國立臺灣大學工學院土木工程學系 博士論文

Department of Civil Engineering College of Engineering

National Taiwan University Doctoral Dissertation

極限狀態的土壤抗剪強度的特徵值

Characteristic Values of Soil Shear Strength for Ultimate Limit States

陶默德

Mohammad Tabarroki

指導教授：卿建業 博士 Advisor: Jianye Ching, Ph.D.

中華民國 110 年 2 月 February, 2021

i

口試委員會審定書

**Acknowledgements **

The book in your hands narrates, in a sense, a chapter of my life. As this chapter is coming to an end, and before embracing the next one, I would like to draw an underline underneath the names of those who have played a meaningful role in this chapter of my life:

Professor Jianye Ching, my advisor, for introducing me to the world of geotechnical reliability, and challenging me with questions that matter not only to scholars but also to practitioners. To say that working with him has been a pleasure is a complete understatement.

My committee members, Professor Ming-Lang Lin and Professor Louis Ge (National Taiwan University), Professor Chih-Ping Lin (National Chiao Tung University), and Professor Jui-Pin Wang (National Central University), for spending their valuable time to review my work, and providing me with insightful comments.

Professor Fu-Shu Jeng for teaching me three courses including Numerical Methods in Geotechnical Engineering, Rock Mechanics, and Constitutive Law of Soil.

His quality teaching goes well beyond the classroom.

All members of our lab. In particular, Ms. Ellen Chiang, our lab secretary, for her kindness and willingness to help. Also, Dr Yu-Gang Hu, Dr Szu-Wei Lee, and Mr.

Ying-Zhong Chen, for their discussions, feedbacks, and of course, friendship.

Professor Gordon Fenton and Professor Vaughan Griffiths for making the source code of their RFEM software freely available to the public. The code is self-explanatory, easy to extend, and is available at http://random.engmath.dal.ca/rfem/.

National Taiwan University, Center for Earthquake Engineering Research (CEER), and Professor Ching, for providing financial support over the course of my PhD work.

iii

**Dedication **

A word is dead when it is said, some say.

I say it just begins to live that day.

– Emily Dickinson (1830-1886)

to my mother, father, and wife for their words of faith, wisdom, and love

**中文摘要 **

歐洲規範 7 (Eurocode 7) 將土壤參數的特徵值定義為“對影響極限狀態發生的值 的謹慎估計”。當前的研究著眼於在存在空間變異性的情況下、決定剪力強度的 特徵值。驅動的剪力強度（即影響極限極限發生的剪力強度）不僅受到沿潛在滑 動面的空間平均的影響，而且還受到尋找弱帶的影響。前者（空間平均）減小了 驅動剪力強度的變異性，後者（弱帶搜索）減小了驅動剪力強度的期望值。前人 文獻中提出了幾個簡單的公式、以決定剪力強度的特徵值，但他們大多只考慮空 間平均，而忽略了弱帶搜索，這可能是不保守的。當前的研究採用最弱路徑模型

（WPM），WPM 雖是一個簡單的模型，卻同時考慮空間平均和弱帶搜索二個因 素、以模擬驅動的剪力強度。由於 WPM 需要校準，因此當前的研究使用隨機有 限元方法（RFEM）來針對一些土壤力學與基礎工程問題對 WPM 進行校準，校 準的 WPM 的行為與 RFEM 的行為相似。基於校準的 WPM，當前的研究提出了 一個簡化公式來決定剪力強度的特徵值，並在實際案例中、驗證該公式的有效性。

驗證結果發現，對於不確定性受空間變異性支配、且破壞面受中、低度約束的問 題，所提出的公式與文獻中現有的簡化公式相比、具有顯著改進。

關鍵字：驅動的特徵值； 空間變異性；隨機有限元；最弱路徑模型；歐洲規範

v

**Abstract **

Eurocode 7 defines the characteristic value of a soil parameter as “a cautious estimate of the value affecting the occurrence of the limit state”. The current study addresses the determination of the characteristic value of shear strength in the presence of spatial variability in the context of the ultimate limit state. The mobilized shear strength, the shear strength affecting the occurrence of the ultimate limit state, is influenced not only by the spatial averaging along the potential slip curve but also by seeking out mechanically admissible weak zones. The former factor (spatial averaging) reduces the variance of the mobilized shear strength, whereas the latter factor (weak-zone seeking) reduces the mean of the mobilized shear strength among others. Several simple formulas have been proposed in the literature for the determination of the characteristic value of shear strength. However, they mostly consider spatial averaging only and ignore the weak-zone seeking, which is potentially unconservative. The current study adopts the weakest-path model (WPM). The WPM is a simple model that can simulate the mobilized shear strength considering both factors of spatial averaging and weak- zone seeking. Since the WPM requires calibration, the current study calibrates the WPM for several geotechnical problems using the random finite element method (RFEM). The behaviors for the calibrated WPM are found to be similar to those for the RFEM. Based on the calibrated WPM, a simplified formula is proposed to determine a mobilization-based characteristic value. The effectiveness of this formula is showcased by real case studies. It is found that for problems whose uncertainty is dominated by spatial variability and whose slip curve is not highly constrained by mechanical admissibility, the proposed formula provides significant improvements over existing simplified formulas in the literature.

Keywords: mobilization-based characteristic value; spatial variability; random finite element; weakest-path model; Eurocode 7

**Table of Contents **

口試委員會審定書 ... i

Acknowledgements ... ii

Dedication ... iii

中文摘要 ... iv

Abstract ... v

Table of Contents ... vi

List of Figures ... xii

List of Tables ... xx

Chapter 1 Introduction ... 1

1.1 Background ... 1

1.2 Objectives of the thesis ... 3

1.3 Structure of the thesis ... 4

Chapter 2 Literature Review... 6

2.1 Introduction ... 6

2.2 Design philosophy in Eurocode 7 ... 6

2.3 Geotechnical design by calculation ... 7

2.4 Partial factors ... 10

2.5 Characteristic values of geotechnical parameters ... 14

2.6 Statistics-based characteristic value ... 19

vii

2.7 Reliability-based characteristic value ... 21

2.8 Modelling spatial variability using random fields ... 24

2.9 Simulating random fields ... 28

2.10 Worst-case SOF ... 30

Chapter 3 Discretization Error in the Random Finite Element Method ... 34

3.1 Introduction ... 34

3.2 Past works ... 35

3.3 Sources of discretization error in RFEM ... 36

3.4 Random field discretization methods ... 37

3.5 Description of investigated problems ... 38

3.6 Random finite element mesh layout ... 40

3.7 Soil column results for a fixed scale of fluctuation ... 42

3.7.1 FE discretization error for a fixed RF mesh ... 44

3.7.2 RF discretization error for a fixed FE mesh ... 45

3.7.3 Discretization error for conventional RFEM ... 47

3.7.4 Influence of FE element type on the conventional RFEM ... 52

3.8 Retaining wall results for a fixed scale of fluctuation ... 55

3.9 Monte Carlo results for various scales of fluctuation ... 60

3.9.1 Discretization error for conventional RFEM for various SOFs ... 60

3.9.2 Suggestions for allowable mesh size ... 62

Chapter 4 Weakest-Path Model for Mobilized Shear Strength ... 65

4.1 Introduction ... 65

4.2 Concepts of WPM ... 65

4.3 Estimating n and Γ^{2} using the maximum likelihood method ... 71

4.4 Capturing the worst-case SOF with n and Г^{2} ... 72

4.5 Description of investigated problems ... 75

4.5.1 Strip footing subjected to vertical loading ... 79

4.5.2 Retaining wall ... 80

4.5.3 Basal heave for excavation in clay ... 81

4.5.4 Soil column subjected to axial compression... 83

4.5.5 Laterally loaded pile ... 84

4.5.6 Friction pile under compression ... 85

4.6 Normalization of SOF ... 86

4.7 Simulating X^{mob} samples by RFEM ... 87

Chapter 5 Calibration Results ... 90

5.1 Introduction ... 90

5.2 Stationary-su scenario ... 90

5.3 Stationary-su*/σ'*v scenario ... 102

5.4 Stationary-tan scenario ... 109

5.5 Influence of distribution type and autocorrelation function on n ... 114

Chapter 6 Mobilization-based Characteristic Value ... 117

6.1 Introduction ... 117

ix

6.2 Simple formula for Xkmob... 118

6.3 Comparison with reliability-based characteristic value... 121

6.4 Comparison with existing statistics-based formulas ... 123

Chapter 7 Real Case Studies ... 126

7.1 Introduction ... 126

7.2 Bolivian experimental site for testing piles ... 126

7.2.1 Site and problem type ... 126

7.2.2 Limit-state function and response of interest ... 128

7.2.3 Ground scenario and input X ... 128

7.2.4 Parameters for X random field ... 128

7.2.5 Slip curve and estimation of (L, L) & ^{2} ... 129

7.2.6 Estimation of various COVs ... 129

7.2.7 Estimation of n ... 130

7.2.8 Estimation of Xkmob... 131

7.2.9 Other calculation results ... 131

7.2.10 Comparison with Schneider and Schneider (2013) ... 132

7.3 Full-scale footing load test at a site in Baytown, United States ... 132

7.3.1 Site and problem type ... 132

7.3.2 Limit-state function ... 134

7.3.3 Ground scenario and input X ... 134

7.3.4 Parameters for X random field ... 134

7.3.5 Slip curve and estimation of (L, L) & ^{2} ... 134

7.3.6 Estimation of various COVs ... 135

7.3.7 Estimation of n ... 135

7.3.8 Estimation of Xkmob... 136

7.3.9 Other calculation results ... 136

7.3.10 Comparison with Schneider and Schneider (2013) ... 136

7.4 Cement-treated soil columns conducted by Building Center of Japan ... 137

7.4.1 Sites and problem type ... 137

7.4.2 Ground scenario and input X ... 138

7.4.3 Parameters for X random field ... 138

7.4.4 Slip curve and estimation of (L, L) & ^{2} ... 140

7.4.5 Estimation of various COVs ... 141

7.4.6 Estimation of n ... 141

7.4.7 Estimation of Xkmob... 141

7.4.8 Other calculation results ... 142

7.4.9 Comparison with Schneider and Schneider (2013) ... 142

Chapter 8 Conclusions and Suggestions ... 145

8.1 Introduction ... 145

8.2 Concluding remarks ... 146

8.2.1 Behavior of the calibrated n and Γ^{2} ... 146

8.2.2 Performance of the calibrated WPM ... 148

xi

8.2.3 Performance of the proposed simple formula... 148

8.3 Suggestions for future work ... 150

References ... 152

Appendix A Calibrated (n, Γ^{2}) for the Stationary-su ... 163

Appendix B Calibrated (n, Γ^{2}) for the Stationary-su*/σ'*v ... 167

Appendix C Calibrated (n, Γ^{2}) for the Stationary-tan ... 171

Appendix D Comparison between Xkmob and XkRFEM ... 175

Appendix E Worst-case SOFs in the literature ... 184

**List of Figures **

Figure 2.1 Outline of design by calculation in Eurocode 7 ... 8 Figure 2.2 Procedure for choosing design values (modified after Orr 2000)... 9 Figure 2.3 Different methods for calibrating partial factors (after EN 1990) ... 12 Figure 2.4 Engineers’ interpretation of the characteristic value for a soil profile.

Solid dots represent data points, while gray lines represent engineers’

interpretations (after Bond and Harris 2008) ... 18 Figure 2.5 Probability distributions of point shear strength and mobilized shear

strength ... 22 Figure 2.6 Statistics-based, mobilization-based, and reliability-based characteristic

values ... 24 Figure 2.7 Inherent (spatial) soil variability (after Phoon and Kulhawy 1999) ... 25 Figure 2.8 A random field realization using separable (left) and radial (right) single

exponential models ... 30 Figure 2.9 Variation of (worst-case mean response)/(nominal response) with respect

to COV: (a) isotropic v.s. anisotropic cases; (b) different problem types;

(c) 2D v.s. 3D soil column (after Vessia et al. 2020) ... 33 Figure 3.1 The FEM model for the (a) soil column problem and (b) retaining wall

problem ... 40 Figure 3.2 (a) Different combinations of FE mesh and RF mesh resolutions for the

*soil column problem. (b) The case with l*FE/Lx *= 1/8 and l*RF/Lx = 1/2
(filled circle in the left) ... 41
Figure 3.3 The normalized mean qu for δ = 12.8 m using (a) LA and (b) MP ... 43
Figure 3.4 *The effect of the FE discretization error for the soil column problem (l*RF

/L fixed at 1/4) ... 44

xiii

Figure 3.5 *The effect of the RF discretization error for the soil column problem (l*FE

is the finest) ... 45 Figure 3.6 A weak spot (the arrow) experiencing different scenarios under LA and

MP methods. The finest FE mesh is adopted. The RF mesh is fine (left) and coarse (right) ... 46 Figure 3.7 The decomposition of the total discretization error for the conventional

RFEM into FE and RF discretization errors. The soil column problem with LA (Figure 3.3a) is used as an example... 48 Figure 3.8 The variation of FE and RF discretization errors in the conventional

*RFEM (l*FE *= l*RF *= l) using (a) LA and (b) MP, for the soil column *
problem with = 12.8 m ... 49
Figure 3.9 *The relative error of the conventional RFEM (l*FE *= l*RF *= l) for the mean *

value of qu ( = 12.8 m) ... 50
Figure 3.10 *The relative error of the conventional RFEM (l*FE *= l*RF *= l) for the *

standard deviation of qu ( = 12.8 m) ... 51 Figure 3.11 The influence of the FE element type on the overly stiff behavior in the

conventional RFEM for (a) LA and (b) MP. The soil column problem is considered with = 12.8 m. ... 53 Figure 3.12 The normalized mean Pa for δ=1 m using (a) LA and (b) MP ... 56 Figure 3.13 The effect of the FE discretization error for the retaining wall problem

*(l*RF/H fixed at 1/2.5) ... 57
Figure 3.14 The effect of the RF discretization error for the retaining wall problem

*(l*FE is the finest) ... 57

Figure 3.15 The variation of FE and RF discretization errors in the conventional
*RFEM (l*FE *= l*RF *= l) using (a) LA and (b) MP for the retaining wall *
problem with = 1 m ... 58
Figure 3.16 *The relative error of the conventional RFEM (l*FE *= l*RF *= l) for the mean *

value of Pa ( = 1 m) ... 59
Figure 3.17 *The relative error of the conventional RFEM (l*FE *= l*RF *= l) for the *

standard deviation of Pa ( = 1 m) ... 59 Figure 3.18 The normalized mean qu for various SOFs for the soil column problem.

*The conventional RFEM (l*FE *= l*RF *= l) is employed with (a) LA and (b) *
MP ... 61
Figure 3.19 The normalized mean Pa for various SOFs for the retaining wall problem.

*The conventional RFEM (l*FE *= l*RF *= l) is employed with (a) LA and (b) *
MP ... 62
Figure 4.1 (a) Two PSCs in a soil column subjected to axial compression, (b) X^{ave}
values for the two PSCc when Δs/SOF = 0.07, (c) X^{ave} values for the two
PSCc when Δs/SOF = 2.1, (d) histograms of X^{ave} values for the two
PSCc when Δs/SOF = 2.1 ... 67
Figure 4.2 Assumptions in the WPM illustrated for the soil column subjected to

compression. (a) all PSCs have the same size and shape, (b) a cluster of
highly correlated PSCs is lumped into one representative PSC ... 69
Figure 4.3 Variation of the negative log-likelihood with respect to n and Г^{2} for a

particular X^{mob} samples ... 72
Figure 4.4 (a) Effect of n and Γ^{2} on PDF of X^{mob} (solid line). The dashed line is

point X with µ = 50 and σ = 15; (b) the phenomenon of the worst-case SOF ... 73

xv

Figure 4.5 Strip footing subjected to vertical loading ... 79

Figure 4.6 Retaining wall ... 81

Figure 4.7 Basal heave for excavation in clay ... 82

Figure 4.8 Definition of FS for the basal heave problem (after Ching et al. 2017a) . 83 Figure 4.9 Soil column subjected to axial compression ... 84

Figure 4.10 Laterally loaded pile ... 85

Figure 4.11 Friction pile under compression... 86

Figure 4.12 su-qu chart for the footing problem ... 88

Figure 4.13 Histogram of the X^{mob} samples (X = su) for the strip footing example
with V = 0.3 and δx = δz = 2 m and the fitted PDF from the WPM ... 89

Figure 5.1 (a) Calibrated n and (b) calibrated Γ^{2} versus δL/L (stationary-su; V = 0.3;
δx/δz = 10)... 91

Figure 5.2 (left) Calibrated n and (right) calibrated Γ^{2} vs. δL/L (retaining wall;
stationary-su; V = 0.3) ... 93

Figure 5.3 (left) Calibrated n and (right) calibrated Γ^{2} vs. δL/L (retaining wall;
stationary-su; δx/δz = 1) ... 94

Figure 5.4 Sample means of X^{mob} (X = su). The solid lines are for the predictions
based on the calibrated WPM. (a) soil column (low constraint), (b) strip
footing (medium constraint), (c) basal heave (medium constraint), (d)
retaining wall (high constraint), (e) friction pile (full constraint) ... 98

Figure 5.5 Sample COVs of X^{mob} (X = su). The solid lines are for the predictions
based on the calibrated WPM. (a) soil column (low constraint), (b) strip
footing (medium constraint), (c) basal heave (medium constraint), (d)
retaining wall (high constraint), (e) friction pile (full constraint) ... 101

Figure 5.6 (left) Calibrated n and (right) calibrated Γ^{2} vs. δL/L for laterally loaded
pile and retaining wall (stationary-su; δx/δz = 1) ... 102
Figure 5.7 (a) Calibrated n and (b) calibrated Γ^{2} vs. δL/L (stationary-su*/σ'*v; V = 0.3;

δx/δz = 10)... 103
Figure 5.8 Sample means of X^{mob} (X = su*/σ'*v). The solid lines are for the predictions

based on the calibrated WPM. (a) strip footing (medium constraint), (b)
basal heave (medium constraint), (c) retaining wall (high constraint), (d)
friction pile (full constraint) ... 106
Figure 5.9 Sample COVs of X^{mob} (X = su*/σ'*v). The solid lines are for the predictions

based on the calibrated WPM. (a) strip footing (medium constraint), (b)
basal heave (medium constraint), (c) retaining wall (high constraint), (d)
friction pile (full constraint) ... 108
Figure 5.10 (a) Calibrated n and (b) calibrated Γ^{2} vs. δL/L (stationary-tan ; V = 0.1;

δx/δz = 10)... 109
Figure 5.11 Sample means of X^{mob} (X = tan ). The solid lines are for the predictions

based on the calibrated WPM. (a) soil column (low constraint), (b) strip
footing (medium constraint), (c) retaining wall (high constraint), (d)
friction pile (full constraint) ... 112
Figure 5.12 Sample COVs of X^{mob} (X = tan ). The solid lines are for the predictions

based on the calibrated WPM. (a) soil column (low constraint), (b) strip footing (medium constraint), (c) retaining wall (high constraint), (d) friction pile (full constraint) ... 114 Figure 5.13 Influence of distribution type (normal vs. lognormal) on the calibrated n

(stationary-su; V = 0.2, δx/δz = 1) ... 115

xvii

Figure 5.14 Influence of autocorrelation function on the calibrated n (stationary-su; V

= 0.3, δx/δz = 1) ... 116
Figure 6.1 Relationship between βηWPM and n ... 118
Figure 6.2 Variation of 0.05WPM versus δL/L (stationary-su; V = 0.3; δx/δz = 10) ... 119
Figure 6.3 Xk for the stationary-su scenario (X = su) (V = 0.3, δx/δz = 10) ... 121
Figure 6.4 Xk for the stationary-su*/σ'*v scenario (X = su*/σ'*v) (V = 0.3, δx/δz = 10) .. 122
Figure 6.5 Xk for the stationary-tan scenario (X = tan ) (V = 0.1, δx/δz = 10) ... 122
Figure 6.6 Xk for the stationary-su scenario for low and medium constraint cases . 125
Figure 7.1 Test results for the Bolivian experimental site: (a) the static load test

*result for the pile B2; (b) tan ϕ profile estimated based on CPTu B2 .... 127*
Figure 7.2 Analysis results for the total axial resistance (R) for the pile case study132
Figure 7.3 Load test result for the square footing case study ... 133
Figure 7.4 Analysis results for the bearing resistance (R) of the square footing case

study ... 137 Figure 7.5 Variation of the normalized Qu of the treated soil columns versus V ... 140 Figure 7.6 Analysis results for Qu of the treated soil columns ... 143

Figure A.1 (n, Γ^{2}) for stationary-su; δx/δz=1 (top) V=0.1, (middle) V=0.3, (bottom)
V=0.5 ... 163
Figure A.2 (n, Γ^{2}) for stationary-su; δx/δz=10 (top) V=0.1, (middle) V=0.3, (bottom)

V=0.5 ... 164
Figure A.3 (n, Γ^{2}) for stationary-su; δx/δz=30 (top) V=0.1, (middle) V=0.3, (bottom)

V=0.5 ... 165
Figure A.4 (n, Γ^{2}) for stationary-su; δx/δz=100 (top) V=0.1, (middle) V=0.3, (bottom)

V=0.5 ... 166

Figure B.1 (n, Γ^{2}) for stationary-su*/σ'*v; δx/δz=1 (top) V=0.1, (middle) V=0.3, (bottom)
V=0.5 ... 167
Figure B.2 (n, Γ^{2}) for stationary-su*/σ'*v; δx/δz=10 (top) V=0.1, (middle) V=0.3,

(bottom) V=0.5 ... 168
Figure B.3 (n, Γ^{2}) for stationary-su*/σ'*v; δx/δz=30 (top) V=0.1, (middle) V=0.3,

(bottom) V=0.5 ... 169
Figure B.4 (n, Γ^{2}) for stationary-su*/σ'*v; δx/δz=100 (top) V=0.1, (middle) V=0.3,

(bottom) V=0.5 ... 170
Figure C.1 (n, Γ^{2}) for stationary-tan ; δx/δz=1 (top) V=0.05, (middle) V=0.1,

(bottom) V=0.2 ... 171
Figure C.2 (n, Γ^{2}) for stationary-tan ; δx/δz=10 (top) V=0.05, (middle) V=0.1,

(bottom) V=0.2 ... 172
Figure C.3 (n, Γ^{2}) for stationary-tan ; δx/δz=30 (top) V=0.05, (middle) V=0.1,

(bottom) V=0.2 ... 173
Figure C.4 (n, Γ^{2}) for stationary-tan ; δx/δz=100 (top) V=0.05, (middle) V=0.1,

(bottom) V=0.2 ... 174 Figure D.1 Xk for the stationary-su scenario (X = su). The solid lines are based on the

proposed simple formula, Xkmob, with calibrated n-equations in Table 5.1.

(a) soil column (low constraint), (b) strip footing (medium constraint), (c)
basal heave (medium constraint), (d) retaining wall (high constraint), (e)
friction pile (full constraint) ... 177
Figure D.2 Xk for the stationary-su*/σ'*v scenario (X = su*/σ'*v). The solid lines are based

on the proposed simple formula, Xkmob, with calibrated n-equations in Table 5.2. (a) strip footing (medium constraint), (b) basal heave (medium

xix

constraint), (c) retaining wall (high constraint), (d) friction pile (full constraint) ... 180 Figure D.3 Xk for the stationary-tan scenario (X = tan ). The solid lines are based

on the proposed simple formula, Xkmob, with calibrated n-equations in Table 5.3. (a) soil column (low constraint), (b) strip footing (medium constraint), (c) retaining wall (high constraint), (d) friction pile (full constraint) ... 183

**List of Tables **

Table 2.1 Definition of consequences classes (after EN 1990) ... 13

Table 2.2 Recommended minimum values of reliability index β for ultimate limit states (modified after EN 1990) ... 14

Table 2.3 Extracts from Section 2.4.5.2 of Eurocode 7 (after Hicks 2013) ... 15

Table 2.4 Top five topics as voted by the UK engineers to be considered for the next revised version of Eurocode 7 (after Orr 2012) ... 19

Table 2.5 Popular autocorrelation functions and their frequency of usage (modified after Cami et al. 2020) ... 26

Table 2.6 Scales of fluctuation in the literature by type of soil (after Cami et al. 2020) ... 27

Table 3.1 Different types of plane strain elements for FE mesh with their names in Abaqus. ... 54

Table 3.2 *Allowable l/, (l/)*allow, for 0.01-tolerance and 0.05-tolerance ... 64

Table 4.1 Parameters for the stationary-su scenario ... 77

Table 4.2 Parameters for the stationary-su*/σ'*v scenario ... 78

Table 4.3 *Parameters for the stationary-tan ϕ scenario ... 78*

Table 5.1 Calibrated n-equations for the stationary-su scenario ... 94

Table 5.2 Calibrated n-equations for the stationary-su*/σ'*v scenario ... 104

Table 5.3 *Calibrated n-equations for the stationary-tan ϕ scenario ... 110*

Table 7.1 Statistics and results for the 13 treated soil columns ... 139

Table E.1 Summary of worst-case SOFs in the literature (after Vessia et al. 2020, prepared for ISSMGE TC304) ... 184

1

**Chapter 1 Introduction **

**1.1 Background **

In design calculations, be it using a global factor of safety, partial factors, or otherwise, the selection of suitable values for soil parameters is a challenging task for engineers.

The challenge stems from special features of geotechnical practice, such as the remarkable variability of soils, and the fact that the soil volume tested is much smaller than the soil volume involved in a design. Due to difficulties in finding a detailed and prescriptive advice, the process of selecting soil parameters is sometimes viewed as a

“black art” (Simpson and Driscoll 1998). The European standard for geotechnical design, Eurocode 7 (CEN 2004), attempts to provide some guidelines on the selection of soil parameters, under Clause 2.4.5.2 “Characteristic values of geotechnical parameters”.

Eurocode 7 defines the characteristic value of a soil parameter as “a cautious estimate of the value affecting the occurrence of the limit state” [Clause 2.4.5.2(2)].

There are two main aspects in the definition of characteristic values: (a) a cautious estimate, and (b) the value affecting the occurrence of the limit state. The first aspect is related to uncertainties or statistics (e.g., spatial variability, transformation uncertainty, statistical uncertainty), while the second aspect is related to mechanics (e.g., spatial variability, force equilibrium, boundary conditions). Note that spatial variability (i.e., point to point variation of soil properties) is influential in both aspects, statistical and mechanical. It is often the spatial average, rather than the property at a certain point (point property), of soil properties over some influence zone that governs the performance of a geotechnical structure (e.g., spatial average of shear strength along a critical slip curve).

The current study focuses on the influence of spatial variability (both statistical
and mechanical aspects) on characteristic values of shear strength for ultimate limit
states. The term “point shear strength” is adopted to refer to the shear strength measured
at a certain point of a spatially variable soil mass, whereas the term “mobilized shear
strength” is adopted to refer to the shear strength that is effectively exhibited by the
spatially variable soil mass as a whole. In other words, the mobilized shear strength is a
value of shear strength that emerges from a back-analysis of a failure, if the failure
occurs. In an ideal and deterministic world, the mobilized shear strength is a single
(deterministic) value. However, in the real world, the mobilized shear strength takes a
range of values in the presence of spatial variability and other sources of uncertainties,
making it a random variable. Considering that the mobilized shear strength affects the
*occurrence of the limit state, it is sensible to adopt a cautious estimate of the mobilized *
*shear strength as the characteristic value of shear strength, following Eurocde 7’s *

definition.

Spatial variability can have both beneficial and detrimental impacts on the mobilized shear strength. There is also an added complexity arising from interactions between these types of impacts. Recall that the performance of a geotechnical structure is often governed by the spatial average over some influence zone. Along this influence zone, high values of point shear strength tend to be balanced by low values of point shear strength, leading to a lower variation of the spatial average. This effect, known as variance reduction (Vanmarcke 1977a), reduces the variance of the mobilized shear strength. This is a beneficial effect of spatial variability, because it reduces uncertainty (smaller variance), which in turn, allows engineers to select a less conservative value for their cautious estimates of the mobilized shear strength. Spatial variability, however, can also lead to failure mechanisms that deviate from the theoretical ones (derived from

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homogenous soils) to seek out weak zones. This reduces the mean of the mobilized shear strength. This is a detrimental effect of spatial variability, because it shifts the mean of the mobilized shear strength to the unconservative side.

Although Eurocode 7 provides a sensible definition for the characteristic values, it does not provide any detailed procedure. Therefore, due to the subjective nature of engineers’ judgment, a wide range of characteristic values may be selected for the same limit state, resulting in designs with different degrees of reliability. To address this, several simple formulas have been proposed to help engineers selecting more objective and consistent characteristic values (e.g., Schneider 1997; Schneider and Schneider 2013; Orr 2017). However, most existing formulas consider only the statistical aspect of characteristic values, and do not fully consider the mechanical aspect. In particular, they almost ignore the detrimental effect of spatial variability (weak-zone seeking), which is potentially unconservative.

**1.2 Objectives of the thesis **

Since existing simple formulas are not fully compliant with Eurocode 7, it is of practical value to develop a simple formula that respects both statistical and mechanical aspects of characteristic values. The weakest-path model (WPM), originally developed by Ching and Phoon (2013a), is a simple model that can simulate the probability distribution of the mobilized shear strength in the presence of spatial variability. The WPM considers both beneficial and detrimental effects of spatial variability. However, the WPM requires calibration using advanced methods such as random finite element method (RFEM). Therefore, the main objectives of the current study are:

1. To calibrate the WPM for several geotechnical problems.

2. To propose a simple formula for determining the characteristic value of shear strength based on the calibrated WPM.

3. To compare the performance of the proposed formula with existing formulas using numerical examples and real cases studies.

**1.3 Structure of the thesis **

The contents of the current thesis are divided into eight chapters, which are briefly described as follows:

Chapter 2 provides an overview of the literature relevant to characteristic values.

The random filed theory, which is usually used for modelling spatial variability, and the phenomenon of the worst-case scale of fluctuation are also reviewed in this chapter.

Chapter 3 is devoted to the issue of the discretization error in the RFEM. Recall that the WPM requires calibration using advanced methods such as the RFEM. It is evident that if implementation errors in the RFEM itself are not controlled properly, they can propagate to the calibration of the WPM. Hence, the aim of Chapter 3 is to make sure that the RFEM is implemented properly.

Chapter 4 presents the concepts of the WPM. It also presents six geotechnical problems, together with three scenarios of ground properties to be used for the calibration of the WPM.

Chapter 5 presents the calibration results for the investigated geotechnical problems. After calibration, the WPM is used to simulate the mean and variance of the mobilized shear, and the results are compared with those from the RFEM. This chapter also demonstrates the performance of the WPM in capturing the worst-case scale of fluctuation.

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Chapter 6 proposes a simple formula based on the WPM for determining the characteristic value of the shear strength. The performance of the proposed formula is verified using the RFEM for the investigated problems. This chapter also provides a comparison with past works, and shows the improvements of the proposed formula over past works in the literature.

Chapter 7 illustrates and verifies the use of the proposed simple formula for three real case studies. The case studies include a pile, a footing, and several cement- treated soil columns.

Chapter 8 concludes this thesis by highlighting the main findings of the current study, and provides some suggestions for future research.

**Chapter 2 Literature Review **

**2.1 Introduction **

This chapter provides an overview of the literature relevant to characteristic values. It starts with the design philosophy in Eurocode 7, limit state design, and briefly reviews ultimate limit states (ULS) and serviceability limit states (SLS). Attention is then paid to “design by calculations” approach, which is one of the approaches for checking that limit states of interest are not exceeded. The characteristic values, among others, lie at the heart of the design by calculations approach. This chapter then highlights important clauses in Eurocode 7 on selection of the characteristic values. Simple formulas proposed in the literature for determination of the characteristic values are also reviewed.

These simple formulas may lead to inappropriate characteristic values that ignore the actual design situation. Hence, this chapter continues with reviewing advanced (but costly) methods for the determination of the characteristic values that fully satisfy the spirits of Eurocode 7. These advanced methods usually incorporate random fields to model soil spatial variability. Therefore, random field theory is reviewed as well. This chapter ends with reviewing the phenomenon of the worst-case scale of fluctuation, which is closely related to the determination of the characteristic values.

**2.2 Design philosophy in Eurocode 7 **

The design philosophy in Eurocode 7 (EN 1997) follows the “head” Eurocode (EN 1990: Basis of structural design), which is based on the principles of limit state design.

EN 1990 defines limit states as “states beyond which the structure no longer fulfils the relevant design criteria” (EN 1990, 1.5.2.12). In the Eurocode suite, limit states are classified into ultimate limit states (ULS) and serviceability limit states (SLS).

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According to EN 1990, ULSs are concerned with “the safety of people, and/or the safety of the structure” [EN 1990, Clause 3.3(1)P], whereas SLSs are concerned with “the functioning of the structure or structural members under normal use, the comfort of people, and the appearance of the construction works” [EN 1990, Clause 3.4(1)P]. An example of ULS is bearing failure of foundations, while settlement of foundations is an example of SLS. Since structures shall be designed “with appropriate degrees of reliability and in an economical way” [EN 1990, Clause 2.1(1)P], the exceedance of SLSs might be occasionally tolerable for sake of economics. However, the exceedance of ULSs must be generally avoided (Simpson and Driscoll 1998).

In order to verify that limit states of interest are not exceeded, Eurocode 7 acknowledges four approaches:

(a) use of calculations (EN 1997, Clause 2.4);

(b) adoption of prescriptive measures (EN 1997, Clause 2.5);

(c) experimental models and load tests (EN 1997, Clause 2.6);

(d) an observational method (EN 1997, Clause 2.7).

The above approaches may be used in combination. For example, load tests (i.e., approach c) may be adopted to determine the pile resistance. Then, the obtained value of pile resistance may be fed into calculations (i.e., approach a) to design the pile foundations (Orr and Farrell 1999). The first approach, use of calculations, is the most common approach in practice (Frank et al. 2004), and hence, many parts of Eurocode 7 are devoted to this approach.

**2.3 Geotechnical design by calculation **

Figure 2.1 depicts some of the main components required for design by calculation.

These components include actions (in Eurocodes, the term “action” roughly implies

loads and imposed displacements), material properties, and geometrical data. These components are then fed into calculation models to verify that limit states of interest are not exceeded. Eurocode 7 recognizes three calculation models: analytical models, semi- empirical models, and numerical models [EN 1997, Clause 2.4.1(5)].

Figure 2.1 Outline of design by calculation in Eurocode 7

One of the main challenges in design by calculation is choosing appropriate values for material properties. In Eurocode’s terminology, this appropriate value is called “design value” of a material property. In other words, the value of a material property chosen to be fed into calculation models is called “design value”. Eurocode 7 presents a four-steps procedure for choosing design values of material properties (Figure 2.2).

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Figure 2.2 Procedure for choosing design values (modified after Orr 2000)

As shown in Figure 2.2, Eurocode 7 assigns a distinct term to the value of material property obtained at each step. These terms are: (1) measured value, (2) derived value, (3) characteristic value, and (4) design value.

(1) A measured value is a value measured in a laboratory or field test. The depth of the water table or CPT cone tip resistance (qc) are examples of measured values.

Sometimes, such as the depth of the water table, the measured value itself is the final parameter required for design, i.e., (design value) = (measured value).

(2) A derived value is a value obtained from measured value by employing theoretical or empirical equations. Cohesion and friction angle obtained from triaxial test using Mohr-Coulomb theory, or undrained shear strength (su) obtained from CPT qc using empirical correlation are examples of derived values.

(3) A characteristic value, in a nutshell, is a value that represents the behavior of ground under limit state of interest. The focus of the current study is on selecting characteristic values, and it will be reviewed in details in Section 2.5.

(4) A design value is obtained by applying a partial factor to a characteristic value.

The values of partial factors are prescribed by the Eurocodes for different scenarios. The next section briefly reviews partial factors.

**2.4 Partial factors **

Parameters involved in a design calculation may not be equally uncertain. Moreover, they may not equally influence the limit state. Hence, in the limit state design framework, the global (traditional) factor of safety is broken up into a sequence of partial factors, one for each parameter (Fenton et al. 2016). The purposes of partial factors have been widely discussed in the literature (e.g., Bolton 1993, Meyerhof 1994, Simpson 2000, Bauduin 2003, Simpson et al. 2009). Simpson (2000) stated that “…it has proved very difficult to reach agreement on where in the calculation process partial factors should be applied”. Nevertheless, the fundamental role of partial factors is to provide a safety margin required by society (ideally, a reliability level as close as possible to the target reliability level. See Table 2.2 for target reliability level). Some other suggested roles of partial factors are as follows:

(1) to account for uncertainties in calculation models (model uncertainty). Note that Eurocode 7 does not prescribe values of partial factors for a specific calculation

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model. Instead, it requires that all calculation models “shall be either accurate or err on the side of safety” [EN 1997, Clause 2.4.1(6)]. Some researchers (e.g., Bauduin 2003) suggest that partial factors should be independent of the adopted calculation model.

(2) to account for robustness. A concise definition of robustness is given by TC205/TC304 Working Group (2016): “ability of a structure to withstand adverse events that are unforeseen but of a magnitude such that society will expect that our designs can accommodate them, having tolerance against mistakes within the design process and during construction”. Although Eurocode 7 does not use the word “robustness”, it considers one aspect of robustness in Clause 2.4.6.3(1): partial factors should “include an allowance for minor variations in geometrical data”.

(3) to account for uncertainty in material properties or actions. This suggested role of partial factors may overlap with the role of the characteristic values. The current study opines that uncertainties in the determination of soil parameters (including spatial variability, measurement errors, transformation uncertainty, and statistical uncertainty) should be handled by the characteristic values.

Annex C of the head Eurocode (EN 1990) reviews various methods for calibrating partial factors (Figure 2.3). According to Clause C4(4), the current partial factors in the Eurocodes are mainly based on the “method ɑ” (historical or empirical methods). Method ɑ ensures that the wealth of past experience is not lost in the (new) limit state design framework. Method b or method c (probabilistic methods) is considered to be used in future developments of the Eurocodes. According to Clause C4(3), when probabilistic methods are used, partial factors should be calibrated in such

a way that the reliability level is as close as possible to the target reliability level (See Table 2.2 for target reliability level).

Figure 2.3 Different methods for calibrating partial factors (after EN 1990)

The choice of the reliability level for a particular structure depends on, among other factors, the possible consequences of failure [EN 1990, Clause 2.2(3)]. Annex B of the head Eurocode defines three consequences classes (CC) as listed in Table 2.1.

Clause B3.1(3) further notes that different components of a structure may have different CCs. For example, following Gulvanessian et al. (2012), consider a hotel building whose public rooms (used for social events) have large spans. In this case, the structural elements supporting the public rooms may be categorized as CC3, while the structural elements supporting the hotel bedrooms may be categorized as CC2. The hotel itself as

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a whole may be categorized as CC2 or CC3, depending on its importance. Similarly, geotechnical components of a structure may have different CCs.

Table 2.1 Definition of consequences classes (after EN 1990) Consequences

Class Description Examples of buildings and civil

engineering works

CC3

**High consequence for loss of human life, **
or economic, social or environmental
**consequences very great **

Grandstands, public buildings where consequences of failure are high (e.g. a concert hall)

CC2

**Medium consequence for loss of human **
life, or economic, social or environmental
**consequences considerable **

Residential buildings, public buildings where consequences of failure are medium (e.g. an office building)

CC1

**Small consequence for loss of human **
life, or economic, social or environmental
**consequences small or negligible **

Agricultural buildings where people do not normally enter (e.g.

storage buildings), greenhouses

The head Eurocode also defines reliability classes (RC) using the reliability index (β) concept. Table 2.2 lists the minimum values of β for ultimate limit states for each RC, as recommended by the head Eurocode. Note that β is related to the probability of failure (Pf) by:

### ( )

*P**f* = − (2.1)

where Φ is the cumulative density function of the standard normal random variable.

Hence, the numbers in parentheses in Table 2.2 are the associated Pf for each β. The
head Eurocode notes that a design using the partial factors given in the Eurocodes “is
considered generally to lead to a structure with a β value greater than 3.8 [i.e., Pf smaller
than 7×10^{-5}] for a 50 year reference period” [EN 1990, note to Table B2].

Table 2.2 Recommended minimum values of reliability index β for ultimate limit states (modified after EN 1990)

Consequences Class

Reliability Class

Minimum values for β (associated probability of failure Pf) 1 year reference period 50 years reference period

CC3 RC3 5.2 (1×10^{-7}) 4.3 (9×10^{-6})

CC2 RC2 4.7 (1×10^{-6}) 3.8 (7×10^{-5})

CC1 RC1 4.2 (1×10^{-5}) 3.3 (5×10^{-4})

Although most of the Eurocodes are devoted to design based on partial factors and characteristic values, the head Eurocode states in Clause 3.5(5) that “as an alternative, a design directly based on probabilistic methods may be used”. However, this is not the focus of the current study. The current study focuses on the determination of characteristic values, which is discussed in the next section.

**2.5 Characteristic values of geotechnical parameters **

Eurocode 7 defines the characteristic value of a geotechnical parameter as “a cautious estimate of the value affecting the occurrence of the limit state” [EN 1997, Clause 2.4.5.2(2)]. This definition is worded to allow experienced designers to exercise their judgment, whilst directing less experienced designers to choose safe and reasonable values (Simpson and Driscoll 1998). Hicks (2013) commented on some clauses in Eurocode 7 regarding characteristic value (Table 2.3).

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Table 2.3 Extracts from Section 2.4.5.2 of Eurocode 7 (after Hicks 2013)

No. Clause Hick’s comments

(4)P The selection of characteristic values for geotechnical parameters shall take account of

the following:

• geological and other background information, such

as data from previous projects; Reduces uncertainty

• the variability of measured property values and other relevant information, e.g. from existing knowledge;

Should account for soil variability

• the extent of the field and laboratory investigation; Affects uncertainty

• the type and number of samples; Affects uncertainty

• the extent of the zone of ground governing the behavior of the geotechnical structure at the limit state being considered;

Spatial aspect of soil variability is important

• the ability of the geotechnical structure to transfer loads from weak to strong zones in the ground

Characteristic values are problem-dependent

(7)

The zone of ground governing the behavior of a geotechnical structure at a limit state is usually much larger than a test sample or the zone of ground affected in an in-situ test. Consequently the value of the

governing parameter is often the mean of the range of values covering a large surface or volume of the ground.

The characteristic value should be a cautious estimate of this mean value

Important to consider the mean over potential failure surfaces; this could be very different to the mean over the domain of influence

(8)

If the behavior of the geotechnical structure at the limit state considered is governed by the lowest or highest value of the ground property, the characteristic value should be a cautious estimate of the lowest or highest value occurring in the zone governing the behavior.

Extreme scenario implying local failure

(11)

If statistical methods are used, the characteristic value should be derived such that the calculated probability of a worse value governing the occurrence of the limit state under consideration is not greater than 5%.

5% refers to probability of failure of the structure; not to parameter values

NOTE: In this respect, a cautious estimate of the mean value is a selection of the mean value of the limited set of geotechnical parameter values, with a confidence level of 95%; where local failure is concerned, a cautious estimate of the low value is a 5% fractile.

Percentages refer to parameter values; not to structure performance

There are two main aspects in the definition of characteristic values: (a) a cautious estimate, and (b) the value affecting the occurrence of the limit state. The first aspect is related to uncertainties, such as spatial variability (Vanmarcke 1977a), transformation uncertainty (Phoon and Kulhawy 1999), statistical uncertainty, and model uncertainty. The second aspect is related to mechanics (e.g., spatial variability, force equilibrium, boundary conditions, etc.).

For the uncertainty (or statistical) aspect, a cautious estimate can be selected based on statistical methods, as Clause 2.4.5.2(11) elaborates that “If statistical methods are used, the characteristic value should be derived such that the calculated probability of a worse value governing the occurrence of the limit state under consideration is not greater than 5%”. Note that the value of 5% (instead of, for example, 10% or 20%) is adopted from the head Eurocode (EN 1990). The determination of the “cautious estimate” requires the statistics (e.g., mean and variance) of various uncertainties. To address this, a growing body of research aims to derive such statistics based on site investigation data (Yang, Xu, and Wang 2017; Zhao et al. 2018; van der Krogt, Schweckendiek, and Kok 2019). It is clear that spatial variability of soil properties plays a role in the statistical aspect of the characteristic value, because spatial variability constitutes one important source of uncertainties.

For the physical or mechanical aspect, Eurocode 7 provides some guidelines on the “value affecting the occurrence of the limit state”. For instance, Clause 2.4.5.2(4) lists factors to account for when selecting the characteristic value, which includes, among other factors, “the extent of the zone of ground governing the behavior of the geotechnical structure at the limit state being considered”. This is well known since Boussinesq established a solution for stress distribution in a homogeneous and isotropic subgrade subjected to a vertical point load on the ground surface in 1885. Clause

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2.4.5.2(7) further states that “the value of the governing parameter is often the mean of a range of values covering a large surface or volume of the ground”. This is in line with the proposition made by Vanmarcke (1977a) that the performance of a geotechnical structure is often governed by the spatial average of soil properties over some influence zone (such as the spatial average of Young’s modulus over a soil volume beneath a footing or the spatial average of shear strength along a critical slip curve) rather than by the property at a certain point, referred to as the “point property”. It is clear that spatial variability of soil properties plays a role in the mechanical aspect of the characteristic value. It is not surprising that spatial variability is influential in both aspects, statistical and mechanical, although the former statistical aspect is more widely studied in the literature. In fact, most existing simplified formulas for characteristic value presented in the next section are based on statistical considerations. The mechanical aspect has not fully been considered. It is important to point out that the two aspects interact very strongly with each other.

Although the definition of characteristic value is sensible, Phoon (in a discussion
in Wang 2017) opined that “there is an under-stated difficulty in making this statement
*[definition of characteristic value] sufficiently concrete for codification”. In fact, there *
is a considerable degree of subjectivity in the selection of characteristic value using
Eurocode 7. As an example, this is demonstrated by Bond and Harris (2008), where
they asked more than a hundred engineers to estimate the characteristic value of
undrained shear strength for a particular soil profile. Figure 2.4 illustrates the outcome.

Solid dots represent data points, while gray lines represent engineers’ estimations. It can be seen that there is a wide range of estimations, and engineers are inconsistent when it comes to selecting the characteristic value.

Figure 2.4 Engineers’ interpretation of the characteristic value for a soil profile. Solid dots represent data points, while gray lines represent engineers’ interpretations (after

Bond and Harris 2008)

As discussed in Orr (2012), based on a questionnaire survey, 94% of engineers asked for more guidance on selecting characteristic soil parameter to be considered for the next revised version of Eurocode 7, which is due for publication in 202x. In fact, as listed in Table 1, it is the second most challenging and hot topic in Eurocode 7.

Considering these difficulties, a number of simplified formulas for determining

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characteristic values have been proposed in the literature, which is reviewed in the next section.

Table 2.4 Top five topics as voted by the UK engineers to be considered for the next revised version of Eurocode 7 (after Orr 2012)

Rank Topic % of respondents

who chose topic

1 Add new parts to Eurocode 7 covering detailed design (e.g.

footings, walls, pile and slopes) 100

2 Improve general guidance on selecting characteristic soil parameter 94

3 Improve guidance on selecting water pressures 70

4 Add new parts to Eurocode 7 covering reinforced soil 59

5 Simplify/reduce number of DAs 43

**2.6 Statistics-based characteristic value **

The determination of the characteristic value for a soil property (X) as defined by
Eurocode 7 has been discussed in the literature. The simplest method is to take the
*characteristic value as the 0.05-fractile of the point property: *

### (

^{1}

^{0.05}

^{N}### )

*X**k* = + *V* (2.2)

where Xk is the characteristic value; is the point mean of X; 0.05N = -1.645 is the 0.05-fractile of the standard normal variable; V is the coefficient of variation (COV) for the spatial variability. This method of determining Xk is considered to be overly conservative (Orr 2017; Varkey et al. 2020). Schneider (1997) proposed the following modified version:

### (

^{1 0.5}

### )

*X**k* = − *V* (2.3)

This equation aims to provide Xk value that is a cautious estimate of the mean value with a confidence level of 0.95 corresponding to 13 test results (Orr 2017).

Schneider and Schneider (2013) proposed the following Xk formula that addresses the spatial averaging of X over an influence zone. This formula also considers other sources of uncertainties, such as measurement errors, transformation uncertainty, and statistical uncertainty:

### ( )

( ^{2} )

0.05

0.05

k ln 1+ 2

1 if is normal

1+ if is lognormal

*N*

*total*

*N*

*total*
*V*

*total*

*V* *X*

*X*

*e*^{} *V* *X*

^{}

+

=

(2.4)

where Vtotal is the COV of the total uncertainty (Phoon and Kulhawy 1999):

2 2 2 2 2

*total* *meas* *trans* *stat*

*V* = *V* +*V* +*V* +*V* (2.5)

^{2} is the variance reduction factor for the spatial averaging; V, Vmeas, Vtrans, and Vstat are
the COVs for the spatial variability, measurement errors, transformation uncertainty,
and statistical uncertainty, respectively.

More recently, an evolution group of the European Committee for Standardization proposed the following formula (Orr 2017):

### ( )

k = − − * _{extr}*

_{v}

_{v}*X* *L* (2.6)

where extr is the expected extreme value of X; v is the vertical scale of fluctuation (SOF); Lv is the vertical dimension of the influence zone; is a factor accounting for the extent and quality of site investigation ( is 0.5 for high quality, 0.75 for average quality, and 1 for no local investigation). Orr (2017) further proposed extr to be 3 standard deviations away (lower) from , which leads to:

k = − 3 _{v}_{v}

*X* *a V* *L* (2.7)

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It is clear that mechanics, specifically how the trajectory (and other possible finer geometrical features) of the critical slip curve is affected by spatial variability, has not been considered. The performance of the above simplified formulas will be compared with the proposed simple formulas in the current study (see Chapter 6). As will be shown later, the proposed new formulas consider both statistics and mechanics and their interactions correctly.

**2.7 Reliability-based characteristic value **

Hicks and Samy (2002) proposed a method of simulating the “effective property” for a
spatially variable soil mass based on the random finite element method (RFEM) (Fenton
and Griffiths 2008). They considered a slope with spatially variable shear strength
modeled by the RFEM. The effective property for a particular random field realization
*is defined as the homogeneous input property that matches the same response as the *
RFEM. For the slope example investigated by Hicks and Samy (2002), the input is the
soil shear strength, and the response is the factor of safety (FS). The concept of effective
property was also adopted by Griffiths et al. (2012) to homogenize a soil cube (input =
soil modulus, response = deformation of the soil cube) and by Ching et al. (2017a) to
homogenize a deep excavation problem (input = shear strength, response = FS for basal
heave). Note that Ching et al. (2017a) adopted the term “mobilized shear strength” for
the effective property. The current study also adopts the term “mobilized shear strength”

in order to maintain consistency with Ching et al. (2017a).

The probability density function (PDF) of the mobilized shear strength is generally different from that of the point shear strength (Figure 2.5). There are two important factors affecting the PDF of the mobilized shear strength: spatial averaging and weak-zone seeking (Hicks and Samy 2002; Ching and Phoon 2013a; Ching et al.

2014; Hicks et al. 2019). The first factor is the spatial averaging effect (Vanmarcke 1977a) along the potential slip curve. This reduces the variance of the mobilized shear strength but preserves the mean (dark dashed PDF in Figure 2.5). The second factor is the tendency for the failure path to seek out mechanically admissible weak zones. The failure path is the solution of a boundary value problem. It goes without saying that the weak-zone seeking mechanism must be constrained by mechanical considerations.

“Weak-zone seeking” is referred in this specific sense from hereon. This reduces the mean of the mobilized shear strength (dark solid curve in Figure 2.5). Both factors are fully addressed by the RFEM (Hicks and Samy 2002; Hicks et al. 2019). The phenomenon of a “worst-case” SOF (Ching et al. 2017a; Zhu et al. 2019) observed in the RFEM is the consequence of the tradeoff between these two factors (Ching et al.

2014). This phenomenon will be reviewed in Section 2.10. It is important to note that the simplified formulas in Eq. (2.2) to Eq. (2.7) ignore the second factor (weak-zone seeking), which is potentially unconservative.

Figure 2.5 Probability distributions of point shear strength and mobilized shear strength

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In essence, the mobilized shear strength is the soil shear strength that is effectively exhibited by a spatially variable soil mass. It is easy to imagine the statistical (random field properties) and mechanical (spatial distribution of soil properties, boundary conditions) factors governing the mobilized shear strength and the added complexity arising from their interactions. The mobilized shear strength for a spatially variable soil can be simulated by the RFEM (to be elaborated in Section 4.7) (Hicks and Samy 2002; Hicks 2013; Hicks et al. 2019). The mobilized shear strength affects the occurrence of the limit state, hence it is sensible to adopt the 0.05-fractile of the mobilized shear strength as the characteristic value of the shear strength. The 0.05- fractile of the mobilized shear strength simulated by the RFEM is called the reliability- based characteristic value (Hicks and Samy 2002; Hicks 2013; Hicks et al. 2019), denoted by XkRFEM in the current study. The reliability-based characteristic value fully satisfies the spirit of Eurocode 7’s Clause 2.4.5.2(2): “a cautious estimate of the value affecting the occurrence of the limit state”. However, the simulation of XkRFEM requires the RFEM which is very costly. In addition, the reliability-based characteristic value is problem specific.

The possibility of expressing XkRFEM by a simplified formula similar in format to those in Eq. (2.2) to Eq. (2.7) is of significant practical interest, because it provides a consistent and more general solution that respects both statistics and mechanics without incurring the RFEM cost. In fact, the value of a characteristic value is diminished under the scenario where a full probabilistic analysis is practical. Figure 2.6 illustrates the relation between conventional statistics-based, mobilization-based (an outcome of the current study based on the weakest-path model, see Chapter 6), and reliability-based characteristic values. The conventional characteristic value is purely related to the input random field. It is cheap to calculate but it is not related to the value affecting the