A comparative study on the horizontal coefficient of consolidation cr obtained from lab and field tests

The key goals of the research are: 1 determine the most reliable methods among the proposed methods for determining the horizontal coefficient of consolidation cr in the literature; 2 de

VIETNAM NATIONAL UNIVESITY, HANOI

VIETNAM JAPAN UNIVERSITY

Hanoi, 2020

TRAN QUYNH GIAO

A COMPARATIVE STUDY ON

THE HORIZONTAL COEFFICIENT OF

FROM LAB TESTS

MASTER’S THESIS

VIETNAM NATIONAL UNIVESITY, HANOI

VIETNAM JAPAN UNIVERSITY

Hanoi, 2020

TRAN QUYNH GIAO

A COMPARATIVE STUDY ON

THE HORIZONTAL COEFFICIENT OF

i

ABSTRACT

When a soft ground is improved by PVDs, consolidation takes place under the condition of drainage in both horizontal and vertical directions Naturally,

horizontal coefficient of consolidation (cr) is larger than the vertical coefficient of

consolidation (cv) by a factor of 3 to 5 The cv value is commonly interpreted from consolidation test using incremental loading method [1] However, up to date, there have not been any similar standards for the consolidation test with horizontal drainage (using incremental loading method)

The key goals of the research are: (1) determine the most reliable methods among the proposed methods for determining the horizontal coefficient of consolidation

(cr) in the literature; (2) determine correlations between cr values obtained from central drain (CD) test and peripheral drain (PD) test; (3) determine correlations

between vertical coefficients of consolidation (cv) and radial cr for a number of test sites in Vietnam

A desk study is carried out to secure the following: (1) a literature review on

equipment used for the test and existing methods used to evaluate the c r value; (2) the thesis using data collected from the following sources literature review and test site in Vietnam

Overall, The most reliable methods for determining the horizontal coefficient of

consolidation (c r ) is non-graphical method and the root t can be used to determine the radial (horizontal) coefficient of consolidation (c r)

cr,PD is less than the cr,CD by a factor of 0.32 to 0.64 from intact samples and 0.33 to 0.58 from remolded samples

cr PD is larger than the cv by a factor of 0.90 to 2.33, cr CD is larger than the cv by a

factor of 2.14 to 5.12 from intact samples cr PD is less than the cv by a factor of 0.35

to 1.01, cr CD is less than the cv by a factor of 0.41 to 0.82 from intact samples

ii

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation for the lecturers of Master of Infrastructure Engineering Program for their help during my undergraduate at Vietnam Japan University (VJU)

My thesis supervisor Dr Nguyen Tine Dung for his enthusiasm, patience, advice and continuous source of ideas for me Dr Dung is always ready to answer my questions His support in professional matters is invaluable

I would like to acknowledge the sincere inspiration from Prof Nguyen Dinh Duc and Prof Hironori Kato Their lectures covered not only specialist knowledge but also the responsibility and mission of a new generation of Vietnam I am grateful to

Dr Phan Le Binh for his support in the last two years since I have studied at Vietnam Japan University Thanks to him, I have learned the professional courtesy

of Japanese people as well as Japanese culture

Finally, I want to spend thank to my parents and friends for their unflinching support in the tough time Their support, spoken or unspoken, has helped me complete my master thesis

iii

TABLE OF CONTENTS

Page

ABSTRACT i

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS iii

LIST OF TABLES vi

LIST OF FIGURES vi

LIST OF ABBREVIATIONS viii

CHAPTER 1 INTRODUCTION 1

1.1 Problem statement 1

1.2 Necessity of study 3

1.3 Objectives 4

1.4 Scope of study 4

1.5 Structure of thesis 4

CHAPTER 2 LITERATURE REVIEW 6

2.1 Introduction 6

2.1.1 Consolidation Theory with Horizontal Drainage 8

2.1.2 Solution of the governing equation (2.2) for a central drain (CD) under equal strain loading (ESL) condition 8

2.1.3 Solution of the governing equation (2.2) for a peripheral drain (PD) under free strain loading (FSL) condition 9

2.1.4 Solution of the governing equation (2.2) for a peripheral drain (PD) under equal strain loading (ESL) condition 9

2.2 Existing methods for determining cr from consolidation test with a peripheral drain using incremental loading 10

2.2.1 Root t method [6] 10

2.2.2 Inflection point method [9] 11

2.2.3 Full – match method [10] 13

2.3 Existing methods for determining cr from consolidation test with a central drain using incremental loading method 15

2.3.1 Root t method [11] 15

2.3.2 Matching log (de2/t) and Ur method [12] 16

2.3.3 Inflection point method [13] 17

2.3.4 Non-graphical method [14] 18

2.3.5 Log - log method [15] 19

2.3.6 Steepest tangent fitting method [16] 20

2.3.7 Log t method [17] 22

2.3.8 Full – match method [10] 24

2.4 Summary of methods for determining cr 25

2.5 Linear regression analysis 25

2.6 Log normal distribution method 26

CHAPTER 3 METHODOLOGY 27

iv

3.1 Introduction 27

3.2 Data collection 28

3.3 Improvement for inflection point methods 28

3.3.1 Theoretical development 28

3.3.2 The procedure for this method 29

3.4 Analysis of Time – Compression curve 29

3.5 Procedure to select the best methods 30

3.6 Procedure to determine ratios of cr PD /cr CD or cr /cv 31

CHAPTER 4 TEST RESULTS & DISCUSSIONS 33

4.1 Introduction 33

4.2 Summary of database 33

4.2.1 Data collected from the literature 33

4.2.2 Data collected from test sites in Vietnam 34

4.2.3 Summary of test data 37

4.3 Evaluation and selection the best methods on intact samples 38

4.3.1 Graph results on intact samples 38

4.3.2 Summary of results on intact samples 40

4.3.3 Summary of rank method on intact samples 47

4.4 Evaluation and selection the best methods on literature data 49

4.4.1 Graph results on literature data 49

4.4.2 Summary of results on literature data 51

4.4.3 Summary of rank method on literature data 52

4.5 Evaluation and selection the best methods on remolded samples 54

4.5.1 Graph results on remolded samples 54

4.5.2 Summary of results on remolded samples 56

4.5.3 Summary of rank method on remolded samples 62

4.6 Comparison of cr CD and cr PD on intact samples 64

4.6.1 Graph results on intact samples 64

4.6.2 Summary of results on intact samples 64

4.7 Comparison of cr CD and cr PD on remolded samples 66

4.7.1 Graph results on remolded samples 66

4.7.2 Summary of results from remolded samples 66

4.8 Comparison of cv and cr PD on intact samples 68

4.8.1 Graph results on intact samples 68

4.8.2 Summary of results on intact samples 68

4.9 Comparison of cv and cr CD on intact samples 70

4.9.1 Graph results on intact samples 70

4.9.2 Summary of results on intact samples 70

4.10 Comparison of cv and cr PD on remolded samples 72

4.10.1 Graph of results on remolded samples 72

4.10.2 Summary of results on remolded samples 72

4.11 Comparison of cv and cr CD on remolded samples 74

4.11.1 Graph results on remolded samples 74

4.11.2 Summary of results on remolded samples 74

v CHAPTER 5 CONCLUSIONS & RECOMMENDATIONS 76 REFERENCES 79

vi

LIST OF TABLES

Page

Table 2.1 Boundary condition 9

Table 2.2 Existing methods for determining cr from radial consolidation 25

Table 4.1 Summary of data from literature for the PD – ESL condition 33

Table 4.2 Summary of data from literature for the CD – ESL condition 34

Table 4.3 Summary of tests done on intact samples 37

Table 4.4 Summary of tests done on remolded samples 37

Table 4.5 Summary of results from PD tests on intact samples 40

Table 4.6 Summary of results from CD tests on intact samples 42

Table 4.7 Rank of each criterion with each pressure from PD tests on intact samples 44

Table 4.8 Rank of each criterion with each pressure for CD case on intact samples 45

Table 4.9 Summary of rank for each method from PD tests on intact samples 47

Table 4.10 Summary of rank on each meth1od from CD tests on intact samples 48

Table 4.11 Summary of results from PD tests on literature for 8 methods 51

Table 4.12 Summary of results from CD tests on literature for 8 methods 52

Table 4.13 Summary of rank on each method from PD tests on literature 52

Table 4.14 Summary of rank on each method from CD tests on literature 53

Table 4.15 Summary results from PD tests on remolded samples for 8 methods 56

Table 4.16 Summary of results from CD tests on remolded samples for 8 methods 58

Table 4.17 Rank of each criterion with each pressure from PD tests on remolded samples for 8 methods 59

Table 4.18 Rank of each criterion with each pressure from CD tests on remolded samples for 8 methods 61

Table 4.19 Summary of rank each method from PD tests on remolded samples 62

Table 4.20 Summary of rank each method from CD tests on remolded samples 63

Table 4.21 Summary of results from PD and CD tests on intact samples 65

Table 4.22 Summary of boundary for PD and CD case on intact samples 65

Table 4.23 Summary of correlations for CD and PD case on remolded samples 67

Table 4.24 Summary of boundary for CD and PD case on remolded samples 67

Table 4.25 Summary of correlations for PD case on intact samples 69

Table 4.26 Summary of boundary for PD case on intact samples 69

Table 4.27 Summary of correlation for CD case on intact samples 71

Table 4.28 Summary of boundary for CD method on intact samples 71

vii

Table 4.29 Summary of correlations for PD case on remolded samples 73

Table 4.30 Summary of boundary for PD case on remolded samples 73

Table 4.31 Summary of correlations for CD method on remolded samples 75

Table 4.32 Summary of boundary for CD method on remolded samples 75

viii

LIST OF FIGURES

Page

Figure 1.1 Map of distribution of major soil types in Indochinese 1

Figure 1.2 Soil phase diagram [3] 2

Figure 1.3 An Illustration of soft ground improved by PVDs 2

Figure 2.1 Research direction of the thesis [5] 7

Figure 2.2 Illustration of flow conditions for equal-strain case [6] 7

Figure 2.3 Time - deformation plot during consolidation for a given load increment [3] 8

Figure 2.4 Consolidation curve relating square - Root time factor to for drainage radially outwards to periphery with equal strain loading [6] 11

Figure 2.5 Log (Ur/Tr) - log Ur relationship [10] 13

Figure 2.6 Determine the value of intersection point in full – match method 14

Figure 2.7 Theoretical log(de2/t) versus Ur curves [12] 16

Figure 2.8 (a) Theretical Ur - log Tr curve and (b) d(Ur)/dlog Tr plot [13] 17

Figure 2.9 Log( - 0) versus log t plot [15] 20

Figure 2.10 Steepest tangent fitting method for determination of cr 21

Figure 3.1 Flow chart of the study

Figure 3.2 Experimental data [9] 28

Figure 3.3 Flowchart of identifying the best methods 30

Figure 3.4 Flowchart of identifying the best methods 31

Figure 4.1 Locations of test sites in Viet Nam (VSIP site, DVIZ site, Kim Chung site) 34

Figure 4.2 Test location at Kim Chung site 35

Figure 4.3 Test location at VSIP site 35

Figure 4.4 Test location at DVIZ site 35

Figure 4.5 Soil profile at DVIZ 36

Figure 4.6 Soil profile at VSIP 36

Figure 4.7 Soil profile at KC 36

Figure 4.8 Results from PD tests on intact samples (at 800 kPa) for 8 methods 38

Figure 4.9 Results from CD tests on intact samples (at 800 kPa) for 8 methods 39

Figure 4.10 Results from PD tests on intact samples (at 800 kPa) for 8 methods 49

Figure 4.11 Results from CD tests on literature for 8 methods 50

Figure 4.12 Results from PD tests on remolded samples (at 800 kPa) for 8 methods 54

Figure 4.13 Results from CD tests on remolded samples for 8 methods 55

Figure 4.14 Comparison of c r CD and c r PD obtained from root t method at all data 64

Figure 4.18 Comparison of cv and cr,PD obtained from root t method at all data 68 Figure 4.19 Comparison of cv and cr,PD obtained from non-graphical method at all data 68

Figure 4.20 Comparison of cv and c r CD, obtained from root t method at all data 70 Figure 4.21 Comparison of cv and cr CD obtained from non-graphical method at all data 70

Figure 4.22 Comparison of cv and cr,PD obtained from root t method at all data 72 Figure 4.23 Comparison of cv and cr,PD obtained from non-graphical method at all data 72

Figure 4.24 Comparison of cv and cr,CD obtained from root t method at all data 74 Figure 4.25 Comparison of cv and cr,CD obtained from Root t method at all data 74

x

LIST OF ABBREVIATIONS

drain (CD) condition

cr,PD

Horizontal coefficient of consolidation under for a peripheral drain (PD) condition

under for a central drain (CD) condition

cr,NG PD

Horizontal coefficient of consolidation form non-graphical

t inf Time at d(Ur) /dlog Tr the maximum

kr PD Permeability coefficient from PD case

kr CD Permeability coefficient from CD case

xi

kv Permeability coefficient from vertical consolidation

mr Soil stiffness from radial consolidation

mv Soil stiffness from vertical consolidation

Figure 1.1 Map of distribution of major soil types in Indochinese

2

In this area, civil constructions and seaports must take measures to treat the ground before construction

The objectives of ground treatment are:

- To increase bearing capacity of the ground

- To decrease the permeability of soil

Therefore, there are many methods used to reinforce or to increase the stiffness of the soft soil, in which consolidating the soft soil is one of the methods According to soil mechanics theory, soil is formed from two or three phases (see Figure 1.2) The voids surrounding the soil particles are filled by water, air or a combination of both Consolidation is the process of contraction of voids under the applied load in association with the process of water drainage

Figure 1.2 Soil phase diagram [3]

Among several common ground improvement methods in practice, ground improvement by Prefabricated Vertical Drain (PVD) is one of the methods most commonly applied in practice Fig 1.3 shows a typical configuration of ground improved by PVDs

Figure 1.3 An Illustration of soft ground improved by PVDs

3

Under the surcharge loading, drainage in the ground improved by PVDs takes place

in two directions (as show in Figure 1.3): vertical direction and horizontal (radial) direction The consolidation settlement of the ground therefore happens due to both vertical and horizontal drains

1.2 Necessity of study

When a soft ground is improved by PVDs, consolidation takes place under the condition of drainage in both horizontal and vertical directions Naturally,

horizontal coefficient of consolidation (cr) is larger than the vertical coefficient of

consolidation (cv) by a factor of 3 to 5 In addition, in many cases, when the soft clay layer is thick, the consolidation would happen mainly due to the horizontal drainage The cr value is therefore very important for the design, sometimes much more important than the cv value

Currently, the cv value is commonly interpreted from consolidation test using incremental loading method [1] This is because the method is simple and applicable in routine laboratories around the world However, up to date, there have not been any similar standards for the consolidation test with horizontal drainage (using incremental loading method) Although cr value might be determined from some Constant Rate of Strain (CRS) tests (e.g., Chung 2019, Sridharan 1996…), the

equipment and test procedures are too complicated to apply in routine tests Thus, cr

value is mostly obtained from empirical correlations, for example from cv value

In the literature, there are about 10 methods suggested to determine cr value obtained from result of the consolidation test with horizontal drainage using incremental loading However, it is unclear as which methods are the best In

addition, there have been no systematic studies on cr value of soft clay in the North

of Vietnam It is therefore very necessary to make a comparative study on the

methods to determine the cr value and the value for soft clay in the North of Vietnam

4

1.3 Objectives

The main objectives of the study are:

1 To determine the most reliable methods among the proposed methods for

determining the horizontal coefficient of consolidation (cr) in the literature;

2 To determine correlations between cr values obtained from central drain (CD) test and peripheral drain (PD) test;

3 To determine correlations between vertical coefficients of consolidation (cv) and

radial cr for a number of test sites in Vietnam

1.4 Scope of study

The scope of the study is limited to the following:

- Collect existing data in the literature and data from experiments of the supervisor‟s research program

- Perform analytical analyses to obtain the three objectives described above Test data on consolidation test with radial drainage (using incremental loading method) are collected from the following sources:

- Existing data from the literature (remolded samples);

- Test site in Kim Chung – Di Trach (Hanoi) (both remolded and intact samples)

- Test site in Dinh Vu Industrial Zone (DVIZ) (Hai Phong) test data (intact samples)

- Test site in Vietnam Singarpore Industrial Park (VSIP) (Hai Phong) (both remolded and intact samples)

1.5 Structure of thesis

The rest of the thesis is organized as follows

- Chapter 2: Find out the principles that have been determined for consolidation theory with Horizontal drainage, existing method for determining from radial consolidation test and theory of comparison method selects the best methods

5

- Chapter 3: Describes the methodologies used to evaluate the coefficients and correlations

- Chapter 4: Methodology provides methods for determining cr values for PD

& CD cases and provides evaluation methods to select the best methods

- Chapter 5: Outlines, discusses the results obtained and describes the conclusion

“A decrease of water content of a saturated soil without replacement of the water by air is called a process of consolidation”

Terzaghi (1943) first suggested the one-dimensional consolidation testing procedure This test performed in a consolidometer (sometimes referred to as an Odometer)

Baron [4] (1948) presented the basic theory of sand drains In key study of sand drains, the author has two fundamental cases

- Free-strain case: When the surcharge applied at the ground surface is of a flexible nature, there will be equal distribution of surface load This will result in an uneven settlement at the surface

- Equal-strain case: When the surcharge applied at the ground surface is rigid, the surface settlement will be the same all over However, this will result in

an unequal distribution of stress

The study in this thesis focuses on equal-strain case During consolidation process, pore water may drain through a Peripheral drain (PD) or a Central drain (CD)

7

Figure 2.1 Research direction of the thesis [5]

A peripheral drain (PD) case A central drain (CD) case

Figure 2.2 Illustration of flow conditions for equal-strain case [6]

To obtain a coefficient of consolidaiton, a curve of time vesus deformation (Figure 2.3) obtained from consolidation test is taken into analysis The curve has three distinct stages described as follows [3]:

- Stage I: Initial compression, which is caused mostly by preloading

- Stage II: Primary consolidation, during which excess pore water pressure gradually is transferred into effective stress because of the expulsion of pore water

- Stage III: Secondary consolidation, which occurs after complete dissipation

of the excess pore water pressure, when some deformation of the specimen takes place because of the plastic readjustment of soil fabric

8

Figure 2.3 Time - deformation plot during consolidation for a given load increment [3]

2.1.1 Consolidation Theory with Horizontal Drainage

Barron [4] (1948) developed the basic theory of consolidation The governing differential equation for the dissipation of excess pore water pressure in both horizontal and vertical drainage directions under the equal strain loading (ESL) condition is:

T u

u

r

8exp11

0

(2.3) where the Time factor (Tr) is defined as follows:

2 2

ln

41

n n

n n

e w

where n = spacing ratio, dw = diameter of the drain

2.1.3 Solution of the governing equation (2.2) for a peripheral drain (PD) under free strain loading (FSL) condition

For a PD under free strain loading (FSL) condition, Silverira [7] (1951) solved the governing equation (Eq 2.2) using the following boundary conditions

Table 2.1 Boundary condition

No Condition Result

141

10

2.2 Existing methods for determining cr from consolidation test with a

peripheral drain using incremental loading

In PD case, slope factor is 1.17 and the value of T90 is 0.288 [6]

The radial (horizontal) coefficient of consolidation is determined in this case:

- Step 2: Drawing a second line with the ratio of the length of vertical axis is

(second line / the straight line in step 1) = 1.17

- Step 3: Find the intersection of a second line (Step 2) and consolidation

curve This point is t90

- Step 4: cr is calculated using Eq (2.9) and Ur = 90%

11

Figure 2.4 Consolidation curve relating square - Root time factor to for drainage

radially outwards to periphery with equal strain loading [6]

Evaluation of the method

2.2.1.3

Advantages

- This method is easy to practice for all engineers

- Determination of cr in this method does not require the determination of 0

and 100

- The definition is a straight line within Ur = 20% to Ur = 60% on the curve

Disadvantages

- cr,90 is influenced by secondary consolidation

2.2.2 Inflection point method [9]

Introduction

2.2.2.1

Ganesalingam [9] (2013) solved the governing equation for the relationship

Ur = f[log (Tr)] The value of Ur maximum is the position of the derivative

d(Ur)/dlog Tr the maximum Chung (2019) redefines the value of Tr with dU/d(lnt) The value of Tr is 1/32 = 0.03125

In thesis, the author recommends that the value of Tr calculated with dU/logTr

Time factor can determine with value of y

13

- Step 2: Plot (Ur - log Tr) to t then find time max value (Ur - log Tr) This is

the value tinf.

- Step 3: cr is calculated using Eq.(2.18)

Evaluation of the method

- There is no method to find tinf from experimental data.

- The accuracy of results depends on the time distance between measurement results

2.2.3 Full – match method [10]

Introduction

2.2.3.1

This method combines two methods: graphical method and non – graphical

matching method The relationship between [log(Ur/Tr) & logUr] studied to characterize for between the two straight lines in primary consolidation and secondary compression

Figure 2.5 Log (Ur/Tr) - log Ur relationship [10]

Ultimate settlement is the value of the settlement between the two straight lines

- Step 2: A selected δ – t data range (0 – δint) is substituted into Eq (2.19) and

the unknowns (δ0 & η) are appropriately determined using the Microsoft

Excel Solver

- Step 3: With η, de ,Fn and cr is calculated using Eq (2.20)

Figure 2.6 Determine the value of intersection point in full – match method

Evaluation of the Full – match method

2.2.3.3

Advantages

- It inherits the advantages of method a graphical method

- Value an ultimate settlement δult is determined exactly

Log  (mm)

Log /t (mm/min)

15

Disadvantages

- The way to choose two straight lines is relative

- Determining the value of δult on the logarithmic coordinate system is often difficult

2.3 Existing methods for determining cr from consolidation test with a central

drain using incremental loading method

2.3.1 Root t method [11]

Introduction

2.3.1.1

The method proposed based on the equation for the equal vertical strain condition

[4] In Eq (2.3) have Ur = f[Tr, F(n)] then the author can find Tr = f[Ur, F(n)]

(n) ln(1 U )

8

r r

Berry (1969) commented that all the curve show linear portions between about 20%

- 60% average degree of consolidation Thus a straight line is drawn through the experimental volume change –t0.5 results between about 20% to 60% consolidation, and a second line is then constructed having an abscissa 1.17 time that of the first [11]

The radial (horizontal) coefficient of consolidation is determined in this case:

16

- Identify a straight line within Ur = 20% to Ur = 60% on the curve

- This method does not need to find value 0 and 100

The method proposed based on the equation for the equal vertical strain condition

[4] This approach solves the equation of Eq (2.3) to find the dependence of Tr on

Ur & F(n) then replaces Tr = f[Ur, F(n)] into Eq.(2.4)

Thus, the radial (horizontal) coefficient of consolidation (cr) is determined

- Step 3: Using graphical or Eq (2.23) can be determined cr

Figure 2.7 Theoretical log(de2/t) versus Ur curves [12]

- Matching between theoretical and experimental curve does not always occur

- The variable Ur needs to determine exact values for 0 &100

2.3.3 Inflection point method [13]

Introduction

2.3.3.1

The method was developed based on [13] and [4] Eq (2.3) can show the

relationship Ur = f[log (Tr)]

According to the mathematical definition, the value of Ur maximum when the

derivative d(Ur)/dlog Tr the maximum

Figure 2.8 (a) Theretical Ur - log Tr curve and (b) d(Ur)/dlog Tr plot [13]

In this case, The degree of consolidation at the inflection point also the same for all

the curves at Ur = Ur,inf = 63,21% with maximum derivative

Thus, the value of Ur = Ur,inf = 63,21% can be calculated by Eq (2.3)

- Step 2: cr can be determined cr by Eq.(2.25)

Evaluation of the method

2.3.3.3

Advantages

- 0, 100 does not need to be identified

- In this method, the author finds tinf value

Disadvantages

- There is no method yet to find tinf from Experimental data

- The accuracy of results depends on the time distance between measurement results

Combine Eq of (2.27) and constant values of de and 100, 0 F(n) can be found by

matching between theoretical and experimental curve Solve r – t curve can find cr

- Matching between theoretical and experimental curve does not always occur

- The value depends on the data range

- Eq of (2.27) variable Ur needs to determine the exact values for 0 and 100

2.3.5 Log - log method [15]

Introduction

2.3.5.1

The value of o can be calculated by selecting two time – settlement in the range

Ur < 20% at experimental data(1, t1) & (2, t2)

20

- Step 2: Plot the time t – corrected settlement ( – 0) in a log – log plot

- Step 3: Identify the initial linear portion and draw line

- Step 4: Identify the linear secondary compression portion by drawing a line and extending it to intersect the initial straight line The time at the point of

intersection (t66) corresponds to a degree of consolidation of 66%

- Step 5: cr can be determined by Eq (2.29)

Figure 2.9 Log( - 0) versus log t plot [15]

Evaluation of the method

2.3.5.3

Advantages

- This method can determine 0 & 66

- Methods Inheriting advantages of graphical method

Disadvantages

- From Experimental data the value of 0 within Ur < 20% is not constant

2.3.6 Steepest tangent fitting method [16]

Introduction

2.3.6.1

The method Inflection point in Section 2.3.3 has disadvantages, Inflection point is difficult to determine exactly with experimental data Vinod (2010) found a straight line through an Inflection point [16]

The equation of tangent through Inflection point on the semi-log graph (Figure 2.10) is determined:

 = b - alog(t)

(2.30)

21

where a, b = constant and (t, ) value of experimental data

One log cycle, the author chooses value (1, t1), (2, t2) on the condition (t1 = 10

time), (t2 = 100 time) & (1 - 2 = h) Substituting (1, t1), (2, t2) into Eq (2.30)

 = [0 + hlog(t0)]- alog(t) = 0 + hlog(t0 / t)

(2.33)

Figure 2.10 Steepest tangent fitting method for determination of cr

Similarly, a straight line through Inflection point on Ur-log Tr and d(Ur) /dlog Tr as

shown Figure 2.8 Function for tangent on Ur - log Tr

Ur = c - Slog(Tr)

(2.34)

where c is constant and (Tr, Ur) value of predicted curve

The value of S is defined by Section 2.3.3

r,inf

0.84763.2%

Substituting b with P(0, t0, Tr,0, Ur,0)

Ur = [Ur,0 + Slog(Tr,0)] - Slog(Tr) = Ur,0 + Slog(Tr,0 / Tr)

(2.36) Then dial reading  corresponding to x % consolidation

- Step 2: Determine o in Eq (2.39)

- Step 3: Draw a tangent PQ to the steepest part of the consolidation curve

- Step 4: Find h, which is the slope of the tangent PQ

- Step 5: Find x use Eq.(2.38)

- Step 6: cr is calculated using Eq (2.35)

Evaluation of the method

2.3.6.3

Advantages

- This method finds the 0 values

- Only conduct experiments to Ur = 60%

- Overcoming method disadvantages Inflection point

24

Tr relationship fits approximately as a straight line within 0 < U r < 20%

The initial compression may be also determined graphically

If the data points are selected such that t2 = 2t1, then Eq (2.28) can show that

2 – 1 = 1 – 0

(2.47) The radial (horizontal) coefficient of consolidation is determined in this case:

- Step 1: Plot the time (t) – settlement () in a log t –  plot

- Step 2: Draw a tangent through the inflection point

- Step 3: identify the asymptotic secondary compression portion, draw a line, and extend it to intersect the tangent line The point of intersection corresponds to a degree of consolidation of 100 % (100)

- Step 4: The value of 0 can be obtained using Eq (2.28) or graphically using Eq.(2.46)

- Step 5: cr is calculated using Eq (2.48)

Evaluation of the method

2.3.7.3

Advantages

- The value 0 can determine in this method

- Only conduct experiments to Ur = 60%

- Overcoming method disadvantages Inflection point

Disadvantages

- Experimental data the value of 0 is not constant

2.3.8 Full – match method [10]

This method is identical for both CD & PD The method is mentioned in the section (2.2.3)

The radial (horizontal) coefficient of consolidation is determined in this case:

2.4 Summary of methods for determining cr

The author summarizes the methods for determining cr as follows

Table 2.2 Existing methods for determining cr from radial consolidation Group No Peripheral drain Central drain

One-

point

5 Inflection point method [9] Inflection point method [13]

Full-match

7 Full – match method [10] Full – match method [10]

2.5 Linear regression analysis

Suppose that from some experiment of n observations, i.e values of a dependent variable y measured at specified values of an independent variable x, have been collected In other words, we have set data points (x1, y1), (x2, y2), (x3, y3),… , (xn, yn)

for i = 1, 2, n The function for linear y = bx with in intercept = 0 through n point can be determined

i i

i

e R

y y

2.6 Log normal distribution method

The bell shaped curve appearing in figure 3.6 is generated using the probability density function [18]

Eq (2.53) is referred to as the normal probability density function It is a

„normalized‟ equation, which is another way of saying that when it is integrated

between -∞ ≤ x ≤ +∞, the value obtained is 1

methods for determining cr values for PD & CD cases and provides evaluation methods to select the best methods

The focus of the research is to evaluate and select the Best methods for determining

the radial (horizontal) coefficient of consolidation and find correlations of cr,CD &

cr,PD as well as correlations between cr,CD (and cr,PD) and cv

Literature review Problem identification

Data collection and analyses