain.................. 28
Chapter 3 ...........................................................................
................
--.................. .... 35
Bottom Drainage Hypothesis ..................................................................................
... 35
3.1 Overview .......................................................................................................
35
3.2 Top Drainage Theories .......................................................................................
36
3.2.1 Advantage of Top Drainage Assumption .........................................................
36
3.2 Bottom Drainage Hypothesis..................................................................................
38
3.2.1 Key Implications of Bottom Drainage.................................................................
38
3.2.2 Base Excess Pore Water Pressure and Bottom Drainage Relationship .......
39
Chapter 4 .......................................................................................................................
42
Numerical M odel for Top and Bottom Drainage.........................................................
42
4.1 General Overview ................................................................................................
42
4.1.1 Structure of M ethod ..........................................................................................
43
4.1.2 Finite Difference Approach in Excel................................................................
45
4.2 Description of Boundary Conditions ...................................................................
47
4.3 Description of Basic Layer Conditions..............................................................
49
4.4 Neutral Plane...........................................................................................................
49
4.5 Discretization of Test Specimen ..........................................................................
51
4.5 Compatibility Criteria..........................................................................................
52
4.5.1 Layer Compatibility..........................................................................................
52
4.5.2 Specimen Compatibility...................................................................................
53
4.5.2.1 Total Deformation Compatibility ..................................................................
53
4.5.2.2 Netural Plane Compatibility .........................................................................
54
5
Numerical Modeling of Constant Rate of Strain Consolidation Tests
4.6 Determination of t and k........................................................................
..... 55
56
4.7 Effects of Time Step and Layer Size ...................................................................
Chapter 5 ................................................................................................------..................
67
Interpretation of Results from Double Drainage Numerical Model............
67
5.1 General Overview ................................................................................................
67
5.1.1 Soil Input Param eters........................................................................................
67
5.1.2 O ther Input Param eters .....................................................................................
68
5.1.3 C onvergence Criteria.......................................................................................
69
5.2 Effects of Tim e Step ........................................................................................
.. 69
5.3 Validation of Excess Pore Water Pressure Distribution .....................................
70
5.2.1 Validation of Stress-Strain Relationship..........................................................
71
5.2.2 Validation of Void Ratio-Hydraulic Conductivity Relationship ......................
72
5.3 Effects of Strain R ates .........................................................................................
72
5.3.1 Effects on Excess Pore Water Pressure Profile.................................................
73
5.3.2 Effects on Maximum and Base Excess Pore Water Pressures............. 73
5.3.3 Effects on Neutral Plane ..................................................................................
74
5.3.4 Effects on Pore Water Pressure Ratio ..............................................................
75
5.4 Effects of Base Compressibility..........................................................................
75
5.5 Comparison With Constant Rate of Strain Consolidation Test Data................... 76
5.6 Application of Wissa Constant Rate of Strain Equations ...................................
77
5.6.1 Variation of Wissa's Relationships with RBBC Stress-Strain Relationship ....... 78
5.6.2 Variation of Wissa's Relationships with RBBC Strain-Hydraulic Conductivity
Relation ship ..................................................................................................................
78
6
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 6 ...................................................................----....
.......------------...................... 90
Conclusion and Future Research ........................................................................
90
6.1 Bottom Drainage...................................................................................
90
6.2 Numerical Simulation of Bottom Drainage ............................................................
91
6.2.1 Validation with Assumed Stress-Strain-Flow Relation ..................
91
6.2.2 Introduction of the Param eter X........................................................................
92
6.3 Recommendation for Further Research ..............................................................
92
References..............................................................................................................----
-
- 95
7
Numerical Modeling of Constant Rate of Strain Consolidation Tests
List of Figures
Figure 2.2.1- 1: Schematic diagram of a Wissa et. al. (1971) constant rate of strain
consolidation device...........................................................29
Figure 2.3.1- 1: Variation in cc with b/s (Smith and Wahls 1969)..........................30
Figure 2.4.1- 1: Deviation of Strain from average as a function of depth for different
time factors (Wissa et. al. 1971)...........................................30
Figure 2.4.1- 2: Curve fitting procedure to determine cv for CRS test during initial stages
when transient component is important (Wissa et. al. 1971)...........31
Figure 2.4.2- 1: Comparison of coefficient of consolidation from linear and nonlinear
theory (Wissa et. al. 1971).................................................
32
Figure 2.4.2- 2: Comparison of effective vertical stress from linear and nonlinear theory
(Gonzalez et. al. 1997).....................................................33
Figure 2.4.2- 3: Comparison of hydraulic conductivity from linear and nonlinear theory
(Gonzalez et. al. 1997).....................................................34
Figure 4.1- 1:
Discretization of test specimen into layers.................................
Figure 4.1- 2: Definitions of boundary conditions.........................................
57
57
Figure 4.1.1- 1; Flowchart of sequence of the first time step.............................58
Figure 4.1.1- 2: Flowchart of neutral plane calculations..................................
59
Figure 4.1.1- 3: Flowchart of layer calculations.............................................
60
Figure 4.1.1- 4: Flowchart of time step calculations.........................................
61
Figure 4.1.2- 1: Flow path of discretized layers............................................
62
8
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Figure 4.1.2- 2: Finite difference scheme................................................63
Figure 4.3- 1:
Schematic diagram of a basic layer with upward flow...................64
Figure 4.4- 1:
Schematic diagram of drainage extremes.................................64
Figure 4.5- 1: Top boundary as reference................................................65
65
Figure 4.5- 2:
Location of neutral plane.....................................................
Figure 4.5- 3:
Separating neutral plane layer based on flow direction.................66
Figure 4.5- 4: Neutral plane as reference....................................................66
Figure 5.2- 1:
CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1%/hr, X=O.01cm/ksc: Distribution of
excess pore water pressure with varying time step.....................80
Figure 5.3- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1%/hr, time step of lhr: Distribution of
excess pore water pressure with specimen.................................80
Figure 5.3.1- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1, 1 and 10%/hr and strain of 10%:
Com pression curve..........................................................81
Figure 5.3.2- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1, 1 and 10%/hr and total strain of
10%: Hydraulic conductivity curve.........................................
81
Figure 5.4.1- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1 %/hr, total strain of 10% and
X=0.0lcm/ksc: Distribution and development of excess pore water
pressure w ithin specim en.....................................................82
9
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Figure 5.4.1- 2: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1 %/hr, total strain of 10% and
X=0.Olcm/ksc: Distribution and development of excess pore water
pressure within specim en..................................................
82
Figure 5.4.1- 3: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1 0%/hr, total strain of 10% and
X=0.Olcm/ksc: Distribution and development of excess pore water
pressure within specimen..................................................
83
Figure 5.4.2- 1: Variation of strain rate on magnitude of maximum and base excess pore
water pressure at 10% strain and X =0.01cm/ksc..........................83
Figure 5.4.2- 2: CRS Test results performed on Model RBBC using numerical
simulation based on strain increments of 1% and X=0.01cm/ksc: Trend
of base excess pore water pressure with strain rates...................84
Figure 5.4.2- 3: CRS Test results performed on Model RBBC using numerical
simulation based on strain increments of 1% and X=0.Olcm/ksc: Trend
of ratio of maximum excess pore water to base excess pore water
pressure with strain rates...................................................84
Figure 5.4.3- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain increment of 1% and X=0.Olcm/ksc: Trend of
neutral plane location with strain rates..................................
85
Figure 5.4.4- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain increments of 1% and ?==0.01cm/ksc: Trend
10
Numerical Modeling of Constant Rate of Strain Consolidation Tests
of ratio of maximum excess pore water to base excess pore water
pressure with strain rates..................................................
Figure 5.5- 1:
85
CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1%/hr and time step of ihour: Trend of
excess pore water pressure distribution with base compressibility..... 86
Figure 5.5- 2:
CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1%/hr and time step of ihour: Trend of
ratio of maximum to base excess pore water pressure with base
com pressibility.................................................................86
Figure 5.5- 3:
CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 1%/hr and time step of Ihour: Trend of
neutral plane location with base compressibility.......................87
Figure 5.6- 1:
CRS Test results performed on RBBC using Wissa apparatus: Pore
w ater pressure ratio............................................................87
Figure 5.7.1- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1, 1 and 5%/hr and X=O.01cm/ksc:
Compression curve with Wissa's linear and non-linear relationships.. 88
Figure 5.7.1- 2: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1, 1, 5 and 20%/hr and
X=0.Olcm/ksc: Enlarged compression curve with Wissa's linear and
non-linear relationships...................................................
88
Figure 5.7.2- 1: CRS Test results performed on Model RBBC using numerical
simulation based on strain rate of 0.1, 1 and 5%/hr and X=0.01cm/ksc:
11
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Hydraulic conductivity curve with Wissa's linear and non-linear
relationships.................................................................
89
12
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 1
Introduction
1.1 Background and Problem Statement
Consolidation testing is a very important aspect of geotechnical engineering. This is
found in many geotechnical designs where settlement of the soil structure is an issue.
Hence to better understand the significance of soil settlement, there has been a reliance
on laboratory testing of consolidation test specimens to extrapolate the effects on true
consolidation by obtaining compressibility coefficients and rates of consolidation.
Consolidation tests are also important to ascertain the overconsolidation ratio (OCR) of a
test specimen and to obtain stress history of a cohesive soil.
Traditionally, consolidation tests are carried out using the conventional oedometer. This
is based on Terzaghi's theory, and is used to obtain consolidation parameters of the soil
in question. Based on the unification of testing methods by ASTM (American Society for
Testing and Materials), conventional oedometer tested are carried out at 24hours loading
intervals and at load increment ratios (LIR) of 1. This effectively doubles the load on the
soil test specimen every 24hours.
13
Numerical Modeling of Constant Rate of Strain Consolidation Tests
In 1969, Smith & Wahls published an approximate solution for the constant rate of strain
consolidation (CRS) process and proposed the CRS test as a consolidation test method. A
rigorous solution which accounts for initial transient effects, was published by Wissa et.
al. (1971). This was processed by a numerical model developed by Gonzalez et. al.
(1997), which found the basis of the research of this thesis. Due to the many advantages
CRS test has over conventional oedometer test, many industries are now adopting the
CRS test as the standard method of consolidation testing.
The CRS test represents an efficient and fast method of determining consolidation
parameters of a given soil. In particular, CRS tests are preferred for soils with non-linear
void ratio and logarithmic vertical effective stress relationship. One important advantage
of the CRS test is the ability to obtain, at real-time intervals, the hydraulic conductivity of
the test specimen. CRS tests avoid problems relating to secondary compression between
load increments, as found in conventional oedometer test.
CRS test is a strain-driven test method as opposed to a conventional oedometer test,
which is stress-driven. This can result in strain rates playing an important role in the test
results obtained. Wissa et. al. (1971) also pointed out that immediately as the piston was
set in motion a transient condition develops, which made predicting the coefficient of
consolidation (cv) and hydraulic conductivity (k) much more difficult. In addition, most
analysis methods of CRS data have assumed that bottom drainage of the test specimen is
not an issue. This is subjected to debate and looked into in detail in this thesis.
14
Numerical Modeling of Constant Rate of Strain Consolidation Tests
1.2 Research Objective
The main objective of this research is to investigate the effects of base compressibility on
excess pore water pressure generation, strain rate, drainage boundaries on Normally
consolidated Resedimented Boston Blue Clay. Using soil parameters obtained from an
actual constant rate of strain test, CRS238, the soil is modeled after the stress-strainhydraulic conductivity relationship from the test CRS238, accounting for the steady state
conditions. This thesis addresses the implications of the location of the neutral plane, a
no-flow boundary in the midst of the test specimen, denoted by p, and the base
compressibility of the testing apparatus, denoted by k.
The theoretical results are compared with some CRSC test data and Wissa's linear and
non-linear theories and discussions are drawn from these comparisons. This research
forms the basis for which future experimental validation, normally consolidated to overly
consolidated and steady state to transient state analysis. This can allow incorporation of
new theories into the interpretation of constant rate of strain consolidation tests.
1.3 Thesis Organization
Chapter 2 provides a literature review of previous research in CRS testing of cohesive
soils and the analysis methods used to interpret CRS data. This Chapter describes the
existing numerical model on CRS test, which is based on top drainage only.
15
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 3 also discusses the proposed hypothesis on bottom drainage in a CRS test cell
and look at the conceptual basis of the hypothesis.
Chapter 4 looks at the formulation of the numerical model based on the proposed bottom
drainage hypothesis and also describes some of the key features in the numerical model.
Chapter 5 presents some of the results obtained based on the numerical method for
bottom drainage and evaluates the validity of these results. Some relationship obtained
based on these results are also discussed in this chapter.
Chapter 6 summaries the research of this thesis and presents the objectives of future
research in this area.
16
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 2
Existing Testing Methods and Theories
2.1 Consolidation Theory
When a stress loading is placed on to a cohesive soil mass, excess pore pressure is
generated as a result of lack of instantaneous dissipation of pore fluid. As excess pore
pressure builds up, water begins to flow due to the hydraulic gradient caused by the
excess pore pressure. The loss in soil volume due to this flow of water out of the soil
mass is defined as consolidation. The mathematical theory describing the dissipation of
excess pore pressures and the associated deformation of the soil is called consolidation
theory.
2.1.1 One-Dimensional Consolidation
Consolidation tests are performed to obtain the consolidation parameters of a soil test
specimen. Such tests are generally carried out allowing flow only in the vertical direction
and also strain only in the vertical direction. The theory behind one-dimensional
consolidation test is Terzaghi's consolidation equation.
17
Numerical Modeling of Constant Rate of Strain Consolidation Tests
2
a Ue
C at
2
aue
~
at
at
[2-1]
where:
cv
coefficient of consolidation
ue
excess pore pressure
total stress
t
time
z
coordinate in the vertical direction (direction of flow and strain)
The main assumptions involved in Terzaghi's equation are the validity of Darcy's Law,
small strain limitation and linear stress-strain relationship.
2.2 Constant Rate of Strain Consolidation Test
Traditionally, one-dimensional consolidation test are carried out using incremental
oedometer test. However, due to the use of a load increment ratio of one, the spacing
between data points is typically large and this can be problematic in determining some of
the consolidation parameters. In order to improve the resolution of the data obtained from
such tests, the load increment ratio will have to be reduced below one. This sparked the
introduction of a series of different tests, which invokes continuously varying load.
18
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Some of these continuously varying load tests include controlled hydraulic gradient test
(Lowe et. al. 1969), constant rate of loading test (Aboshi et. al. 1970) and constant rate of
strain test (Smith and Wahls 1969; Wissa et. al. 1971).
The constant rate of strain (CRS) test is a fast and efficient method of obtaining
consolidation parameters of cohesive soils. These include the stress history,
compressibility, hydraulic conductivity and rate of consolidation of the soil in question.
The CRS test has many advantages over the traditional incremental oedometer test. The
unexhausted list include (1) the CRS test can be easily automated due to a constant strain
implementation and is less labor intensive; (2) the CRS test is able to provide more
meaningful test data by taking data at a far more frequent rate than an incremental
oedometer test; (3) the CRS test generally takes a lesser time than an standard
incremental oedometer test as the CRS test does not allow any time for secondary
compression as in the case of incremental oedometer; (4) the CRS test apparatus is
designed to be able to back pressure saturate test specimens.
2.2.1 Constant rate of strain testing apparatus
The typical apparatus setup for a constant rate of strain test is shown in Figure 2-1. The
figure depicts the schematic of the general-purpose consolidometer developed by Wissa
at MIT. The test specimen is held inside a stainless steel ring, which rests directly on a
fine ceramic porous stone epoxied into the base. The specimen is typically loaded at a
19
Numerical Modeling of Constant Rate of Strain Consolidation Tests
constant rate of strain by means of moving the piston with a gear driven load frame.
During the test, a transducer is connected through the porous stone and is used to measure
the excess pore pressure at the base of the specimen. The chamber pressure is measured
through another transducer outside the apparatus. At the same time, the vertical load,
which is a reaction caused by the specimen stiffness, is measured by an external load cell.
The imposed displacement is measure by a linear voltage displacement transducer
(LVDT) attached to the piston.
2.3 Constant Rate of Strain Theory
The theory for the constant rate of strain loading has been developed in order to
determinate important consolidation and stress-strain parameters from the tests. These
include the determination of average effective stress ('), void ratio (e), coefficient of
consolidation (cv), and hydraulic conductivity (k). The two most popular theories for
interpreting CRS data would be the Smith and Wahls (1969) linear theory and Wissa et.
al. (1971) non-linear theory. These theories are looked at in detail in proceeding sections.
2.3.1 Smith and Wahl's Linear Theory
Smith and Wahls (1969) had developed the governing equation for constant rate of strain
consolidation similar to that developed by Terzaghi (equation [2-1]). The theory is based
on the following assumptions:
*
The soil is both homogenous and saturated.
20
Numerical Modeling of Constant Rate of Strain Consolidation Tests
"
The water and soil solids are both incompressible; i.e. consolidation can only
occur as a result of loss in voids.
* Darcy's law is valid for flow through the soil.
*
Drainage and strain can only occur in the vertical direction.
"
The total and effective stresses are assumed to be uniform along a horizontal
plane; i.e. stress differentials occur only between different horizontal planes.
Based on the continuity of flow through a soil element, the basic equation of
consolidation is:
a (kau,
z (Y z
1 ae
[2-2]
1+e at
where:
k
hydraulic conductivity
yw
unit weight of water
e
void ratio
By assuming that the hydraulic conductivity of the soil is a function of the average void
ratio and that the change in void ratio with time is so small that infinitesimal strains exist,
equation [2-2] can be simplified. In addition, since there is no lateral strain, the
volumetric strain is equal to the axial strain. Hence, the equation can be simplified to the
expression below, making the excess pore water pressure at the base of the test specimen
the subject.
21
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Au -Ws-H2 1 _b[~1[23
k(1 + eavg)(2 s _12_
where:
Aub
base excess pore water pressure
s
rate of change of average void ratio,
H
test specimen height
eavg
average void ratio
b
constant that depends on the variation in void ratio with depth and
Ae
Av
At
time
In order to obtain the void ratio versus effective stress relationship, an expression for
average effective stress that involves variables measured in the testing procedure was
developed.
a-', = o- - a -Aub
[2-4]
where:
a'V
average vertical effective stress
a
the ratio of the average pore water pressure, uavg, to the pore water
pressure at the base, Aub
Wahls and de Godoy (1965) noted that in an incremental loading test a is initially 1.0.
With time, a was found to reach a steady value of about 0.63. For linearly increasing
22
Numerical Modeling of Constant Rate of Strain Consolidation Tests
stress, they had estimated that a starts initially at about 2/3 and progressively decrease to
a steady value of 0.64. The variations of a with b/s are shown in Figure 2.3.1-1.
Based in Terzaghi's definition of the coefficient of consolidation (cv),
k
[2-5]
k (1+ e)
where
Ae
mv
coefficient of volume compressibility,
av
coefficient of volume compressibility, Ac
-
Solving equations [2-3] and [2-5],
V= s-[H 2
1
b 1
a, Aub _2-s-12_
[2-6]
An important observation made with regards to Smith and Wahls' solution is that the
equation appeared to be valid for all strain rates. This is however untrue and the critical
strain rate was proposed by the authors to be defined empirically by limiting the ratio of
excess pore water pressure in the undrained face to the total vertical stress.
2.4 Wissa's Theories
Wissa proposed the first complete solution for constant rate of strain consolidation. The
solution predicts initial transient conditions generated in the soil before moving into
steady state conditions. The assumptions of his solution is listed below:
23
Numerical Modeling of Constant Rate of Strain Consolidation Tests
* Infinitesimal strains exist.
* The coefficient of consolidation (c,) is constant with any variation of hydraulic
conductivity and coefficient of volumetric compressibility (mv).
*
Deformation and flow occur only in the vertical direction.
*
The soil is both homogenous and saturated.
*
The water and soil solids are both incompressible.
The governing equation for constant rate of strain consolidation by Wissa (1971) is
defined as:
C
VaZ
[2-7]
-=
2
at
where:
E-
vertical strain
Wissa et. al. (1971) had showed that there are two components of strain - transient state
conditions and steady state conditions. The transient state conditions refer to the decay of
initial discontinuities setup at the start of the test. As the test progresses, the conditions
shift to steady state. The steady state conditions represents the average strain is increasing
and the variation in strain with position is constant throughout the specimen. As a results,
the strains are constant in time.
24
Numerical Modeling of Constant Rate of Strain Consolidation Tests
2.4.1 Transient State Conditions
Wissa et. al. (1971) solved the governing equation for consolidation (equation [2-7]),
giving strain as a function of time factor Tv, and dimensionless spatial variable X. It is
expressed in the following form:
c(X,Tv)
= 8
[2-8]
t[ 1+F(X,Tv)]
where
H
specimen height
X
dimensionless spatial variable = z/H; z = 0 at the upper end and z = H at the
bottom of the specimen
8
strain rate
Tv
'*2
time factor, 9H2
F(X, T,)=
C -t
1
6T,
(2 -6X +3X
[2-9]
2
)-
2
7r T n1
cos nrX
e _nE TV
2
2
[2-10]
n
The first component of equation[2-10] represents the steady state conditions and the
second component of equation [2-10] represents the transient state conditions. Figure
2.4.1-1 shows the deviation from the average strain as a function of depth for various
times in the test.
For transient state conditions in linear theory, Wissa showed that transient component is
significant in the initial stages of loading, and Tv can be found as a function of F3 .
25
Numerical Modeling of Constant Rate of Strain Consolidation Tests
__
F3
(U-V
(-
Aub)
-
-
As)-o''5
(V
-(
v,(t=O)
[-1
[2-11]
v,(t=O)
Similarly, for non-linear theory,
F = log(oC
-
Aub) - log(ov,(O))
log(Ov,(t-O))
[2-12]
3 -
T, can be determined by using Figure 2.4.1-2 or by regression analysis equation by
Sheahan et. al. (1997).
Tv = 4.78(F 3)3 - 3.2 1(F 3 )2+1.63F 3+0.0356
[2-13]
After Tv is determined, for Tv < 0.5, transient states is assumed and for Tv>0.5, steady
state is assumed.
2.4.2.1 Modified Linear Theory
When steady state conditions are established, Wissa et. al. (1971) simplified Smith and
Wahls' linear stress-strain relationship equations by assuming b/s=O, ct=2/3 and eavg=eo.
The average effective stress and hydraulic conductivity are simplified to the following:
a'=I
k
=
-
c-H
2
Aub
[2-14]
[2-15]
.Aub
26
Numerical Modeling of Constant Rate of Strain Consolidation Tests
By substituting Terzaghi's definition of c, (equation [2-5]) and the following:
At
___
[2-16]
m = A =r
Aa
AoBased on Wissa's modified linear theory,
C = H2
[2-17]
gv.J
S2 Aub ( At
2.4.2.2 Wissa's Non-linear Theory
The non-linear solution of Wissa et. al. (1971) is based on the non-linear consolidation
theory of Davis and Raymond (1965). The stress-strain relationship equations for a'y, cv
and k are given below:
- 2
,=-
- Au2
- Au, +v
)/3
[2-18]
H 2 - log 1v21
LV Auv
k =-
0.434
4rA
2.-o-' -log
2
- At
-
[2-19]
[2-20]
Au
When the ratio of the excess pore water pressure at the base to the total vertical stress,
Aub/av is small, the results from linear and non-linear theories are very similar. However,
as Aub/av increases, the divergence of results from different methods becomes more
apparent.
27
Numerical Modeling of Constant Rate of Strain Consolidation Tests
2.4.3 Comparison of Wissa's Linear and Nonlinear Theory
The comparison of results for coefficient of consolidation cy, effective vertical stress a'v
and hydraulic conductivity k, are shown in Figures 2.4.3-1 to 2.4.3-3 respectively. The
deviation of all three parameters increases significantly with increase in pore water
pressure ratio Aub/v. The deviation can be seen in further detail in Chapter 5, when these
equations are compared with the results from the numerical simulation.
2.5 Previous Research on 1-D Consolidation at Constant Rate of Strain
Gonzalez et. al. (1997) has produced a numerical simulation of a model soil with known
constitutive properties tested under a constant rate of strain consolidation. The numerical
simulation was used to facilitate:
*
A method to check various CRS equations
" Large strain simulations
*
The elimination of the assumption of a constant cv
The soil was simulated by splitting up the specimen into twenty small layers, where an
iterative procedure was used to solve the simultaneous equations for the application of an
increment of displacement at the top (z=O) surface over a fixed amount of time. The
iteration was continued until a unique e versus log a'v and e versus log kv was observed
for all layers within the specimen. The program used a semi analytical approximation to
converge to a final value.
28
Numerical Modeling of Constant Rate of Strain Consolidation Tests
LOAD CELL
PISTON ASSEMBLY
CELL
PRESSURE
AIR VENT
CELL
ROLLING
CHAMBER
DIAPHRAGMSEAL
COARSE
POROUS
STONE
FINE
CERAMIC
POROUS
STONE
PRESSURE
TRANSDUCER
TO BACK
PRESSURE
(DRAINAGE)
Figure 2.2.1- 1: Schematic diagram of a Wissa et. al. (1971) constant rate of strain consolidation
device
29
Numerical Modeling of Constant Rate of Strain Consolidation Tests
b/s
c
0.0
0.5
1.0
1.5
2.0
0.667
0.682
0.700
0.722
0.750
Figure 2.3.1- 1: Variation in a with b/s (Smith and Wahls 1969)
aR~
0.0
-
-1-
-
0.2
e
XQ4
Steady
State
x 0.6
1.0
---
-.3
-. 2
-. 1
.1
1r
cosnWZ/H
n2
I z Z 2
FxTV= - -- +---3 H2H2 12
.2
.3
-
.4
TH
TV
nia
Figure 2.4.1- 1: Deviation of Strain from average as a function of depth for different time factors
(Wissa et. al. 1971)
30
Numerical Modeling of Constant Rate of Strain Consolidation Tests
0.9
.,il]H
O(18
(Ir ,-Ub)- V, so
V .'TWO'
1:SLkWW
t.
V. No.-LbWbq*
r
-
CsL40X 1
6-
WVI. '4040
ae
Non -Lkssor
0,5(1
Tcmr
C
/
.
Tmu
eIeae
0.2
0.1
0
~1~~~.
2'/
I
V
4
0
--..--.
F* Q4
Q4
U4
F
Q6
II
ww 0.020
cQwI
QO
O
003
04
t/H t
I I I I
L05
6
OLO
0.08
09
Figure 2.4.1- 2: Curve fitting procedure to determine c, for CRS test during initial stages when
transient component is important (Wissa et. al. 1971)
31
Numerical Modeling of Constant Rate of Strain Consolidation Tests
N.6 -LINEAR ThEORY. WbbogTeconstA
LINEAR TWORY:Wb l - ccnstnt
2.---
-
2
b 2.
-~
-
-~~~
-
-
-
ro)
0.%
z 0
Z
-I-~~
1.0
0
.1
.2
.3
.4
.5
.6
~
.7
.8
.9
LO
Figure 2.4.2- 1: Comparison of coefficient of consolidation from linear and nonlinear theory
(Wissa et. al. 1971)
32
Numerical Modeling of Constant Rate of Strain Consolidation Tests
3.50
-
-
3.25
-
~~~~-
3.00
--
2.75
-
- - ..
- .
- .
-
- - - - -- - - -- --- - - ---- ----
--
- -
. .--
-
-
-- - - -".
-
-
-
- - -
-- -
-
- -
--
- - - - --
---
--
-- -
--- - ----- - - - - - - --- ---
-- -
- -- -
-
-
2.50
Ib
-
2.25
-
- -- - - - - ---- ----
2.00
- - - -
1.75
-
-
- - -
- --
---
---
1.50
1.25
- - - - - - - - - - - - -- - -- --- - - ------ - - ---
-
- -
-.
- - - -- - -
-
1.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Pore-Water Press=e Ratio; AU b/av
Figure 2.4.2- 2: Comparison of effective vertical stress from linear and nonlinear theory
(Gonzalez et. al. 1997)
33
Numerical Modeling of Constant Rate of Strain Consolidation Tests
3.50
3.25
--
- --
--
- --- - --- -- -
- - - ----
---- -- - - - - - --
3.00
S2.75
- -- - ----
- - ---
--- - ---
--
2.50
2.25 - -
-----
- - --- -- - - - - --- - - - - - - - --
- -
- -
- -
- -
- - - -
2.00
1.75
- --
- - -
-
-
-
---
--- -
- - -- - --- - ---- - - ----
1.50
-- -- - - -- -- -
- - - - ----
-
- - --
----
--
1.25
1.00
0.00
0.10
0.20
0.30
0.40
Pore-Wat
0.50
0.60
0.70
0.80
0.90
1.00
Pressure Ratiou /orv
Figure 2.4.2- 3: Comparison of hydraulic conductivity from linear and nonlinear theory
(Gonzalez et. al. 1997)
34
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 3
Bottom Drainage Hypothesis
3.1 Overview
A standard constant rate of strain test apparatus has a coarse porous stone placed at the
top of the test specimen and is connected to a backpressure valve. The bottom of the test
specimen sits on another porous stone, which is connected to a pressure transducer in
order to measure the excess pore pressure at the base of the test specimen. This can be
seen in Figure 2-1.
When the test specimen undergoes consolidation, pore fluid, in most cases, water, is
forced out of the test specimen due to the increasing strain. The soil grains are generally
assumed to be incompressible. Hence any change in volume is accommodated by the
removal of water. It is generally assumed that the bottom flow boundary is perfectly rigid
and hence all flow is through the top porous stone. However, flow is required to increase
pressure in the bottom pore pressure measuring system. This chapter looks at the
implications of each boundary and discuss on the concept of accounting for drainage flow
through the bottom porous stone.
35
Numerical Modeling of Constant Rate of Strain Consolidation Tests
3.2 Top Drainage Theories
The existing theories reviewed in the previous chapter, Wissa et. al.(1971) and Gonzalez
et. al. (1997), were all based on one common assumption. Pore fluid that is forced out of
the test specimen during consolidation is assumed to only flow upwards, through the top
porous stone and the backpressure pipe. It is assumed that there is no flow through the
base porous stone. In other words, all loss in voids is taken to have occur through top
drainage.
When all drainage occurs through the top drainage, the excess pore water pressure is
expected to increase down the test specimen. This is because the least amount of pore
water flow occurs at the base of the specimen. This in turn results in minimal excess pore
water pressure relief at the base.
3.2.1 Advantage of Top Drainage Assumption
The advantage of this assumption is that the computation needed to model constant rate
of strain consolidation becomes simplified. Any change in strain can be directly related to
the amount of top drainage, as seen in equation [3-1]
VOltopflow - AH . A = A. - Hiniial 'A =
Sinitial
Ae
1
+ e0
. Hinitial -A
[3-1]
where
voltopflow
volume of pore water escaping from top drainage
AH
change in height of test specimen
36
Numerical Modeling of Constant Rate of Strain Consolidation Tests
A
cross-sectional area of test specimen
AF,
change in strain, Ae = C. t
C
strain rate
t
time of consolidation
Hinitial
Initial height of test specimen
Ae
change in void ratio due to consolidation
eo
initial void ratio
In addition, by assuming no bottom drainage, the excess pore pressure measured at the
base, ub, can be deduced to be at the maximum. The base excess pore pressure measured
can then be used to calculate the hydraulic conductivity, and the average effective vertical
stress of the test specimen.
By looking at the setup of the constant rate of strain consolidation test apparatus,
however, it can be seen that the simplifying assumption of top drainage may not be
always valid. The amount of bottom drainage may be significant if the base
compressibility of the base porous stone connection is high. Hence, there is a need to
improved on the top drainage model to account for cases where the bottom drainage is no
longer negligible.
37
Numerical Modeling of Constant Rate of Strain Consolidation Tests
3.2 Bottom Drainage Hypothesis
The base setup of a constant rate of strain consolidation test connects a porous stone at
the base of the test specimen with a pressure transducer. As excess pore water pressure
builds up at the base, it can be expected that some flow must occur through the base
porous stone. This flow, in reality, exerts a pressure, which is needed to produce a
reading in the base pressure transducer. Furthermore, it is expected that the amount of
flow is dependent on the magnitude of the pressure at the base. At higher base excess
pore water pressure, the flow will increase.
3.2.1 Key Implications of Bottom Drainage
Due to the existence of bottom drainage, the loss in void ratio or the change in strain is no
longer dependent on top drainage alone. Instead, the change in void ratio or strain is now
dependent on the sum of both top and bottom drainage. This increases the complexity of
the problem.
By accounting for bottom drainage, one of the important implications is that the excess
pore water pressure generated no longer increases from zero at the top boundary to a
maximum at the base of the test specimen. Instead, the excess pore water pressure
generated increases from zero at the top boundary to a maximum somewhere within the
test specimen and decreases to a value at the bottom boundary.
38
Numerical Modeling of Constant Rate of Strain Consolidation Tests
In addition, this new excess pore water pressure profile results in a slightly varied method
of computing the effective vertical stress exerted by the piston on the test specimen.
Based on Gonzalez's et. al. (1997) numerical model, the effective stress is calculated
based on the void ratio of the base layer, where the excess pore water pressure is a
maximum. In this hypothesis, the effective vertical stress is computed based on the void
ratio of the layer where the excess pore water pressure is the maximum. Due to the
distribution of the excess pore water pressure in a both drainage model, this maximum
pressure elevation may not be at the lowest layer of the model. Hence, there is a need to
determine the elevation with the maximum excess pore water pressure and calculate the
effective vertical pressure based on this elevation.
Another significance of this maximum excess pore water pressure elevation is that this
elevation is the neutral plane of the test specimen. The neutral plane of the test specimen
serves as the division line between top drainage and bottom drainage. This implies that a
water molecule at this elevation would have no preference to flow towards the top or
bottom drainage. Other implications include (1) no flow can pass through the neutral
plane and (2) the base pore water pressure is used to compute average effective vertical
stress and the hydraulic conductivity.
3.2.2 Base Excess Pore Water Pressure and Bottom Drainage Relationship
When a constant rate of strain consolidation test is performed, pore water drainage occurs
through both top and bottom porous stone. At the top drainage, the pipe connecting to the
39
Numerical Modeling of Constant Rate of Strain Consolidation Tests
backpressure cell acts as a full drainage with no resistance. At the base of the test
specimen however, the bottom drainage acts only as a partial drainage. Hence, there is a
need to look at the effects of the partial drainage.
The amount of partial drainage is hypothesized to be dependent on the excess pore water
pressure developed at the base of the test specimen. As the base excess pore water
pressure increases, the amount of bottom drainage is expected to increase. Based on this
understanding, the relationship between base excess pore water pressure and the amount
of bottom drainage is defined as shown:
Aub= 2AHbotom
[3-2]
where:
Aub
excess pore water pressure developed at the base
AHbottom
change in axial height of test specimen due to bottom drainage
X
correlation constant
The principle of the relationship is described in detail in Chapter 4. As seen in equation
[3-2], the relationship between the base excess pore water pressure and the change in
axial height due to bottom drainage is assumed to have a linear relation. Since there is no
lateral strain in a CRS test cell, the change in axial height is directly proportional to the
change in volume or void ratio.
40
Numerical Modeling of Constant Rate of Strain Consolidation Tests
The correlation constant X can be seen as a device parameter. X is dependent on the
coarseness and size of the porous stone, the stiffness of the base pressure transducers and
other components connected to the base porous stone and the compressibility of water.
As X is defined as a device parameter, the validity of bottom drainage hypothesis may be
verified by means of CRS testing using different apparatus setup.
41
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 4
Numerical Model for Top and Bottom
Drainage
4.1 General Overview
This thesis investigates the effects of apparatus pore pressure system compressibility on
the integrated stress-strain-hydraulic conductivity of a soil under the constant rate of
strain tests by means of a numerical model. This numerical model is a mathematical
simulation of a constant rate of strain consolidation test using a finite difference scheme
using Microsoft Excel. The modeled test specimen is divided in 20 layers, shown in
Figure 4.1-1. The top and bottom drainage conditions are defined as a full drainage and
partial drainage boundary respectively, shown in Figure 4.1-2. The partial drainage is
defined as a function of the base measuring stiffness of the apparatus.
The model has the following assumptions:
*
Darcy's Law
*
Flow and strain is one- dimensional
*
Large strain analyses
42
Numerical Modeling of Constant Rate of Strain Consolidation Tests
"
No secondary compression
* Pressures generated are constant along any horizontal plane
* Linear approximation within time step
The model uses a finite difference approach to obtain excess pore water pressure
solutions that satisfies the stress-strain-hydraulic conductivity relationships of the soil,
namely the void ratio versus hydraulic conductivity and void ratio versus effective
vertical stress equations.
4.1.1 Structure of Method
The numerical model simulates the steady state conditions of a constant rate of strain test
at specified time intervals. There are two sets of inputs that have to be defined in the
Excel spreadsheet. The algorithm of the first time step simulation is shown in Figure
4.1.1-1.
The first set of input defines the constant rate of strain test parameters, which include:
*
Strain rate, e
"
Time step, t
*
Test specimen height, Ho
* Height of Soil, Hs
*
And unit weight of pore fluid (usually water, yw=0.01kg/m 3)
43
Numerical Modeling of Constant Rate of Strain Consolidation Tests
The second set of input defines the soil behavior parameters of the test specimen, which
includes the slope and intercepts of:
*
void ratio, e, versus logarithm of hydraulic conductivity, k
*
void ratio, e, versus logarithm of effective vertical stress, a',
The next step in the simulation is to calculate the neutral plane, shown in Figure 4.1.1-2.
An assumed value of pt is used to define the neutral plane, shown in Figure 4.1-2 . The
neutral plane is used determine if top and/or bottom flow occurs in a particular defined
layer, shown in Figure 4.1.1-3. The numerical model computes at each layer:
*
Layer deformation
*
Average void ratio
*
Hydraulic conductivity based on average layer void ratio, e
" Excess pore water pressure at layer boundary
*
Average effective vertical stress for layer
Once the results for each layer are obtained for the first time step, the void ratios are used
to calculate the above-listed parameters for the subsequent time step. The algorithm is
shown in Figure 4.1.1-4. The process is repeated until the last time step required is
solved.
44
Numerical Modeling of Constant Rate of Strain Consolidation Tests
4.1.2 Finite Difference Approach in Excel
The numerical model analyzes a constant rate of strain consolidation test by using a finite
difference approach in Excel. The approach is used to give an approximate value of the
excess pore water pressure at each layer boundary, bounded by the boundary conditions
of the top and bottom drainage.
There are several steps in the finite difference approach, which relates the void ratio and
excess pore water pressure of each layer. The first step is to relate the amount of flow
with the change in void ratio. The amount of flow in a single layer is dependent on the
net change in void ratio of all layers between the neutral plane and the layer in
consideration. This is because all flow from these layers must pass through the layer in
question, as seen in Figure 4.1.2-1. For example, flow through layer 1 is due to the
change in void ratio of layers 1 through 7, and flow through layer 7 is due to the change
in void ratio of layer 7 only. The equation relating the change in void ratios with the
Darcy's constant is shown below.
i=n
Z
i=k
Ae - H1
-v.
[4.1]
t
where
Aej
change in void ratio at ith layer
Hi
height of ith layer
t
time step
Vi
Darcy's constant for ith layer
45
Numerical Modeling of Constant Rate of Strain Consolidation Tests
number of layers with the same flow direction
n
The next step is to obtain the hydraulic conductivity of each layer, using equation 4.2.
e1 -eO
k
=1O
Ck
[4.2]
where
k
hydraulic conductivity of ith layer
eo
void ratio at k = 1 cm/s
ei
void ratio of ith layer at end of time step
ck
coefficient of log hydraulic conductivity versus void ratio
Using Darcy's Law, the change in excess pore water pressure between two boundaries is
calculated using the equation below.
ui - ui- = Hi,r, vi
ki
[4.3 ]
where
ui
excess pore water pressure at the bottom of ith layer
ui 1
excess pore water pressure at the top of ith layer
7W
unit weight of water
The finite difference scheme starts with the topmost layer (i.e. the first layer), bounded by
the top drainage, where the excess pore water pressure, uO, is zero. The excess pore water
46
Numerical Modeling of Constant Rate of Strain Consolidation Tests
pressure at the base of the first layer ui is dependent on the Darcy's constant, vI, of the
first layer, shown in Figure 4.1.2-2.
Once the excess pore water pressure at the base of the first layer, ui, is obtained, the
excess pore water pressure at the base of the second layer, u2 , can be obtained as a
function of ul and v2 . This process is repeated until the last layer or neutral plane. The
excess pore water pressure at the bottom boundary can be determined from the Darcy's
constant of the last layer vn, the excess pore water pressure at the top of the last layer u,.and the base measuring stiffness, which is a function of k.
4.2 Description of Boundary Conditions
The test specimen in a constant rate of strain consolidation test is bounded by three
boundaries - radial, top and bottom. The conditions of these boundaries are critical in the
analysis of the consolidation process. This section describes the main characteristics of
the three boundaries.
The radial boundary is defined as a rigid boundary. This results in a one-dimensional
consolidation test as no lateral deformation of the test specimen is possible. The radial
boundary is also non-permeable. This forces all drainage to flow in the vertical direction,
towards the top and bottom boundaries.
47
Numerical Modeling of Constant Rate of Strain Consolidation Tests
The top boundary is defined as a full drainage boundary. This implies that the top
boundary does not offer any flow resistance for flow to move out of the top. As a result,
there can be no excess pore water pressure at the top - the excess pore water pressure
measured at the top is always zero.
The bottom boundary is defined as a partial drainage boundary. This is to simulate the
response of the real testing devices. For the course of this thesis, however, there are no
data available, so a range of values will be assumed. During a constant rate of strain
consolidation test, excess pore water pressure builds up at the bottom boundary. This
pressure acts to push water out of the test specimen to pressurize the measuring system.
Based on Henderson et. al.(1994), the volume compressibility is defined to be
A Vol
A o Pre . For the purpose of this application, there is no lateral deformation. Hence
A Pr essure
the volume compressibility can be modified to
.
AHL- Area..
, if the volume is taken in
A Pressure
terms of the specimen area. The volume compressibility is further simplified using
equation [3-1] defined below:
[3-2]
Aub = AJIbottom
This allows the volume compressibility to be reduced to
Area
.
48
Numerical Modeling of Constant Rate of Strain Consolidation Tests
4.3 Description of Basic Layer Conditions
A basic layer comprises a soil element of fixed area and varying height, subjected to
upper and lower boundary conditions, seen in Figure 4.3-1. Excess pore water pressure
begins to develop on both the upper and lower boundaries of the layer under constant rate
of strain loading. Due to the difference drainage conditions of the top and bottom
drainage of the test specimen, the developed excess pore water pressures are unequal.
The difference in excess pore water pressure results in pore water flow, which in turn
results in the change in height of the layer.
The amount of excess pore water pressure and deformation is dependent on the stressstrain-hydraulic conductivity of the soil. These are defined by soil behavior parameter
inputs stated in Chapter 4.1.1 and presented in the equations below.
ei -eO
k 1 =10
Ck
[4.2]
e1 -eO
- =10
CR
[4.4]
where
&-'
effective vertical stress experienced at the middle of ith layer
CR
compression ratio
4.4 Neutral Plane
Flow can only occur in one direction at any given point during a time step. The
separation between upward and downward flow is defined as the neutral plane. The
49
Numerical Modeling of Constant Rate of Strain Consolidation Tests
numerical model assumes a neutral plane of the test specimen under constant rate of
strain, shown in Figure 4.1-2. This neutral plane is defined by a ratio between the height
of the test specimen undergoing bottom drainage and the total height of the test specimen
below:
H bottom
Htota
[4-5]
where:
Hbottom height of test specimen undergoing bottom drainage
Htotal
total height of test specimen
p
correlation constant
The numerical model has to account for the boundary conditions of the top and bottom
drainage. However, as the finite difference method runs from the highest layer
downwards, it is difficult to iterate the neutral plane location by means of applying both
boundary conditions of top and bottom drainage. The numerical model is unable to
identify at which location the drainage flow direction switches from upward to
downward. Hence, by making use of the relationship between p and k, the neutral plane
location is varied until the required base measuring stiffhess is obtained.
The elevation of the neutral plane from the base of the test specimen decreases with the
flow resistance of the bottom drainage. In one extreme, when the compressibility is zero,
there will be no bottom drainage at all. The neutral axis is exactly at the base of the test
specimen. In the other extreme, when the compressibility is infinite, there is full drainage
50
Numerical Modeling of Constant Rate of Strain Consolidation Tests
at the base of the test specimen. When this occurs, the flow towards both the top and
bottom drainage will be identical. Hence, a water molecule at the center of the test
specimen will have no preference to flow upwards or downwards. Both extremes are
shown in Figure 4.4-1.
Numerically, when p approaches zero, k is expected to approach zero. Conversely, when
p approaches
V,
k is expected to approach infinite.
4.5 Discretization of Test Specimen
The numerical model divides the test specimen into 20 layers, the coordinates of each
layer boundary is referenced to the top specimen boundary, defined as the absolute
coordinate system seen in Figure 4.5-1. Based on the assumed location of the neutral
plane, the layer where the neutral plane is identified. This is done by comparing the
absolute vertical coordinate of the assumed neutral plane with the absolute vertical
coordinate of each layer boundary, seen in Figure 4.5-2.
Once this neutral plane layer is identified, this layer will be then subdivided into 2 sublayers - the top layer is defined to undergo top drainage and the bottom layer, bottom
drainage, seen in Figure 4.5-3. Hence, all the layers above the neutral plane, including the
top sub-layer, will be analyzed based on top drainage. Similarly, all the layers below the
neutral plane, including the bottom sub-layer, will be analyzed based on bottom drainage.
51
Numerical Modeling of Constant Rate of Strain Consolidation Tests
During the finite difference computation, the relative vertical coordinate system, taking
the neutral plane as reference, is used for the determination of the consolidation
deformation of each layer boundary, as seen in Figure 4.5-4. By taking the neutral plane
as reference, the deformation due to top and bottom drainage can be distinctly separated.
The neutral plane is perceived to have no movement and the soil above and below the
neutral plane deforms towards the neutral plane. The reason for switching coordinate
systems is to directly relate the top and bottom drainages with the deformation due to top
and bottom drainages respectively. This enables the simplification of the computations
involved within the numerical model.
4.5 Compatibility Criteria
Every finite difference computation has a series of compatibility criteria, which the
model needs to satisfy to ensure that the solution is meaningful. This numerical model is
no exception. The compatibility criteria are divided into three types, listed below.
" Layer compatibility criteria
*
Specimen compatibility criteria
*
Soil compatibility criteria
4.5.1 Layer Compatibility
Layer compatibility criteria need to be satisfied between individual layers. This is to
ensure that the individual layers can be combined to behave as a single entity. The first
layer criterion is the matching of excess pore water pressure at the interface between
52
Numerical Modeling of Constant Rate of Strain Consolidation Tests
layers. This is important to ensure that the behavior at interfaces are continuous and has
are representative of the entire specimen.
The second layer criterion is continuity of flow between layers. Flow originating from a
layer further from the drainage boundary has to pass through subsequent layers towards
the drainage boundary, seen in Figure 4.1.2-1.
4.5.2 Specimen Compatibility
Specimen compatibility criteria are criteria that the specimen has to satisfy as a whole.
These include total deformation criterion and neutral plane compatibility.
4.5.2.1 Total Deformation Compatibility
The first compatibility criterion is the total deformation of the test specimen. Under a
constant rate of strain loading, the total deformation is dependent only on the strain rate
and time step. In other words, the numerical sum of the deformation due to top and
bottom drainage should yield the total deformation of the test specimen:
total
-
AHt
+ AHbottom
-
Hinitial t
[4-2]
where:
AHinitial
total deformation of the test specimen
AHtop
deformation of the test speci men due to top drainage
AHbottom
deformation of the test speci men due to bottom drainage
6
strain rate
53
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Htotal
initial total height of test specimen
t
time
4.5.2.2 Netural Plane Compatibility
When excess pore water pressure is generated as a result of the constant rate of strain
induced loading, the maximum excess pore water pressure is expected to occur at the
neutral plane. Since the neutral plane is the defined no flow boundary between top and
bottom layers, the calculation of excess pore water pressure from both the top and bottom
drainage analysis have to be equal.
U
Oeurl=Au bottomneutral
[4-3]
where:
AUtOP
excess pore water pressure at neutral plane based on top drainage
Aubottomet,,
excess pore water pressure at neutral plane based on bottom drainage
By accounting for the compatibility of the excess pore water pressure at the neutral plane,
the loss in excess pore water pressure at the base can be correctly determined. The base
excess pore water pressure and bottom drainage is dependent on the excess pore water
pressure at the neutral plane. As the excess pore water pressure at the neutral plane
increase, the base excess pore water pressure also increases. This in turn increases the
bottom drainage. The increase in bottom drainage will results in an increase of the loss in
base excess pore water pressure.
54
Numerical Modeling of Constant Rate of Strain Consolidation Tests
In addition, the excess pore water pressure at the neutral plane is needed to determine the
vertical effective stress exerted on the test specimen due to the constant rate of strain
loading. The vertical effective stress at the neutral plane is then used to calculate the total
vertical stress, which is constant through the specimen.
4.6 Determination of p and A
The parameter pt is a measure of the position of the neutral plane in a double drainage
constant rate of strain test. The parameter X is a measure of the compressibility of the
bottom drainage boundary. Both i and k are interrelated. As the compressibility of the
bottom drainage increases, the neutral plane begins to migrate upwards from the bottom
boundary. This is most likely true for a given soil stiffness.
While there is no clear indication that the relationship is linear or log-linear, there is a
corresponding value of pt for every value of ?. Mathematically, this relation implies a
one-one and on-to function relating p and ?,. This in turns implies that the inverse of the
function exist. Hence, it is valid to use p to determine X and vice versa.
In this numerical model, the research has concluded that it is simpler to use p to obtain k
than the other way around. Hence, the model assumes a neutral plane and iterates to find
the required k of the testing device.
55
Numerical Modeling of Constant Rate of Strain Consolidation Tests
4.7 Effects of Time Step and Layer Size
The numerical model takes snapshot views of the consolidation process under constant
rate of strain at defined time steps. These time steps will provide a progressive look at the
migration of excess pore water pressure from zero at time zero to a resultant distribution
dependent on the drainage. The model has to, therefore, be able to produce the same
response regardless of the number of time steps taken to reach a particular time. In other
words, the distribution of excess pore water pressure from a single time step of 1 hour
should be identical to the distribution from two time steps of
2 hours
each, or from four
time steps of % hours each.
The model discretized the specimen into 20 layers. The parameters of each layer are then
solved by the finite difference method described earlier. In any finite difference method,
the number of elements are expected to affect the precision of the results but not the
accuracy. The increase in number of elements will reduce the layer size. This in turn
brings the solution closer to a partial differential solution. The effects of time step and
will be looked into in detail in Chapter 5.
56
Numerical Modeling of Constant Rate of Strain Consolidation Tests
4190
Top drainage conditions
0
Test specimen
discretized into
2tdayers
Bottom drainage boundary
Figure 4.1- 1: Discretization of test specimen into layers
m-
f(base measuring stiffness)
Bottom drainage conditions
Figure 4.1- 2: Definitions of boundary conditions
57
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Start of the first
time step
Input strain rates, time step,
test specimen height, and
height of solids
Calculate total deformation
of time step
ratio vs effective
stress, and void ratio vs
hydraulic conductivity
relationships
Input void
Neutral plane calculations
(See Figure 4.1.1-2)
End of the first
time step
Figure 4.1.1- 1 : Flowchart of sequence of the first time step
58
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Begin neutral
plane calculations
Input p,
required X
Calculate neutral
plane
No
Layer calculations
(See Figure 4.1.1-3)
Calculate X
equired value?
Yes
End neutral plane
calculations
Figure 4.1.1- 2: Flowchart of neutral plane calculations
59
Numerical Modeling of Constant Rate of Strain Consolidation Tests
SBegin layer
calculations
Top only?
Assume a void
ratio
etermine flow
Top and Bottom?
Bottom only?
Assume a void
ratio
Yes
stere inexo
Step 1: calculate layer deformation
Step 2: calculate hydraulic conductivity
Step 3: calculate excess pore pressure
layer ?
Replace assumed
void ratio withy
calculated void
ratio
No
a
Identify maximum
excess pore
pressure generated
Step 1: calculate effective stress at max. u
Step 2: calculate total stress due to CRS loading
Step 3: calculate effective stress at each layer
Step 4: calculate void ratio at effective stress
Step 5: calculate sum of differences between
calculated and assumed void ratio
No
'
su h
fdifferences at a
minimal?7
Yes
End layer
calculations
Figure 4.1.1- 3: Flowchart of layer calculations
60
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Start of subsequent time
steps
Begin n time step
calculations
Carry forward void ratio data from
n-1 time step
No
Layer calculations
(See Figure 4.1.1-3)
Is this the last
time step?
Yes
End n time step
calculations
Figure 4.1.1- 4: Flowchart of time step calculations
61
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Top drainage
A
A
Neutral
"
-
.
-
.
.
U
.
Plane
v
V
Bottom drainage
Figure 4.1.2- 1: Flow path of discretized layers
62
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Top boundary condition
ftI
At
I I, tkfl
u(= 0
I
Ul
= f(vI)
u, = f(vyu,)
I
I
u
= f(v ,u )
Un., = f(Vn-1,,Un.)
Un = f(vn,u,. 1, X)
Bottom boundary condition
I - measure of base measuring stiffness
Figure 4.1.2- 2: Finite difference scheme
63
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Constant rate of
strain loading
ut
Upper boundary
HO
H
Pore water flow
Lower boundary
U'
Figure 4.3- 1: Schematic diagram of a basic layer with upward flow
Top and bottom
drainage
Distribution of excess
pore water pressure
Top drainage
only
Figure 4.4- 1: Schematic diagram of drainage extremes
64
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Top boundary taken as reference
0
_
I*Hninita1/ 2 0
2*Hninital/ 20
3*Hninit /20
4*Hninita/ 2 0
Hinitial
Hinitiat
Figure 4.5- 1: Top boundary as reference
Top boundary taken as reference
jneutral plane
=*HinIta
Figure 4.5- 2: Location of neutral plane
65
Numerical Modeling of Constant Rate of Strain Consolidation Tests
N
U
m
U
E
-
U
u
m
*
-
U
-
Figure 4.5- 3: Separating neutral plane layer based on flow direction
Initial Htop
+z
drainage
U
Initial Hbottom
-z
drainage
F
Figure 4.5- 4: Neutral plane as reference
66
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 5
Interpretation of Results from Double
Drainage Numerical Model
5.1 General Overview
The numerical model is performed by applying soil parameters from the CRS 238 (strain
rate=0.84%/hr) performed on Resedimented Boston Blue Clay (RBBC) by Gonzalez et.
al. (1997). Other input data on the test specimen are also listed. These specific parameters
from the slow test, which is postulated to be correct, is used in the normally consolidated
(NC) range to study the effects of rate. The base pore water pressure and the standard
CRS equations are used to evaluate the errors in test results.
5.1.1 Soil Input Parameters
The stress-strain relationship is defined by the relationship between the void ratio or
strain and the log of effective vertical stress. Based on test results obtained from CRS
238, the assumed e versus log a'v equation is given below in the NC range:
e = 1.1776 - 0.3523 -log(a',)
[5-1]
67
Numerical Modeling of Constant Rate of Strain Consolidation Tests
where:
e
void ratio of the test specimen
a'
average effective vertical stress of the test specimen
The assumed relationship between void ratio and hydraulic conductivity based on
CRS238 is shown in the equation below also in the NC range.
k = 10(2.2331e-9.4058)
[5-2]
where:
k
hydraulic conductivity of the test specimen
e
void ratio of the test specimen
5.1.2 Other Input Parameters
In order to fully model the Resedimented Boston Blue Clay in the numerical simulation,
the following parameters are assumed for the purpose of this research.
"
Hinitiai
*
HS0 1i =1.175 cm
*
eo=l
=
2.35 cm
y, = 0.001 kg/cm 3
68
Numerical Modeling of Constant Rate of Strain Consolidation Tests
5.1.3 Convergence Criteria
The numerical model takes an input void ratio for each layer and computes the Darcy's
constant, hydraulic conductivity, boundary excess pore water pressures and the average
effective vertical stress. From the average effective vertical stress, the corrected void ratio
at each layer is calculated using equation [5-1]. The difference in the input and corrected
void ratio of every layer is added together. The iteration process continues until this sum
of difference in void ratio reaches a minimum, usually in the order of 10-6
-
10-9 in error.
From the numerical simulations, it is noted that the error increases with strain rate. It is
also noted the error decreases with the increase in the number of discretized layers.
5.2 Effects of Time Step
The finite difference method assumes linear approximation calculations within a time
step. As a result, the magnitude of the time step in which the iteration is carried out will
affect the solutions of the iteration. The effect of time step is investigated by modeling
the excess pore water pressure distribution at the end of a 1%/hr strain rate for 2hours, for
=0.1hr/ksc. The time steps taken include a single step of 2 hours, two steps of 1 hour,
four steps of
/2hour
and eight steps of 1/4 hour. The trend of excess pore water pressure
with time step is shown in Figure 5.2-1. The variation in the excess pore water pressure
distribution decreases with time step and is expected to approach a single value. For the
course of this thesis however, the time step used will be to satisfy the strain of 1% per
time step, i.e. time step of 1 hour for strain rate of 1%/hr, 2hour for strain rate of 0.5%/hr
and so on.
69
Numerical Modeling of Constant Rate of Strain Consolidation Tests
5.3 Validation of Excess Pore Water Pressure Distribution
In considering both top and bottom drainage in the numerical model, it is necessary to
first determine the validity of the numerical model. As discussed in preceding chapters, it
is important that the numerical model is able to simulate both top drainage only as well as
full top and bottom drainage. The two cases are selected as they represent the limiting
conditions of the constant rate of strain consolidation test apparatus. The limiting
conditions dictate the maximum and minimum flow resistance of the bottom boundary.
The minimum in compressibility implies no flow possible while the minimum implies
unimpeded flow.
The numerical simulation is performed at a strain rate of 1%/hr and a time step of 1 hour.
The results of the numerical model are shown in Figure 5.2-1. The curves are plotted with
varying values of pt, which is a measure of the location of the neutral plane. When the
neutral plane is set at the base of the test specimen - t is zero, the excess pore water
pressure generated increases with distance from the top drainage. This is in agreement
with Gonzalez's solutions. Similarly, when the neutral plane is set at the middle of the
test specimen, the model is able to produce a symmetrical isochrone of the excess pore
water pressure generated, as expected from a full double drainage problem, when p is
0.5.
The rest of the curves depict the migration of the distribution of excess pore water
pressure from a single drainage to a double drainage base condition. As the
70
Numerical Modeling of Constant Rate of Strain Consolidation Tests
compressibility of the bottom drainage decreases, the excess pore water pressure buildup
increases. This is expected as the increase in base stiffness results in less drainage and
causes the pore water to take an increasing amount of the total stress exerted onto the test
specimen. As seen in figure 5.2-1, the isochrone for [=0.5 and p=0.4 intersect each other.
As the differences are not very large, it is likely that the intersections are due to the
discretization of the test specimen.
5.2.1 Validation of Stress-Strain Relationship
The model considers the test specimen to be normally consolidated. In other words,
equation [5-1] represents the virgin compression line. All consolidation of the test
specimen using the numerical model is based on the navigation of data along the virgin
compression line. The convergence criteria requires that each layer remains on the line.
The results of the numerical model relationship at 0.1, 1 and 10%/hr strain rate, along
with the assumed input relationship is shown in figures 5.2.1-1, for k=0.01cm/ksc. The
plots of the data from the numerical model are in agreement with the assumed stressstrain relationship. These data include all the void ratio and vertical effective stress
experienced by every layer at each time step, with a total of 20 data points for each strain
rate. The void ratio data is the average of a layer and the vertical effective stress data is
the average of the top and bottom of a layer. To be absolutely correct, the logarithmic
average should be taken. This can be addressed in future work.
71
Numerical Modeling of Constant Rate of Strain Consolidation Tests
5.2.2 Validation of Void Ratio-Hydraulic Conductivity Relationship
The hydraulic conductivity of the test specimen decreases as the void ratio decreases. As
with the stress-strain data, the hydraulic conductivity and the void ratio of every
discretized layer is plotted in figure 5.2.2-1 along with the assumed input relationship, for
),=0.01cm/ksc. The results are based on a strain rate of 0.1,1 and 10%/hr. The results
obtained from the numerical model are in good agreement with the assumed relationship.
5.3 Effects of Strain Rates
The numerical model is used to simulate different strain rates. The strain rates modeled
include 0.1%/hr, 0.5%/hr, 0.8%/hr, 1.0%/hr, 3%/hr, 5%/hr, 10%/hr and 20%/hr. All these
results are based on X=0.01 cm/ksc used in equation [3-1]. When different strain rates are
used, the time steps are adjusted so that the amount of strain remains the same. Therefore,
p is varied to keep X constant.
Aub = AAbottom
[3-1]
From the results of the numerical simulation, this thesis addresses the effects of strain
rate on the following:
*
Excess pore water pressure profile
*
Base excess pore water pressure
*
Ratio of maximum excess pore water pressure to base excess pore water pressure
op7
72
Numerical Modeling of Constant Rate of Strain Consolidation Tests
0
Ratio of base excess pore water pressure to total vertical stress
5.3.1 Effects on Excess Pore Water Pressure Profile
The isochrone distributions of excess pore water pressure at some of the strain rates are
plotted in Figures 5.3.1-1, 5.3.1-2 and 5.3.1-3 of strain rates 0.1%/hr, 1%/hr and 10%/hr
respectively. The X in all three cases is O.Olcm/ksc and the total strain imposed is 10%.
The isochrone profile of the excess pore water pressure deviates significantly from a top
drainage only profile as the strain rate increases, seen in Figure 4.4-1. This would imply
the assumption of impermeable bottom boundary becomes less reliable as the strain rate
increases.
5.3.2 Effects on Maximum and Base Excess Pore Water Pressures
The amount of excess pore water pressure generated is expected to increase significantly
as the strain rate increases. As the strain rate increases, the test specimen has less time to
dissipate pore water for the same amount of strain. This results in more excess pore water
pressure buildup in the specimen. At 10% strain, the maximum and base excess pore
water pressure developed for various strain rates are shown in Figure 5.3.2-1, for
X=0.Olcm/ksc.
The trend of base excess pore water pressure with strain rate is shown in Figure 5.3.2-2,
for X=0.01cm/ksc. The results of base excess pore water pressure is determined at 1%
73
Numerical Modeling of Constant Rate of Strain Consolidation Tests
strain increment up to a total strain of 10%. The results appear to be in agreement with
expected trends. For a given strain rate, the base excess pore water pressure increases
with amount of strain. In addition, the amount of base excess pore water pressure
increases with strain rate.
The trend of ratio of maximum to base excess pore water pressure with strain rate is
shown in Figure 5.3.2-3, for X=0.Olcm/ksc. The ratio of maximum to base excess pore
water pressure is a measure of the deviation of excess pore water pressure isochrone from
top drainage only isochrone. As the amount of deviation increases, the ratio of maximum
to base excess pore water pressure increases. The ratio also increases with amount of
strain. For the initial data point for 10% and 20% strain rate, there appears to be a change
in gradient of the plots at 1-2% strain. This is likely reflecting the shift from transient
state condition to steady state condition. When the Tv is calculated for both 10 and 20%
strain rates, T, is found to be slightly above 0.5, 0.507 and 0.539 respectively, which is
defined as the boundary value between transient and steady state conditions (Wissa et. al
1971).
5.3.3 Effects on Neutral Plane
The trend of strain rates with neutral plane is shown in Figure 5.3.3-1, for X=0.01cm/ksc.
The base compressibility appears to increase asymptotically with strain rate. The amount
of base excess pore water pressure generated increases with strain rate and in turn
increases the amount of downward drainage. This will result in the neutral plane
74
Numerical Modeling of Constant Rate of Strain Consolidation Tests
migrating upwards. At large strain rates, the neutral plane moves asymptotically towards
pt=0.5. At very small strain rates, little pore water pressure is generated and the neutral
plane approaches p=0.0. However, even when the strain rates are slow, the base excess
pore water pressure measured can still be below the correct value. This can result in
errors in calculations of hydraulic conductivity.
5.3.4 Effects on Pore Water Pressure Ratio
The trend of pore water pressure ratio Aub/av with strain rate is plotted in Figure 5.3.4-1,
for X=O.Olcm/ksc. The pore water pressure ratio at a given strain rate is expected to
decrease asymptotically to a fixed value with increase in strain. As the total strain of the
specimen increases, the pore water pressure ratio stabilizes as the soil is behaving in
steady state conditions.
The pore water pressure ratio increases with increasing strain rate. This is also anticipated
as the amount of excess pore water pressure developed is proportional to the strain rate.
The increase in excess pore water pressure results in a corresponding increase in the pore
water pressure ratio.
5.4 Effects of Base Compressibility
The base compressibility is characterized by the parameter X. By varying X, the effects of
base compressibility on excess pore water pressure distribution are shown in Figure 5.475
Numerical Modeling of Constant Rate of Strain Consolidation Tests
1. When the base compressibility is high, X=l, the excess pore water pressure distribution
is almost like double drainage profile. When base compressibility is low, X=O, the
distribution is similar to single drainage profile. The trend of ratio of maximum to base
excess pore water pressure is also shown in Figure 5.4-2. This further illustrates the effect
of base compressibility on excess pore water pressure distribution.
The effect of base compressibility on the neutral plane location is shown in Figure 5.4-3.
As the base compressibility increases, the neutral plane location moves from the base
(p=O) to the middle of the test specimen (t=0.5). This can be seen from the asymptotic
approaches of the graph towards t=O and 0.5.
5.5 Comparison With Constant Rate of Strain Consolidation Test Data
The results from actual constant rate of strain consolidation tests by Gonzalez et. al.
(1997) are shown in Figure 5.5-1. These results are compared with the simulations from
the numerical model, shown in Figure 5.3.4-1. The excess pore water pressure generated
from the numerical model are less than the actual CRS results. This discrepancy could be
due to the assumed bottom drainage resistance factor k of 0.Olcm/ksc.
In addition, in actual CRSC tests, the processes to arrive at the base pore water pressure
and hydraulic conductivity are different. If there is a base drainage effect in the test, then
using Wissa's equations with measured base pore water pressure to obtain hydraulic
76
Numerical Modeling of Constant Rate of Strain Consolidation Tests
conductivity, and using equation [5-2] with the model to obtain the base pore water
pressure should results in different solutions.
5.6 Application of Wissa Constant Rate of Strain Equations
The Wissa's linear and non-linear equations [2-8] and [2-12] are used to determine the
average vertical effective stress from the computed base excess pore water pressure from
the numerical model.
= a-2
a-'V
-'=
(o
v3
[2-8]
Au3
- 2U2 . Aub + a, . Au2 )1/3
[2-12]
Based on Wissa's linear and non-linear consolidation theory, the hydraulic conductivity
can be obtained from equations [2-9] and [2-14] shown below.
k=
.r
2 - Aub
k =-
0.434 r-H
2--,, -log1
[2-9]
y
-
At
[2-14]
Aub
77
Numerical Modeling of Constant Rate of Strain Consolidation Tests
5.6.1 Variation of Wissa's Relationships with RBBC Stress-Strain
Relationship
The deviation of Wissa's linear and non-linear equations, relating effective vertical stress
and void ratio from the model stress-strain relationship (Equation [5-1]) is not significant
as seen in Figure 5.6.1-1, for k=0.01cm/ksc. The scale of the plot is enlarged, the
logarithmic function removed from effective vertical stress and results from 20%/hr
strain rate was added. Figure 5.6.1-2 illustrates that the deviation from the assumed
relationship based on Wissa's theories increases as the strain rate increases. There is a
definite shift of the compression curve to the right. The error is small because the prore
pressure ratio is only 5%. At about 7 ksc, the base excess pore water pressure is about
0.55 ksc and the error in acv is 0.5 ksc (Figure 5.6.1-2). This error is huge relative to the
amount of base excess pore water pressure.
5.6.2 Variation of Wissa's Relationships with RBBC Strain-Hydraulic
Conductivity Relationship
The results obtained from Wissa's linear and non-linear consolidation theory at various
strain rates is shown in Figure 5.6.2-1. Wissa's relationships appear to deviate
significantly from the modeled relationship (Equation [5-2]). The error is due to Wissa's
simplifying theory of the relationship between the base excess pore water pressure and
the distribution of the excess pore water pressure itself. Wissa assumed that the
distribution is approximately triangular in shape. Under one-way drainage, top drainage,
the triangular distribution resembles the actual excess pore water pressure distribution. In
78
Numerical Modeling of Constant Rate of Strain Consolidation Tests
the case of double drainage however, the triangular approximation deviates significantly
from the actual distribution. Due to the fact that the base excess pore water pressure is
always lower than that predicted by Wissa, the hydraulic conductivity is always over
predicted. The error is significant for extremely slow tests with very low excess pore
water pressures. Hence, Wissa's theories result in inaccurate representation of the soil
behavior.
79
Numerical Modeling of Constant Rate of Strain Consolidation Tests
AUb
0
0.02
0.04
(ksc)
0.06
0.08
0.1
0.12
0
0.1
0 .2
-
-
--
0.3
-
-
0.4
---- -
- - -
--
-
--------------
0.5
- -
0.6 - - - - -
--
0 .7--
0.8
- - - - - - - - - - - ----- - - - --- --
----------
---
-
--
- - -
- -- -
-
- --
-
----
-----
- - -- - - - -
- -- - --
--------
----
--------
-
- - - -----
time step
-
- -
2hrtimestep
0.5hr t ime st ep
---
0.9
- -
-
----
-
-Baseline
7r
-.
-
-
-
0.25hr time step
-
-
---
Figure 5.2- 1: CRS Test results performed on Model RBBC using numerical simulation based on
strain rate of I %/hr,A=0. 01cm/ksc: Distribution of excess pore water pressure with varying time
step
Au (ksc)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0
+-p=0.1
0.5
-
~~-
-
+ p=0.3
--
p=0.4
p=0.5
-+- Gonzalez (1997)
1..
--
-------
---
---
-
-
N
15
--
-
-
--
-
----
-
- -
Figure 5.3- 1: CRS Test results performed on Model RBBC using numerical simulation based on
strain rate of 1%/hr, time step of 1hr: Distribution of excess pore water pressure with specimen
80
Numerical Modeling of Constant Rate of Strain Consolidation Tests
1*
Numerical Simulation at
I
strain rate of 10%/hr
Simulation at
I
+ Numerical
0.9-
-
--
-
*
A
Numerical Simulation at
---
Assumed Relationship,
0.8
-
0.1%/hr
Eqn. [5-11
I
I
strain rate of
II
-
--
-
---
--
-
-
--
-----
L
0.75
1
10
100
log a', (ksc)
Figure 5.3.1- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0.1, 1 and 10%/hr and strain of 10%: Compression curve
1
'7
Numerical Simulation at
strain rate of 10%/hr
*
0.95
Numerical Simulation at
strain rate of 1%/hr
Numerical Simulation at
strain rate of 0.1%/hr
-
0.9
Assumed Relationship,
Eqn. [5-2]
0.85
0.8
0.75
4-1.00E-07
1.00E-08
log k
Figure 5.3.2- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0.1, 1 and 10%/hr and total strain of 10%: Hydraulic conductivity curve
81
Numerical Modeling of Constant Rate of Strain Consolidation Tests
0
u (ksc)
0.01
0.005
0.015
0.025
0.02
0
-+t=10hr
0 .1 -
t=2Ohr
t=3 h
t=4Ohr
-K- t=5Ohr
+- t=Ohr
o
- - - - - - - - - --- - - - - - - - - - - - -- -- - - - - -- -- -- -- ---
-
0.2 - - - - - - - - -
-
-------
-
--
----
--
0.3
-
-- -
-
- - ---
- - - --
-
------------
---------
-
-
-
-
-
-
0.6 -- ---------------
--------
0.8-
-
-
-- -- -- -- -- - ---------------
----- -----
0.9------
- ------
-
-
---
-
-
t9 h
- --------------- - -
--- -----------------------
E 0.5 --
-
t=7Ohr
-----
- ------
----
-
--
---
Figure 5.4.1- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0. 1%/hr, total strain of 10% and A=0.Olcm/ksc: Distribution and development of
excess pore water pressure within specimen
0
0.02
0.06
0.04
u (ksc)
01
0.08
0.14
0.12
0.18
0.16
0.2
0
0.1
0.2
- -
-
--- - - -
-
-.--
- - - - - - --- -- ------ -- -- - ------- - - - - - - - - - -
--
-
-
-
-
-
-
-
-
-
-t=1hr
-
-4-t=2hr
-
-----
-
-
--
t=3hr
-N.-
0.3----------------------
-
-
-
-
-
-
-
-
---- -
-
-
-
-
-t=4hr
-
IN-t=5hr
_
_
-4t=7hr
0.4 -- - - - - --------
-.--
-0.5-L.-
0.7-------
0.8-----------
.-
----
---
-------
-
-
----
-
-
-
--
---
--
--
t=9hr
-4tOhr
--
---
-
.-
-
- - -
- -
-
0.8------ -----------
- - - - -
- -t=8hr
r-------.-----
-
--
-
-
--------
-
-
Figure 5.4.1- 2: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 1%/hr, total strain of 10% and A=0.Olcm/ksc: Distribution and development of
excess pore water pressure within specimen
82
Numerical Modeling of Constant Rate of Strain Consolidation Tests
u (ksc)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
-4-t=6min
0.1
--
- - - -
- -- -
-
--
- -L
- - - - - -- -- - --L- - - - - - - --- - - - ----
- --
-4--t=2min
t=24min
0.2 ---
-
-
----
----
-
-
---
- - ----
0.3
-- --- - - - - - - - - - - -
- -
-
-
- -
-
-
t=42min
-t=48min
- - - - - - - - - - -
t=54min
------ - --
0.4 --------E
- ----
0.5
0.6
---
--------
-
- --
-------
-- -
----
-- -
-
----
-
0I
-
-
-
-
0.9
- - --- --- ---- - - - ---
- -
-
-
-
-
-
-
-
-
---
- - -
-
7
-
---
------
-
------
--
-
-
-
-
-
---
--
--
----
Figure 5.4.1- 3: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 10%/hr, total strain of 10% and A=0.01cm/ksc: Distribution and development of
excess pore water pressure within specimen
Strain Rate 6
Maximum Excess Pore Water
Base Excess Pore Water
(%/hr)
Pressure umx (ksc)
Pressure Ub (ksc)
0.1
0.0220
0.0217
0.5
0.120
0.118
0.8
0.161
0.155
1
0.176
0.162
3
0.364
0.321
5
0.560
0.454
10
0.837
0.530
20
1.333
0.592
Figure 5.4.2- 1: Variation of strain rate on magnitude of maximum and base excess pore water
pressure at 10% strain and A =0.01cm/ksc
83
Numerical Modeling of Constant Rate of Strain Consolidation Tests
0.7 -Strain
i
0.6-
Strain
-
0.5
-
rate=20%
Strain rate=10%
rate=5%
-N--Strain rate=3%
-4--Strain rate=.%
-0.4
--
-- -
-- - -------
-
- - - - - - - .. . ---.
Strain rate=0.8%
0.3
2Strain
rate=.5%
Strain rate=O.1I%
--
0.2
--
---
-
0.1
0
0
6
a, (ksc)
4
2
10
8
12
Figure 5.4.2- 2: CRS Test results performed on Model RBBC using numerical simulation based
on strain increments of 1% and A=0.01cm/ksc: Trend of base excess pore water pressure with
strain rates
2.4
2.2
--
Strain rate=20%
--
Strain rate=10 %
-
2-
Strain rate=5%
-
-WI-e-
1.8
Strain rate=1%
Strain rate=0.8%
--
UI/U,
----- ----------
Strain rate=3 %
1.6
-
---
Strain rate=0.5%
Strain rate=0.1%
1.4
1.2
..
=
LL
1
0
2
4W
4
p
p
-M
p
W
)K
--
,'M
6
I
8
10
ML
12
d, (ksc)
Figure 5.4.2- 3: CRS Test results performed on Model RBBC using numerical simulation based
on strain increments of 1% and A=0. 01cm/ksc: Trend of ratio of maximum excess pore water to
base excess pore water pressure with strain rates
84
Numerical Modeling of Constant Rate of Strain Consolidation Tests
25
20 1-
Numerical sim Lation dataF
+
-Best
fit trend line
------
15 -
% 10
-----------------.-
{
0
0
------
-
----
- ----------4- -------
0.I.5
0.1
0 0.0
0.05
04
0204
0.15
0.2
0.25
0.3
0.4
0.35
0.45
Figure 5.4.3- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain increment of 1% and A=O.Olcm/ksc: Trend of neutral plane location with strain rates
0.7
-Strain
0.6
+
rate=20%
Strain rate=10%
-4-- Strain rate=5%
- ---
0.5
---
-0.4
-- -
-
-
rate=3%
-*-Strain
Strain rate=1%
-4--Strain rate=0.8%
-
- -- - -- ---
--Strain rate=0.5%
-- *-Strain rate=O.1%
0.2
-
0.1
- -
0
0
2
4
6
8
10
12
a', (ksc)
Figure 5.4.4- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain increments of 1% and A=O.01cm/ksc: Trend of ratio of maximum excess pore water to
base excess pore water pressure with strain rates
85
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Au (ksc)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0
0.5
1
--
-
-
- -
--
---
-- - --- -------
---
-
------
-
-
--- -- - - -
- - - -- -
-
-
--
X=4).025!
-
-----------
NI
-
1.5 -------
-
---
-
- -----
-----
-
-
--
21
Figure 5.5- 1: CRS Test results performed on Model RBBC using numerical simulation based on
strain rate of 1%/hr and time step of Ihour: Trend of excess pore water pressure distribution with
base compressibility
2.5
225
2.5
-
-
-
I
------------
-
I
r----
-
I
rr------i--
II
I-
II
I
I
- e
0.001
I
--I - T--
I
I
--I
- - -t-- I
I
II
- - - -
- - - - - - - - - - - - - -
0.0---
- - ----
II
I
I
-
I
I
-I -
I
I
|
I
I
I
u---m---c
0.----------
-s-n
I
I
-
I
I
- --
I
I
I
0.5
I
I
I
I
I
I
- - - --
- -- - - - -
I
I
I
1.2r
0.0
- -
------
I
----------------- - - - - - -
I
I
I
I
1.5
- - - - -
-- ----1
I
i--t-
I
1
-a-
I
I
I
I
I
I
I
I
I
I
0.01
0.11
Figure 5.5- 2: CRS Test results performed on Model RBBC using numerical simulation based on
strain rate of 1%/hr and time step of Ihour: Trend of ratio of maximum to base excess pore water
pressure with base compressibility
86
Numerical Modeling of Constant Rate of Strain Consolidation Tests
0.6
I
II
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
II
I
J
II1
I
I
I
I
I
I
I
L
- - - - - I
1.1
I
-
-
liii
1 1
1
11111
I
4
0.5
0.4
0.3-
- - - -- - - --- -
-
-
-
-
- -- -
- -
0.01
0.1
0.01
0.1
--
- --
-
- --- i i 100
10
0.2
0.1
0
0.0001
0.001
100
10
1
Figure 5.5- 3: CRS Test results performed on Model RBBC using numerical simulation based on
strain rate of 1%/hr and time step of 1 hour: Trend of neutral plane location with base
compressibility
0.90
-
> 0.80
.I0 .70
-
- -- - 12.71Whrl
0.60
24
- -
74.10%/hr
- - - - -48.10 %/hr
-- -- -
0.40
-- --
--
0.50.
-
-
--
- -
- - -
-
-
... ...
- - --
- - --
---
-
-
- - - --
---
--
----
- -- - --
0.84 %/hr
1.35 %6/hr
0.30
----------
0.20
-- - -
--
--
-\-%
3.04% /hr--
-
-- 4.02 & 3.76 %/hr 7.79 & 7.83 %/br
- --
0.10
0.00
.. .. r--- -O
0.0
2.0
4.0
6.0
8.0
10.0
- - --- - -
12.0
Effective Vertical Stress,
14.0-
16.0
18.0
20.0.
a', (ksc)
Figure 5.6- 1: CRS Test results performed on RBBC using Wissa apparatus: Pore water pressure
ratio
87
Numerical Modeling of Constant Rate of Strain Consolidation Tests
1.05
1
-
-
--
0.9-
-
---
-
*
Strain rate 5%/hr (non-linear)
I
Strain rate 5%/hr (linear)
-
Strain rate 1%/hr (non-linear)
-- - - - -- - - -- - -
---
--
0.95-
-
--
-
-
0.85
-
-- - - - -
-
-
- -- -
-
x
Strain rate 1%/hr (linear)
X
Strain rate 0.1%/hr (non-linear)
I
Strain rate 0.1%/hr (linear)
Relationship
SI-+-Assumed
0.8.
-.
0.75-
-----
.
.
-
-
- -
----
1-
1-
-
-.--
-
-
- - - -
--
-
-
--
0.7
1
10
100
log ce, (ksc)
Figure 5.7.1- 1: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0.1, 1 and 5%/hr and A=0.O1cm/ksc: Compression curve with Wissa's linear and
non-linear relationships
I
o
I
Strain rate 20/dwhr (non-linear)
Strain rate 20 Ohr (linear)
-
I
*
Strain rate 50hr (non-linear)
U
Strain rate 5%hr (linear)
Strain rate Nhr (non-linear)
0.95
x
Strain rate 1%hr (linear)
x
Strain rate 0.1%hr (non-linear)
e
Strain rate0.1%hr (linear)
-
0.9
-
-
I
0.85
--
I
-------
-
Assumed Relationship
-
-
-
IL
I
4
6
8
a', (ksc)
Figure 5.7.1- 2: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0.1, 1, 5 and 20%/hr and A=0.Olcm/ksc: Enlarged compression curve with
Wissa's linear and non-linear relationships
88
Numerical Modeling of Constant Rate of Strain Consolidation Tests
1.05
1
0.95
-
-
0 .9 --
-
0.75
Strain rate5V
I ix
0I--I -la -
- --
m
0.85
0.8
-
-
-
-
aStrain
--I
-
(non-inear)
rate 10Yr (non-linear)
x
-----
hr
--
E Strain rat e 5%hr (Iinear)
4
Strain rate 1%hr (linear)
x
Strain rate O.1%hr (non-linear)
I
Strain rateO.1/dhr (linear)
x
--
-
AssuTed RelationsNp
I
0.7
1.OOE-08
1.OOE-07
1.OOE-06
log k
Figure 5.7.2- 3:: CRS Test results performed on Model RBBC using numerical simulation based
on strain rate of 0.1, 1 and 5%/hr and A=0.01cm/ksc: Hydraulic conductivity curve with Wissa's
linear and non-linear relationships
89
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Chapter 6
Conclusion and Future Research
6.1 Bottom Drainage
This research looks at the significance of bottom drainage due to compressibility of pore
pressure measuring system in constant rate of strain consolidation tests. Traditionally, the
bottom boundary is assumed to behave as a non-drainage boundary and the excess pore
water pressure measured by the transducer connected to the base porous stone is assumed
to be at a maximum. This is seen in detail from research done by Wissa et. al. (1971) and
Gonzalez et. al. (1997).
In reality, the bottom drainage does not behave as a non-drainage boundary, rather it
behaves as a partial drainage boundary. The key hypothesis of this research is the fluid
needs to drain into the base to cause a pressure change result is an under measurement of
the true pore water pressure. This creates significant error in test interpretation.
90
Numerical Modeling of Constant Rate of Strain Consolidation Tests
6.2 Numerical Simulation of Bottom Drainage
This thesis is based on previous research by Gonzalez et. al. (1997), in which a top
drainage finite difference numerical simulation was carried out. The numerical model
developed in this research extends the Gonzalez simulation to have partial drainage to the
base which is modeled using a constant system compressibility. The partial drainage is
implemented by assuming the flow of pore water through the test specimen to be
distinctly divided by a horizontal plane, termed the neutral plane, to top and bottom
drainage. The neutral plane location, represented by the term jt, is also postulated to be
related to the amount of bottom boundary flow resistance, represented by the term k. In
order to validate the results, the numerical model is tested on the two well-understood
behaviors - single and double drainage response. As discussed in detail in Chapter 5, the
numerical model appears to have good agreement with the anticipated single and double
drainage response.
6.2.1 Validation with Assumed Stress-Strain-Flow Relation
The numerical model is run based on the actual CRS results obtained from CRS 238 on
Resedimented Boston Blue Clay by Gonzalez et. al. (1997) in MIT. The model test
specimen is discretized into 20 layers and the results of the stress-strain-flow at each
layer is in agreement with the assumed relationship from the actual test results.
The numerical model has also produced results based on Wissa et. al. (1971) linear and
non-linear theory on constant rate of strain consolidation tests. Deviation in stress-strain91
Numerical Modeling of Constant Rate of Strain Consolidation Tests
flow results generated by the model and by Wissa's equations are observed and are due to
the deviation of the excess pore water pressure distribution from that assumed by Wissa.
The deviation of the stress-strain relationship between the numerical model is small for
the conditions simulated. The deviation of the strain-flow relationship is more notable
and may require more research to study on the issues relating to the deviation.
6.2.2 Introduction of the Parameter A
The parameter X is defined as a measure of the amount of volume compressibility of the
bottom boundary of a constant rate of strain consolidation apparatus. There is no prior
research on the effects on bottom drainage in CRS testing. Hence, it is common practice
to assume that the bottom boundary is a no-flow boundary.
In the course of the numerical modeling, it is apparent that there is a need to identify
typical values of X. The model is able to postulate that variations in X can result in
drastically different results.
6.3 Recommendation for Further Research
This research deals entirely on the postulation and numerical modeling aspect of the
bottom drainage issue. The relationship of bottom drainage with base excess pore water
pressure is assumed and analyzed. The next step is to look into actual constant rate of
92
Numerical Modeling of Constant Rate of Strain Consolidation Tests
strain testing of Resedimented Boston Blue Clay and other soils under different strain
rates. This is achieved by means of actual CRSC testing under various strain rates, using
experimental results to postulate the model.
X is an apparatus dependent parameter. Hence, there is a need to measure the realistic
values of X which will likely vary between testing apparatus, like the Trautwein device.
By obtaining actual values of X, the numerical model may be able to better predict actual
CRS test results. X will also be able to determine whether certain soil types will have
drastically different results when tested in different testing apparatus. The next step is to
look at the implications of X on soil testing, by acting as a stiffness comparison parameter
for the apparatus and soil stiffness.
Another important area of further research is the expansion of the model to account of
overconsolidated soils as well. This will allow an understanding of the full spectrum of
soil specimens. There is opportunity to investigate changes in shape of stress-strainhydraulic conductivity for overconsolidated clays. This thesis has emphasized on steady
state conditions. Hence, the numerical model can also be expanded to investigate
transient state, for Tv<0.5, F3<0.4.
The final stage of research is the incorporation of the numerical simulation with
experimental validation and expansion of model into the interpretation of constant rate of
strain test results. This research approaches a new aspect of the constant rate of strain
93
Numerical Modeling of Constant Rate of Strain Consolidation Tests
consolidation test. There is great potential for improvement in this area and the actual
testing of soil specimen will likely open a wide door for further research.
94
Numerical Modeling of Constant Rate of Strain Consolidation Tests
References
Casagrande, A. "The Determination of the Preconsolidation Load and Its Practical
Significance." Proceedingsof the FirstInternationalConference on Soil
Mechanics and FoundationEngineering,Cambridge, MA, 60-64
Davis, E. H. (1965). "A Non-Linear Theory of Consolidation." Geotechnique, 15(2), 161173
Gibson, R. E., England, G. L., and Hussey, M. J. L. (1967). "The theory of onedimensional consolidation of saturated clays." Geotechnique, 17, 261-273
Gonzalez, J. H. (1997). "Experimental and Theoretical Investigation of Constant Rate of
Strain Consolidation." M.S., MIT, Cambridge, MA
Henderson, E. (1994). "Evaluation of the Time Response of Pore Pressure
Measurements," M.S., MIT, Cambridge, MA
Ladd, C. C. (2003)."1.322 Soil Behavior (MIT): Class Notes." MIT, Cambridge, MA
Lambe, W. T. and Whitman, R. V. (1969). Soil Mechanics, John Wiley & Sons Inc, New
York
Olson, R. E. "State of the Art: Consolidation Testing." Consolidationof Soils: Testing
and Evaluation, Fort Lauderdale, FL, USA, 7-68
Sheahan, T. C., and Watters, P. J. (1996). "Experimental Verification of CRS
consolidation theory." Journalof Geotechnicaland Geoenvironmental
Engineering, 123(5), 430-437
95
Numerical Modeling of Constant Rate of Strain Consolidation Tests
Smith, R. E. and Wahls, H. E. (1969). "Consolidation Under Constant Rate of Strain,"
Journalof the Soil Mechanics and FoundationDivision, 95(SM2), 519-539
Wahls, H. E. and deGodoy, N. S. (1965). "Discussion." Journal of the Soil Mechanics
and FoundationDivision, Proceedingsof ASCE, 91(SM3), 147-152
Wissa, A. E. Z., Christian, J. T., Davis, E. H. and Heiberg, S. (1971). "Consolidation
Testing at Constant Rate of Strain." ASCE Journal of Soil Mechanics and
FoundationsDivision, 97(10), 1393-1413
96