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 amongthe proposed methods for determining the horizontal coefficient of consolidation cr in the literature; 2 det

VIETNAM NATIONAL UNIVESITY, HANOI

VIETNAM JAPAN UNIVERSITY

TRAN QUYNH GIAO

A COMPARATIVE STUDY ON

THE HORIZONTAL COEFFICIENT OF

FROM LAB TESTS

MASTER’S THESIS

Hanoi, 2020

VIETNAM NATIONAL UNIVESITY, HANOI

VIETNAM JAPAN UNIVERSITY

TRAN QUYNH GIAO

A COMPARATIVE STUDY ON

THE HORIZONTAL COEFFICIENT OF

When a soft ground is improved by PVDs, consolidation takes place under thecondition 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 fromconsolidation test using incremental loading method [1] However, up to date, therehave not been any similar standards for the consolidation test with horizontaldrainage (using incremental loading method)

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

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

between vertical coefficients of consolidation (cv) and radial cr for a number of testsites 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 testsite 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

i

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

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

I would like to acknowledge the sincere inspiration from Prof Nguyen Dinh Ducand Prof Hironori Kato Their lectures covered not only specialist knowledge butalso 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 atVietnam 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 unflinchingsupport in the tough time Their support, spoken or unspoken, has helped mecomplete my master thesis

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

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

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

2.1.4Solution 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

iii

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

CHAPTER 5 CONCLUSIONS & RECOMMENDATIONS 76 REFERENCES 79

v

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

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

vii

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 cr CD and cr PD obtained from root t method at all data 64

Figure 4.15 Comparison of cr CD and cr PD obtained from non-graphical method at alldata 64

Figure 4.16 Comparison of c r,CD and cr,PD obtained from root t method at all data 66 Figure 4.17 Comparison of cr CD and cr PD obtained from non-graphical method atall data 66

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 alldata 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 alldata 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 alldata 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

ix

Horizontal coefficient of consolidation under for aperipheral drain (PD) condition

Horizontal coefficient of consolidation form root t method under for a central drain (CD) condition

Horizontal coefficient of consolidation form non-graphical method under for a peripheral drain (PD) condition Vertical coefficient of consolidation

Diameter of the soil sampleDrain diameter

Source/sink term; function; cyclic load natural frequencyRatio of influence radius to drain radius

Radial coordinateTime

Time required to reach 50% consolidationTime required to reach 90% consolidationTime required to reach 66% consolidation

Time at d(Ur) /dlog Tr the maximum

Time factor for horizontal consolidationTime factor for horizontal consolidation to reach 90% consolidation

Time factor for horizontal consolidation to reach 66% consolidation

Time factor for vertical consolidationDegree of consolidation

Pore-water pressureChange in pore pressureInitial settlement

Finally settlement at Primary consolidationSettlement at time t

Predicted settlementMeasured settlementPermeability coefficient from PD casePermeability coefficient from CD case

x

m

w Permeability coefficient from vertical consolidation

Soil stiffness from radial consolidationSoil stiffness from vertical consolidationWater unit weight

xi

CHAPTER 1 INTRODUCTION

1.1 Problem statement

Fig 1.1 shows a typical map of distribution of major soil types in Viet Nam Amongthe soil types, the soft and young deposits distributed in major deltas in Vietnam(Red River Delta, Mekong Delta and Saigon – Dongnai River delta) and along thecoast are very much concerned in construction of the infrastructure system

Figure 1.1 Map of distribution of major soil types in Indochinese

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 ofthe soft soil, in which consolidating the soft soil is one of the methods According tosoil mechanics theory, soil is formed from two or three phases (see Figure 1.2) Thevoids 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 inassociation with the process of water drainage

Figure 1.2 Soil phase diagram [3]

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

Figure 1.3 An Illustration of soft ground improved by PVDs

2

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 bothvertical and horizontal drains

1.2 Necessity of study

When a soft ground is improved by PVDs, consolidation takes place under thecondition 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 softclay layer is thick, the consolidation would happen mainly due to the horizontaldrainage The cr value is therefore very important for the design, sometimes muchmore important than the cv value

Currently, the cv value is commonly interpreted from consolidation test usingincremental 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 beenany similar standards for the consolidation test with horizontal drainage (usingincremental loading method) Although cr value might be determined from someConstant 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 valueobtained from result of the consolidation test with horizontal drainage usingincremental 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 ofVietnam

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

The rest of the thesis is organized as follows

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

4

- 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

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction

When a soil layer is subjected to a compressive stress, such as during theconstruction of a structure, it will exhibit a certain amount of compression Thiscompression is achieved through a number of ways, including rearrangement of thesoil solids or extrusion of the pore air and/or water Terzaghi (1943) recommends,

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

Terzaghi (1943) first suggested the one-dimensional consolidation testingprocedure This test performed in a consolidometer (sometimes referred to as anOdometer)

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

- Free-strain case: When the surcharge applied at the ground surface is of aflexible nature, there will be equal distribution of surface load This will result in anuneven 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)

6

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 (Figure2.3) obtained from consolidation test is taken into analysis The curve has threedistinct stages described as follows [3]:

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

- Stage II: Primary consolidation, during which excess pore water pressuregradually 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 placebecause of the plastic readjustment of soil fabric

7

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 governingdifferential equation for the dissipation of excess pore water pressure in bothhorizontal and vertical drainage directions under the equal strain loading (ESL)condition is:

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

2.1.4 Solution of the governing equation (2.2) for a peripheral drain (PD) under

equal strain loading (ESL) condition

For a PD under the equal strain loading (ESL) condition,

governing equation is expressed as follows [8]

U 1  u

1  exp  32T 

the solution of the

(2.8)

9

2.2 Existing methods for determining cr from consolidation test with a

peripheral drain using incremental loading

2.2.1 Root t method [6]

2.2.1.1 Introduction

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

[4] Settlement and volume – change measurements govern by the deformation ofthe sample, as a whole analysis is dependent on an overall “average” behavior Somemethod of “curve fitting” is necessary for graph base on these measurements relate to theconditions at a particular point [6]

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:

2.2.1.2 The procedure for determine c r

- Step 1: Graph with  - t0.5 then finding a straight line within Ur = 20% to

- 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%

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

radially outwards to periphery with equal strain loading [6]

2.2.1.3 Evaluation of the method

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]

2.2.2.1 Introduction

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.

∆t

then findin

g the best fittingcurve

Substituting of Eq (2.4) from Eq (2.10) The settlement () collateral t is expressed

    1  exp( 32.10 y )

Derivative of v = -32.10y

(2.18)

12

- 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)

2.2.2.3 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]

2.2.3.1 Introduction

This method combines two methods: graphical method and

matching method The relationship between [log(Ur/Tr) &

characterize for between the two straight lines in primary

secondary compression

non – graphical

logUr] studied toconsolidation and

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

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

The theoretical solutions (PD case and CD case) are rearranged in terms of time:

2.2.3.2 The procedure for determine c r from Full – match method

- Step 1: A log (δ/t) – log δ graph is plotted using the monitored data, and an

intersection (δint) between two straight lines is determined

- 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)

Log /tt (mm/min)

Log  (mm)Figure 2.6 Determine the value of intersection point in full – match method

2.2.3.3 Evaluation of the Full – match method

Advantages

- It inherits the advantages of method a graphical method

- Value an ultimate settlement δult is determined exactly

14

- 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]

2.3.1.1 Introduction

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)].

8

T90 is the time factor at 90% average degree of consolidation so the value of T90 can calculated by F(n), Ur

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 theexperimental volume change –t0.5 results between about 20% to 60% consolidation, and asecond 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:

2.3.1.2 The procedure for determine c r

The steps for determining the radial (horizontal) coefficient are the same asdescribed in section.2.2.1.2

2.3.1.3 Evaluation of the method

Advantages

- This method is easy to practice for all engineers 15

- 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

t   F ( n) ln(1  U r ) re

2.3.2.2 The procedure for determine c r

- Step 1: Plot log (de2/t) versus on

Figure 2.7

- Step 2: Identify a zone where the

theoretical curves

(2.23)

Ur curve considering the δ - t data

the experimental curve is parallel to

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

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

16

2.3.2.3 Evaluation of the method

Advantages

- cr can determine easily by the graph.

Disadvantages

- 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]

2.3.3.1 Introduction

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)

2.3.3.2 The procedure for determine c r

- Step 1: Plot (Ur - log t) to t then finding time at max value (Ur - log t) This is the value tinf

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

2.3.3.3 Evaluation of the method

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

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

18

2.3.4.2 The procedure for determine c r

Finding cr with100, 0, F(n), de and r – t curve by A source code or program on Eq.

(2.27)

2.3.4.3 Evaluation of the method

Advantages

- Use a coding program to resolve results independent of the implementer

Therefore, the value has high accuracy

- Results processing time is fast

Disadvantages

- 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]

2.3.5.1 Introduction

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

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

consolidation is determined in this case:

2.3.5.2 The procedure for determine c r

- Step 1: Calculate the initial compression (0) using Eq (2.28) from the time

– compression data by choosing two points in the early stages of

consolidation The value of 0 can be calculated by selecting two time –

settlement data within Ur < 20%

19

- 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] 2.3.5.3 Evaluation of the method

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]

2.3.6.1 Introduction

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

The equation of tangent through Inflection point on the semi-log graph (Figure

2.10) is determined:

20

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)

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.

2.3.6.2 The procedure for determine c r

- Step 1: Plot the dial reading against time on semi log graph as show in Figure2.10

- 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)

2.3.6.3 Evaluation of the method

Advantages

- This method finds the 0 values

- Only conduct experiments to Ur = 60%

- Overcoming method disadvantages Inflection point

10 0

(2