The effects of multi-directional loading
on soil liquefaction
Mengchen Sun
Department of Engineering
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
St Edmund’s College May 2019

To my loving family . . .

Declaration
I hereby declare that except where specific reference is made to the work of others, the
contents of this dissertation are original and have not been submitted in whole or in part
for consideration for any other degree or qualification in this, or any other university. This
dissertation is my own work and contains nothing which is the outcome of work done in
collaboration with others, except as specified in the text and Acknowledgements. This
dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes,
tables and equations and has fewer than 150 figures.
Mengchen Sun
May 2019

Acknowledgements
I would like to express my deep gratitude to my supervisor, Dr. Giovanna Biscontin, for her
insightful suggestions and enthusiastic supervision on this research work, which inspired me
a lot and convinced me that the excitement of research is anything but exaggerated.
I would like to thank technicians John Chandler, Kristian Pether, Mark Smith, Chris
McGinnie, Richard Adams at the Schofield centre, who helped me to install instruments, test
the apparatus and machine testing accessories. I would like to thank Cooper Technology for
modifying the testing apparatus, and New Era Machining for machining the redesigned top
caps.
I wish to extend my thanks to the lovely colleagues in the research group: Dr. Yingyan
Jin, Dr. Jeffery Lau, Dr. Stephan van Eeden, Dr. Ramesh Kannan Kandasami, Dr. Ying Chen,
Gianmario Sorrentino, Christopher Wilkes, Charalampos Konstantinou, Eman Al-Swaity, Dr.
Khalid Alhaj Abdalla, and lately Jikai Guo, Helen Sims for making this group so amazing. I
wish to thank Prof. Gopal Madabhushi, Dr. April Bowmen, Dr. Gue Chang Ye, Dr. Njemile
Faustin, Dr. Stefan Ritter, Dr. Orestis Adamidis, Dr. Srikanth Madabhushi, Dr. Talia da Silva,
Dr. Raz Jabary, Dr. Peter Kirkwood, for their very kind encouragement and discussions at the
early stage of my study. I would like to thank Mrs Magdalena Charytoniuk, Chuhan Deng,
Fiona Hughes, Thejesh Kumar Garala, Samy Garcia Torres, Deryck Y.K. Chan, Jad Boksmati,
Andrei Dobrisan and all the colleagues in Schofield centre for making our laboratory such a
lovely place. I would also like to acknowledge the great colleagues who were not based at
Schofield centre, Dr. Yuze Wang, Bingyu Zhao, Dr. Xiang Sun, and Tsubasa Sasaki, to name
a few.
I would also like to thank all my friends for their friendship and help, which made my
life in Cambridge marvelous and memorable. My gratitude goes to Dr. Hongbo Sun, Dr. Qi
Wang, Dr. Luoke Hu, Zhiyou Wu, Xiao Xiao, Luming Fan, Kaiqi Hu, Wenkai Guo, Dr. Jinqi
Fu, Dr. Jianzhao Geng, Dr. Shanshan Yan, Chao Li, Zixin Huang, Kexin Zhang, Benyi Cao,
Tao Zhang, Yunhui Zhang, Wenting Mao, to mention but a few.
And I would like to thank my parents for their everlasting support.

Abstract
Four decades after it was investigated experimentally for the first time, the influence of multi-
directional shearing on soil behaviour, especially the undrained behaviour of saturated sands,
remains one of the least understood areas in geotechnics. This study developed a new test
database on Hostun sand with the modified multi-directional simple shear testing apparatus.
The results of uni-directional tests were examined under available analysis frameworks to
validate the performance of the adopted testing techniques. Multi-directional tests were
analysed and compared with their uni-directional counterparts to illustrate the remarkable
change in soil response when loading paths change from uni-directional to multi-directional.
A three-dimensional τ−σ ′v space was established as the extrapolation of the 2D τ−σ ′v
plane, with a 3D conical failure surface derived from 2D failure lines. The concept of the
failure cone was validated by the tests with various loading paths. The failure cone hypothesis
explain the unsolved questions, including why bi-directional linear stress paths do not reach
the failure line in the 2D τ −σ ′v plane and why an inverse correlation exists between the
limiting excess pore pressure ratio and shear stress amplitude.
A bimodal phenomenon was identified in terms of the phase transformation of sand under
multi-directional shearing conditions. Phase transformation states form a conical surface,
like failure states, in the tests with circular, oval and figure-8 loading paths but do not fall on
a cone in uni-directional and bi-directional linear tests.
The effects of multi-directional loading on soil liquefaction were further investigated
from the perspective of liquefaction criteria, liquefaction resistance, development of excess
pore pressure and shear strain, as well as the degradation of shear modulus. The definitions
of shear stress, shear strain and shear modulus were re-examined in multi-directional loading
scenarios. Liquefaction resistance assessment based on N−CSR curves was examined and
potential alternative discussed. The shear modulus degradation predictions of Hardin and
Drnevich (1972) were found to fit with the newly-proposed multi-directional secant shear
modulus.
This study provides novel perspectives to understand the undrained soil behaviour under
multi-directional loading and has useful implications for future potential modelling research.

Table of contents
List of figures xv
List of tables xxi
Nomenclature xxiii
1 INTRODUCTION 1
1.1 Motivation of the research . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Organisation of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . 5
2 LITERATURE REVIEW 7
2.1 Fundamentals of soil liquefaction . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Undrained monotonic response of saturated sands . . . . . . . . . . 7
2.1.2 Undrained cyclic response of saturated sands without fines . . . . . 10
2.1.3 Interpretation of undrained monotonic behaviour of saturated sand
in Critical State Theory . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Element-level laboratory testing for soil liquefaction . . . . . . . . . . . . 18
2.2.1 Triaxial testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Simple shear testing . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Torsional shear testing . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Multi-directional loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING 37
3.1 Material tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Review on specimen preparation technique . . . . . . . . . . . . . . . . . 38
3.2.1 Moist Tamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Air Pluviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Under-compaction Moist Tamping . . . . . . . . . . . . . . . . . . 41
xii Table of contents
3.2.4 Water Pluviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.5 Slurry Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.6 Air pluviation with CO2 flushing . . . . . . . . . . . . . . . . . . . 44
3.3 Inflation problem in undrained simple shear tests . . . . . . . . . . . . . . 44
3.4 Testing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Multi-directional direct simple shear apparatus . . . . . . . . . . . 49
3.4.2 Specimen preparation procedures . . . . . . . . . . . . . . . . . . 50
3.4.3 Simple shear testing procedures . . . . . . . . . . . . . . . . . . . 54
3.5 Data correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Liquefaction criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7 Parameter definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8 Summary of tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS 67
4.1 Uni-directional linear tests . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Uni-directional tests without static shearing . . . . . . . . . . . . . 67
4.1.2 Uni-directional tests with static shearing . . . . . . . . . . . . . . . 74
4.2 Liquefaction resistance of uni-directional tests . . . . . . . . . . . . . . . . 84
4.3 Excess pore pressure generation in uni-directional tests . . . . . . . . . . . 89
4.4 Shear modulus degradation in uni-directional tests . . . . . . . . . . . . . . 92
4.5 Discussion on uni-directional tests . . . . . . . . . . . . . . . . . . . . . . 96
5 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS 99
5.1 Bilinear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Multi-directional tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.1 Tests with circular loading path . . . . . . . . . . . . . . . . . . . 111
5.2.2 Tests with oval loading path . . . . . . . . . . . . . . . . . . . . . 125
5.2.3 Test with figure-8 loading path . . . . . . . . . . . . . . . . . . . . 130
5.2.4 Discussions on the interpretation of multi-directional tests . . . . . 133
6 THREE-DIMENSIONAL τ−σ ′v SPACE 137
6.1 Failure cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.1 The concept of failure cone . . . . . . . . . . . . . . . . . . . . . . 137
6.1.2 Failure states of bi-directional linear tests and multi-directional tests 143
6.1.3 Determination of failure states with failure cone . . . . . . . . . . . 150
6.2 Phase transformation in 3D τ−σ ′v space . . . . . . . . . . . . . . . . . . . 151
6.2.1 The validity of phase transformation cone in multi-directional tests 152
Table of contents xiii
6.2.2 The invalidity of phase transformation cone in bi-directional linear
tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.3 Discussions on the phase transformation in multi-directional loading
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING 165
7.1 Liquefaction criterion for multi-directional simple shear tests . . . . . . . . 166
7.2 The effects of multi-directional loading on liquefaction resistance . . . . . 174
7.3 The development of excess pore pressure in multi-directional simple shear tests185
7.3.1 Maximum excess pore pressure ratios in multi-directional simple
shear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.3.2 Development of excess pore pressure in multi-directional simple
shear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.4 The development of shear strain in multi-directional simple shear tests . . . 194
7.5 The degradation of Shear modulus in multi-directional simple shear tests . . 205
8 CONCLUSIONS AND FUTURE WORK 211
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
References 217

List of figures
2.1 Different types of undrained monotonic response of sands (Castro, 1969) . . 8
2.2 Characteristic states of undrained shearing behaviour of sand (Murthy et al.,
2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Stress-controlled cyclic triaxial test results for loose and dense sands (modi-
fied from Seed and Lee (1966)) . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Stress parth and stress-strain curve for loose sand (Ishihara, 1985) . . . . . 12
2.5 Stress parth and stress-strain curve for dense sand (Ishihara, 1985) . . . . . 13
2.6 Typical records of the measurements of shear stress and excess pore pressure
(Ishihara and Yasuda, 1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Schematic diagram of critical state theory (Yamamuro and Lade, 1998) . . . 15
2.8 Undrained monotonic behaviour of sands with different initial confining
pressure (Yamamuro and Covert, 2001) . . . . . . . . . . . . . . . . . . . 16
2.9 Defination of state parameter ψ (Been and Jefferies, 1985) . . . . . . . . . 17
2.10 Idealized field loading condition (Seed, 1979) . . . . . . . . . . . . . . . . 19
2.11 Stress conditions for triaxial test on saturated sand under simulated earth-
quake loading conditions (Seed and Lee, 1966) . . . . . . . . . . . . . . . 20
2.12 A hollow cylinder specimen under torsional shear (Ampadu and Tatsuoka,
1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.13 Seismometer traces recorded in Dry Canyon Dam during 1971 San Fernando
Earthquake (Pyke et al., 1975) . . . . . . . . . . . . . . . . . . . . . . . . 25
2.14 The schematic diagram of gyratory loading path and settlement of sands
(Pyke et al., 1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.15 Pore pressure development (Seed et al., 1978) . . . . . . . . . . . . . . . . 26
2.16 Comparison between multidirectional and unidirectional shaking using the
proposed method (Tokimatsu and Yoshimi, 1982) . . . . . . . . . . . . . . 28
2.17 Schematic illustration of idealized multi-directional loading (Kammerer et al.,
2004b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
xvi List of figures
2.18 Relationship between shear stress ratio and limiting excess pore pressure
ratio (Kammerer, 2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Shape of Hostun Sand particles (Ezaoui and Benedetto, 2009) . . . . . . . 38
3.2 Particle size distribution curve of tested Hostun Sands . . . . . . . . . . . . 39
3.3 Inflation of the rubber membrane (Ishihara and Yamazaki, 1980) . . . . . . 45
3.4 The photo of water pouch forming in a sample which has outward water
pressure gradient of 60 kPa . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 The photo of reinforced membrane deformed by outward water pressure
gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 The schematics of a soil sample in preparation and after preparation . . . . 48
3.7 A photo of the multi-directional direct simple shear testing apparatus . . . . 51
3.8 A photo of the testing device with a sample mounted . . . . . . . . . . . . 52
3.9 The measurements of horizontal load and displacement load in a pilot test . 56
3.10 Filtered load versus X displacement . . . . . . . . . . . . . . . . . . . . . 57
3.11 Graphical illustration of a figure-8 loading path and the definition of direc-
tions and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 The stress paths of the four monotonic uni-directional tests . . . . . . . . . 68
4.2 The development of excess pore pressure in the four monotonic uni-directional
tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 The stress-strain relationships of the four monotonic uni-directional tests . . 70
4.4 The development of shear strain and excess pore pressure of test LC10 . . . 71
4.5 The stress path of test LC10 . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 The 4-way plot of the 17th and 18th cycle of test LC10 . . . . . . . . . . . 73
4.7 The stress-strain response of test LC10 . . . . . . . . . . . . . . . . . . . . 74
4.8 The development of shear strain and excess pore pressure of test LS7 . . . . 75
4.9 The variation of vertical stress and vertical LVDT with time in test LS7 . . 76
4.10 The stress path of test LS7 . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.11 The 4-way plot of the 25th and 26th cycle of test LS7 . . . . . . . . . . . . 79
4.12 The stress-strain response of test LS7 . . . . . . . . . . . . . . . . . . . . 80
4.13 The stress path of test LS11 . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.14 The stress-strain response of test LS11 . . . . . . . . . . . . . . . . . . . . 82
4.15 The stress path of test LS10 . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.16 The stress-strain response of test LS10 . . . . . . . . . . . . . . . . . . . . 84
4.17 Number of cycles to liquefaction versus cyclic stress ratio in uni-directional
cyclic tests with no static shearing . . . . . . . . . . . . . . . . . . . . . . 85
List of figures xvii
4.18 The correction factor Kα versus initial static shear stress ratio α . . . . . . 88
4.19 The development of excess pore pressure ratio with normalized cycle number 90
4.20 The development of excess pore pressure ratio with normalized cycle number,
with two groups recognised . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.21 The relationship between the excess pore pressure ratio in the first cycle and
the number of cycles to liquefaction in uni-directional tests . . . . . . . . . 92
4.22 The shear modulus degradation with double-amplitude shear strain . . . . . 94
4.23 The degradation of shear modulus with excess pore pressure ratio . . . . . 95
5.1 The plan view of shear strain development of bi-directional test Bilinear-3,
Bilinear-5 and Bilinear-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Strike-direction stress-strain curves of test Bilinear-3, Bilinear-5 and Bilinear-8103
5.3 The generation of excess pore pressure in test Bilinear-3, Bilinear-5 and
Bilinear-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Stress paths of test Bilinear-3, Bilinear-5 and Bilinear-8 . . . . . . . . . . . 105
5.5 Stress paths of test Bilinear-6, Bilinear-7 and Bilinear-8 . . . . . . . . . . . 106
5.6 The relationship between the maximum excess pore pressure ratio and static
shear stress ratio for bi-directional linear tests, with the schematic for deriving
equation 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7 The effects of static shear stress ratio α on Kα in uni-directional tests and
bi-directional linear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.8 The relationship between the number of cycles to liquefaction and the gener-
ated excess pore pressure ratio in the first shearing cycle . . . . . . . . . . 109
5.9 Strike-direction Shear modulus degradation curves in bi-directional linear tests110
5.10 The plan view of shear stress and shear strain of test C20-3 . . . . . . . . . 112
5.11 The 4-way plot of test C20-3 . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.12 3D development of excess pore pressure ratio of test C20-3 . . . . . . . . . 114
5.13 The shear stress and excess pore pressure ratio of test C20-3 in polar coordi-
nate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.14 The development of excess pore pressure ratio of test C20-1 and C20-5 in
polar coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.15 The plan view of shear stress and shear strain of test C20-X20-1 . . . . . . 118
5.16 3D development of excess pore pressure ratio of test C20-X20-1 . . . . . . 118
5.17 The shear stress and excess pore pressure ratio of test C20-X20-1 in polar
coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.18 The 4-way plot of test C20-X20-1 . . . . . . . . . . . . . . . . . . . . . . 120
xviii List of figures
5.19 Maximum excess pore pressure ratio versus minimum total shear shear stress
in all circular tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.20 The development of excess pore pressure with normalised number of cycles
in all circular tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.21 The excess pore pressure ratio generated in the first cycle versus the number
of cycles required to trigger liquefaction in all circular tests . . . . . . . . . 123
5.22 The projections of stress paths and stress-strain relationships of test C20-3
on X and Y planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.23 The projections of stress paths and stress-strain relationships of test C20-
X20-1 on X and Y planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.24 The plan view of shear stress and shear strain of test E2010-1 . . . . . . . . 126
5.25 The 4-way plot of test E2010-1 . . . . . . . . . . . . . . . . . . . . . . . . 127
5.26 The projections of stress paths and stress-strain relationships of test E2010-1
on X and Y planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.27 3D development of excess pore pressure ratio of test E2010-1 . . . . . . . . 129
5.28 The plan view of shear stress and shear strain of test F8-3 . . . . . . . . . . 131
5.29 The projections of stress paths and stress-strain relationships of test F8-3 on
X and Y planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.30 3D development of excess pore pressure ratio of test F8-3 . . . . . . . . . . 133
5.31 The four-way plot of test F8-3 . . . . . . . . . . . . . . . . . . . . . . . . 134
6.1 The schematic of friction cone . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 The schematic of failure cone in 3D SSR−σ ′v space . . . . . . . . . . . . . 141
6.3 The 4-way plot of test C20-X20-1 (C.F. Figure 5.18) . . . . . . . . . . . . 142
6.4 The schematic of the failure cone intersected by a plane . . . . . . . . . . . 144
6.5 Stress paths of test Bilinear-8 . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.6 The stress path of test Bilinear-8 and hyperbolic conic section in 3D τ−σ ′v
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 The stress path of circular test C20-3 in 3D τ−σ ′v space . . . . . . . . . . 148
6.8 The stress path of circular test C20-X20-1 in 3D τ−σ ′v space . . . . . . . . 148
6.9 The stress path of circular test E2010-1 in 3D τ−σ ′v space . . . . . . . . . 149
6.10 The stress path of circular test F8-3 in 3D τ−σ ′v space . . . . . . . . . . . 149
6.11 The verification of phase-transformation cone hypothesis with 3D stress path
of test C20-X20-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.12 The verification of phase-transformation cone with 3D stress path of test
E2010-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.13 The verification of phase-transformation cone with 3D stress path of test F8-3157
List of figures xix
6.14 The strike-direction stress path of test Bilinear-6, Bilinear-7 and Bilinear-8
with the hypothesized hyperbolic phase-transformation envelopes . . . . . 159
6.15 The strike-direction stress path of test Bilinear-4, Bilinear-5 and Bilinear-13
with the hypothesized hyperbolic phase-transformation envelopes . . . . . 160
6.16 The results of test ms50cyck of Kammerer (2002) . . . . . . . . . . . . . . 161
7.1 The development of γtotal with cycle number in all tests in this study . . . . 169
7.2 The development of γtotal,DA with cycle number in all tests in this study . . 171
7.3 The development of γdir,DA with cycle number in all tests in this study . . . 173
7.4 Liquefaction resistance reflected in the form of NL−CSRdir . . . . . . . . . 176
7.5 The coefficient of multi-directional loading Kµ versus CSRdir . . . . . . . . 177
7.6 The coefficient of multi-directional loading Kµ versus aperture ratio AR for
tests with reversal parameter at 0 and 0.5 . . . . . . . . . . . . . . . . . . . 179
7.7 The coefficient of multi-directional loading Kµ versus reversal parameter RP
for tests with aperture ratio at 0 and 1 . . . . . . . . . . . . . . . . . . . . 180
7.8 Liquefaction resistance reflected in the form of NL−SSRtotal,max . . . . . . 182
7.9 Comparison of Kµ derived from NL−CSR and NL−SSRtotal,max relationships183
7.10 Relationships between limiting excess pore pressure ratios and total shear
stress ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.11 Correlation between limiting excess pore pressure ratios and total shear stress
ratios (modified from Kammerer (2002)) . . . . . . . . . . . . . . . . . . . 189
7.12 The development of normalised excess pore pressure ratio with normalised
cycle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.13 The development of normalised excess pore pressure ratio with normalised
cycle number categorised into three groups . . . . . . . . . . . . . . . . . 193
7.14 The relationship between the excess pore pressure ratios in the first cycle and
the number of cycles to liquefaction in all tests in this study . . . . . . . . . 195
7.15 Comparison of uni-directional linear, oval and circular tests with the same
CSRdir at 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.16 Comparison of three figure-8 tests with the same CSRdir at 0.15 . . . . . . . 198
7.17 Plan-view shear strain analysis of the 58th cycle in test E2010-3 . . . . . . 199
7.18 The increment of total shear strain versus excess pore pressure ratio in all tests202
7.19 Comparison of three circular tests with different initial static shearing . . . 204
7.20 The degradation of multi-directional secant shear modulus with multi-directional
double-amplitude shear strain . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.21 The degradation of shear modulus with excess pore pressure ratio . . . . . 208

List of tables
2.1 Idealized in situ boundary conditions for earthquake-induced soil liquefaction
(After Kammerer (2002)) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Properties of Hostun S28 Sand (Mitrani, 2006) . . . . . . . . . . . . . . . 37
3.2 The summary of transducer replacement . . . . . . . . . . . . . . . . . . . 49
3.3 Summary of the conducted tests . . . . . . . . . . . . . . . . . . . . . . . 62
4.1 Fitting parameters of NL−CSR curves . . . . . . . . . . . . . . . . . . . . 86

Nomenclature
AR Aperture ratio
CO2 Carbon Dioxide
CSR Cyclic shear stress ratio
CSRX Cyclic shear stress ratio in X direction
CSRY Cyclic shear stress ratio in Y direction
CSRα Cyclic shear stress ratio in the direction of static shearing
CSRdir The maximum CSR across all directions
CSR f ield Standardised liquefaction resistance
CSRperp Cyclic shear stress ratio in the direction perpendicular to that of CSRdir
CM Liquefaction resistance correction factor for earthquake magnitude
Dr Relative density
D50 Median particle diameter
G Secant shear modulus
G2 The secant shear modulus of the second loading cycle
Gs Specific gravity
Gmax Small-strain shear modulus
Is State index
K0 Lateral stress coefficient
xxiv Nomenclature
Kα Liquefaction resistance correction factor for initial static shearing
Kµ Liquefaction resistance correction factor for multi-directional loading
Kσ Liquefaction resistance correction factor for consolidation stress
K0,min The minimum K0 in a simple shear test
N Number of cycles
NL Number of cycles to liquefaction
PT Phase transformation
RP Reversal parameter
Rc Relative contractiveness
SSR Shear stress ratio
SSRX Shear stress ratio in X direction
SSRY Shear stress ratio in Y direction
SSRX ,0 Initial shear stress ratio in X direction
SSRY,0 Initial shear stress ratio in Y direction
SSRcyclic Shear stress ratio in the direction perpendicular to that of initial static shear
stress
SSRdip Shear strain ratio in the dip direction
SSRstrike Shear strain ratio in the strike direction
SSRtotal,min Maximum resultant shear stress ratio
SSRtotal,min Minimum resultant shear stress ratio
SSRtotal Resultant shear stress ratio
∆u Excess pore pressure
α Initial static shear stress ratio
δγtotal Resultant strain increment
Nomenclature xxv
εa Axial strain
γ Shear strain
γX Shear strain in X direction
γY Shear strain in Y direction
γDA Uni-directional double-amplitude shear strain
γdip Shear strain in the dip direction
γdir,DA Maximum double-amplitude shear strain across all directions
γdir Shear strain in a certain direction
γstrike Shear strain in the strike direction
γtotal,DA Double-amplitude total shear strain
γtotal Resultant shear strain
µs Coefficient of kinetic friction
φ ′ Angle of internal friction
φFL Angle of the failure line
φPT Angle of the phase-transformation line
ψ State parameter
σ1 Major principal stress
σ3 Minor principal stress
σv Vertical stress
σ ′v Effective vertical stress
σv,i Initial vertical stress
τ Shear stress
τX Shear stress in X direction
τY Shear stress in Y direction
xxvi Nomenclature
τcyclic Shear stress in the direction perpendicular to that of initial static shear stress
τdir,DA Maximum double-amplitude shear stress across all directions
τdir Shear stress in a certain direction
τstatic Initial static shear stress
τtotal Resultant shear stress
a,b Curve fitting parameters
c Cohesion
e Void ratio
eλ Void ratio from isotropic consolidation line
ecs Void ratio at the critical state
emax Maximum void ratio
emin Minimum void ratio
p′ Mean effective stress
q Deviatoric stress
ru Excess pore pressure ratio
ru,1 The maximum excess pore pressure in the first loading cycle
ru,max The maximum excess pore pressure ratio in the liquefaction cycle
ru,min The minimum excess pore pressure ratio in the liquefaction cycle
B-value Skempton’s B-value
Chapter 1
INTRODUCTION
1.1 Motivation of the research
Soil liquefaction is a seismic phenomenon experienced by sands, which results in severe
damage including foundation failure, sinking and overturning of structures as well as destruc-
tion of infrastructure. The loss of life and economic damage induced by soil liquefaction
is costly, so there has been much effort towards understanding the cause of liquefaction
and the behaviour of saturated sands under cyclic shearing. Early studies focusing on static
undrained tests have shown that the occurrence of soil liquefaction is closely related to
strain-softening undrained response of sands, which may lead to either unlimited or limited
runaway deformation (Castro, 1969). However, it has been widely demonstrated that strain
softening is not the exclusive cause of the excessive deformation under cyclic undrained
conditions. Cyclic accumulation of plastic strain, termed cyclic mobility, also occurs with or
without transient states of zero effective stress states (Castro, 1975; Seed, 1979; Vaid and
Chern, 1985; Vaid and Thomas, 1995). It is evident that the type and path of shearing can
actively influence the response of sands.
In terms of laboratory research with respect to soil liquefaction, direct simple shear
apparatuses, which have been utilised successfully for decades in characterising static and
dynamic soil properties, are often preferred, especially when it is desirable to produce a
condition where the rotation of the principal stress directions occurs continuously. Most
2 INTRODUCTION
existing simple shear apparatuses can only shear specimens in a single horizontal direction,
and thus preclude the consideration of multi-directional loading paths in experimental work.
However, it is widely recognised that earthquakes apply multi-directional cyclic shearing
on soils. The differences between the realistic loading conditions and their simplified
counterparts hampered the investigation of soil behaviour in actual earthquakes.
To improve the situation, several multi-directional direct shear devices have been devel-
oped and a limited number of multi-directional shearing tests on sands have been conducted.
Casagrande and Rendon (1978) developed a gyratory shear apparatus and observed that sand
behaviours differ greatly when multi-directional loading is involved. Ishihara and Yamazaki
(1980) developed another pioneering multi-directional simple shear apparatuses to investigate
the effects of the change of loading directions on liquefaction resistance. Subsequently, some
researchers used this apparatus to perform multi-directional simple shear tests, concluding
that rotation of stress directions tends to result in a lower liquefaction resistance (Fukutake
and Matsuoka, 1989; Ishihara and Nagase, 1988; Nagase and Ishihara, 1987; Tokimatsu
and Yoshimi, 1982; Yamazaki et al., 1985). Later, DeGroot (1992) developed a new type
of multi-directional direct simple shear (MDSS) apparatus at Massachusetts Institute of
Technology. Although they used this device to investigate the effect of different directions
of static shear loading and ice shear loading on the response of Boston Blue Clay, their
observation that the peak undrained resistance of clay was affected significantly by the angle
between the two shear forces is similar to the preceding results for sand.
Simultaneously, Boulanger and his colleagues, developed another type of multi-directional
simple shear apparatus, the UC Berkeley bi-directional simple shear device (UCB-2D) to
study the effects of multi-directional loading on the strength of modified Sacramento River
Sand (Boulanger, 1990; Boulanger et al., 1993; Boulanger and Seed, 1995). They observed
that stress reversal can exert significant influence on generation of pore water pressure and
liquefaction resistance. One of the most comprehensive and systematic investigations into
the effect of multi-directional loading and initial static driving shear on the response of sands
and clay was conducted with this device by Kammerer (2002), Wu (2002), Biscontin et al.
(2004) and colleagues. They confirmed an inverse linear relationship between shear stress
1.1 Motivation of the research 3
amplitude and limiting pore pressure ratio reported by Boulanger (1990). They also proposed
that stress rotation aids particle rearrangement and densification of soil, and that the initial
static shear affects the extent of stress rotation. Moreover, the stress rotation appears to be
even more important than cyclic stress amplitude in terms of pore pressure generation and
shear strain.
Rutherford and Biscontin (2013) developed a new multi-directional direct simple shear
apparatus (TAMU-MDSS), which Rutherford (2012) used to investigate the cyclic shear
response of Gulf of Mexico Clay. Later, Bernhardt (2013) used this device to validate the
numerical simulation with the discrete element method (DEM).
More multi-directional simple shear testing database was developed with the variable
direction dynamic cyclic simple shear device (VDDCSS) by GDS (Li et al., 2016; Mirbaha,
2017; Rudolph et al., 2014). However, they conducted equivalent undrained tests rather than
actual undrained tests. They adopted constant height conditions, regarding the reduction of
vertical stress as the generation of excess pore pressure. Although such a boundary condition
is accepted as an effective alternative to actual undrained condition, the actual discrepancies
between the methods are difficult to estimate.
Despite all these efforts, multi-directional simple shear test data are still scarce and the
effects of multi-directional loading are far from being fully understood. Firstly, there has
not been a theoretical framework under which multi-directional simple shear testing results
can be interpreted consistently. Furthermore, whether the concepts of failure line and phase
transformation line derived for τ−σ ′v plane can be extrapolated into 3D τ−σ ′v space has not
been investigated. The reason why the stress paths of bi-directional linear tests cannot reach
the failure line remains unanswered. The mechanism responsible for the inverse correlation
between limiting excess pore pressure and shear stress amplitude has not been elucidated.
Secondly, the widely adopted liquefaction assessment method proposed by Seed and
Harder (1990) has not been assessed in multi-directional loading scenarios. Although
the consensus reached by previous studies is that multi-directional loading tends to result
in a lower liquefaction resistance, the claimed degree of reduction varies from study to
study. Moreover, cyclic shear stress ratio cannot be conveniently defined in multi-directional
4 INTRODUCTION
shearing conditions, which is challenging for the assessment of liquefaction resistance
through N−CSR curves.
Thirdly, there has been little discussion of excess pore pressure development, strain
accumulation and shear modulus degradation in multi-directional simple shear loading. The
definition of stress, strain and shear modulus needs to be re-explored and modified to provide
a compatible interpretation of multi-directional testing results.
Consequently, this study was designed to gain insights into these unsolved questions,
using the multi-directional simple shear testing apparatus developed by Rutherford and
Biscontin (2013) as the experimental platform with which to develop a new multi-directional
test database. The test results were then analysed and discussed to further our understanding
on the mechanical response of saturated sand to multi-directional shearing.
1.2 Scope of work
The objective of this project was to investigate the effects of multi-directional loading on soil
liquefaction. A series of undrained multi-directional simple shear tests were conducted on
saturated Hostun sand with the modified multi-directional simple shear testing apparatus.
The top cap of the sample assembly and sample preparation procedures were redesigned to
overcome membrane problems caused by the outward water pressure gradient. The desired
physical confinement and satisfactory saturation were achieved with the modifications.
Firstly, the uni-directional test results were analysed and compared against available
theories regarding undrained behaviour of sands, not only to validate the performance of
the modified testing techniques, but also to provide the baseline for further comparisons.
Multi-directional tests were then analysed to illustrate how soil behaviour is altered when
loading paths change from uni-directional to multi-directional.
The 2D τ −σ ′v plane was extrapolated into the 3D τ −σ ′v space, with the failure line
transformed to a conical failure surface. The concept of the failure cone was verified by the
tests with bi-directional linear, circular, oval and figure-8 loading paths. The reason why the
stress paths of bi-directional linear tests cannot reach the failure line in the 2D τ−σ ′v plane
1.3 Organisation of the dissertation 5
was elaborated. The inverse correlation between limiting excess pore pressure ratio and shear
stress ratio was also explained with the newly-proposed failure cone.
A bimodal phenomenon was identified in the phase transformation of sands. For circular,
oval and figure-8 tests, phase transformation states fall onto a conical surface. For uni-
directional and bi-directional linear tests, however, phase transformation states do not form a
cone.
The effects of multi-directional loading on soil liquefaction were investigated from
several aspects of liquefaction assessment and the definition of liquefaction was reassessed.
The influence of multi-directional loading on liquefaction resistance was investigated and
a potential alternative to the conventional N−CSR based liquefaction assessment method
was discussed. The development of excess pore pressure with multi-directional shearing was
explored and compared with that in uni-directional tests. The accumulation of shear strain
was investigated and its relationship with excess pore pressure clarified. The definition of
secant shear modulus was re-examined and a newly-proposed multi-directional secant shear
modulus was found to follow the degradation prediction of Hardin and Drnevich (1972) at
large strain.
The findings of this study provide new perspectives to understand soil behaviour un-
der multi-directional shearing and have useful implications for potential work including
numerical modelling and liquefaction assessment modifications.
1.3 Organisation of the dissertation
Chapter one introduces soil liquefaction and the importance of research into the effects
of multi-directional loading. A literature review on the fundamentals of soil liquefaction,
element-level testing, as well as the effects of multi-directional loading is presented in chapter
two to provide an overview of the current progress of research on these topics.
Chapter three details the simple shear testing techniques used in this study, the properties
of Hostun sands, the testing apparatus and discusses the sample preparation methods and
membrane problems. The parameters and a summary of the performed tests are also included.
6 INTRODUCTION
The results of the uni-directional and multi-directional tests are analysed in chapter four
and five. Uni-directional test results are compared against the well-established theories to
validate the performance of the testing techniques in this study. Multi-directional test results
are analysed to determine the differences between the soil behaviour under multi-directional
and uni-directional loading conditions.
Chapter six introduces 3D τ−σ ′v space and the concept of a conical failure surface. It
demonstrates that the phenomenon that the stress paths of bi-directional linear tests cannot
reach the failure line in 2D τ−σ ′v plane and the inverse correlation between limiting excess
pore pressure ratio and shear stress ratio can be explained effectively by the failure cone.
Conversely, phase transformation exhibits bimodal characteristics, with phase transformation
states forming a conical surface in the tests with circular, oval and figure-8 loading paths, but
do not produce a cone in uni-directional and bi-directional linear tests.
Chapter seven examines the liquefaction criterion, liquefaction resistance assessment,
development of excess pore pressure and shear strain as well as the degradation of shear
modulus in multi-directional shearing condition. The challenges are discussed, and the
implications are scrutinised.
Lastly, the insights obtained in this study are summarised in chapter eight. The sugges-
tions for future work are also outlined.
Chapter 2
LITERATURE REVIEW
2.1 Fundamentals of soil liquefaction
2.1.1 Undrained monotonic response of saturated sands
Since soil liquefaction was recognised to be a major cause of the damage in the 1964 Niigata
Earthquake and Alaska Earthquake, much effort has been made towards developing in-
depth understanding of the mechanisms of this phenomenon. It is well established that soil
liquefaction is caused by the generation of positive excess pore pressure during shearing.
When saturated loose sand is sheared, it can exhibit a tendency to contract, which will lead
to a decrease in volume if excess pore water can dissipate instantly. However, the permeability
of sands is usually not large enough to allow instant excess pore water dissipation when
subjected to fast earthquake loading. Since pore water is not dissipated rapidly, the volume
change of sands is inhibited and positive excess pore pressure is hence generated.
Early studies focusing on monotonic tests showed that the undrained response of satu-
rated clean sands can be generally categorised into three types: flow liquefaction (or strain
softening), dilation (or strain hardening) and limited liquefaction (Castro, 1969), as shown in
Figure 2.1.
Flow-liquefaction behaviour is commonly observed in very loose sands or the sands
under extremely high confining pressures. Pore water pressure accumulates rapidly and can
8 LITERATURE REVIEW
Fig. 2.1 Different types of undrained monotonic response of sands. a: liquefaction; b: limited
liquefaction; c: dilation (Castro, 1969)
become equal to the initial confining pressure as a result of the contractile tendency of sand.
Sand specimens can thus be softened to such an extent where large strain can occur suddenly
and sand cannot bear any increase in shear stress, indicating that the critical state of sands
has been reached. In q− p′ space, both deviator stress (q) and mean effective stress (p′)
decreases rapidly after a peak deviator stress is reached, and no regain in either q or p′ will be
seen thereafter. Similar trends can also be observed in τ−σ ′v space, since both the deviator
stress q and shear stress τ are the indicators of the shear strength of sands.
On the contrary, the undrained response of dense sands under moderate confining pressure
is dilative. Excess pore water pressure increases in the initial stage of shearing but decreases
after a small strain level is accumulated. In q− p′ space, mean effective stress may or may
not decrease at the early stage depending on the properties of sands, but the deviator stress
does not always reduce during the whole process of shearing. Sand never loses its shear
strength completely in this case. This dilative behaviour can also be reflected by q− εa plots,
2.1 Fundamentals of soil liquefaction 9
in which deviator stress keeps increasing with the accumulation of axial strain. Critical state
cannot be reached within a small range of deviator stress and axial strain.
Limited liquefaction behaviour, by contrast, involves a significant transformation of the
sand behaviour. In the early stage of shearing, sand exhibits liquefaction-like response. Posi-
tive excess pore pressure is generated, deviator stress drops and a stage of strain developing
rapidly without increase in deviator stress is encountered. Nevertheless, as shearing continues,
excess pore pressure will hit the peak and then decreases, which marks the transition in the
response of sands from contraction to dilation, and liquefaction behaviour is terminated at
this point as sand starts to dilate. In q− p′ space, this transformation can be reflected by an
abrupt turn on stress paths, beyond which a regain in both q and p′ is seen.
Fig. 2.2 Characteristic states of undrained shearing behaviour of sand: critical state (CS),
phase transformation state (PTS), quasi-steady state (QSS), and undrained instability state
(UIS).(Murthy et al., 2007)
Conventionally, limited liquefaction behaviour is of the most interest to geotechnical
engineers and researchers partly because it is much more dangerous than dilation behaviour
but more common than flow-liquefaction, and partly because it contains the features of
both of the two other types of soil behaviour. Four distinguishable states can be determined
for limited liquefaction behaviour: undrained instability state, quasi-steady state, phase
transformation state and critical state, as shown in Figure 2.2 (Murthy et al., 2007). The
undrained instability state is the point where the deviator stress reaches a local peak for the
first time and liquefaction initiates. After this point, deviator stress keeps decreasing until
the quasi-steady state is reached, where a local minimum deviator stress is obtained. It is
10 LITERATURE REVIEW
termed quasi-steady state because the inclination angle of the q− εa curve becomes zero
at this point, suggesting that strain develops without change in stress, just as it does at the
critical state. But this is not the true critical state point because the constant deviator stress
does not continue to larger strain.
Deviator stress begins to increase again after the quasi-steady state point, while mean
effective stress continues to decrease until a local minimum value is reached at the phase
transformation state (Ishihara et al., 1975). The minimum mean effective stress is a result of
the maximum excess pore pressure, which indicates the undrained response of sands changes
from contraction into dilation. In q− p′ space, quasi-steady state and phase transformation
state are so closely located that they are usually regarded as the same point, but it should be
noted that these two states are not the same and their differences become noticeable in q− εa
plots. As shearing proceeds thereafter, sands exhibit a dilative response until the critical state
is reached, beyond which strain can develop without change of stress. Given the limitation of
testing devices, critical state may not be reached if sands exhibit dilative behaviour, but this
rarely affects the interpretation of the undrained response of saturated sands. The definitions
of the four distinct states are applicable to flow-liquefaction response and dilative behaviour
as well, though the former does not show quasi-steady state and phase transformation state,
while the latter does not have quasi-steady state.
2.1.2 Undrained cyclic response of saturated sands without fines
As soil liquefaction is a phenomenon that is most commonly triggered by seismic loading,
understanding the undrained behaviour of sands subjected to fast cyclic shearing becomes
important. Much research has been conducted towards this goal and huge progress has been
made. Generally, seismic soil liquefaction results from the generation of positive excess pore
pressure induced by cyclic ground motion. When pore water pressure increases to be equal to
the initial confining pressure, soil liquefaction is initiated and flow-type deformation ensues
(Seed and Lee, 1966).
The earliest laboratory attempt to investigate cyclic soil liquefaction was made by Seed
and Lee (1966). They conducted cyclic triaxial tests on sands with different densities to
2.1 Fundamentals of soil liquefaction 11
observe and compare undrained response of loose and dense sands. The stress-controlled
testing results for loose and dense sands from Seed and Lee (1966) are presented in Figure 2.3.
For loose sands, both the peak and residual pore pressure of each cycle accumulate smoothly
in the pre-liquefaction cycles until a sudden jump takes place and peak pore pressure equals
the initial confining pressure. Large or unlimited shear strain will suddenly occur almost
simultaneously with the abrupt increase of pore pressure. For dense sands, by contrast, excess
pore pressure also increases within the pre-liquefaction cycles but with a lower accumulation
rate. After pore pressure equals the initial confining pressure, however, there is not a sudden
increase in shear strain; instead, shear strain develops gradually.
Fig. 2.3 Stress-controlled cyclic triaxial test results for loose and dense sands (modified from
Seed and Lee (1966))
Much research has been carried out since Seed and Lee published their pioneering paper.
An important feature of the strain development of dense sands under cyclic loading was
unveiled: there appears to be a cyclic strain level which cannot be exceeded no matter how
many cycles of loading are applied. It suggests dense sands can withstand unlimited number
of cycles of a given stress amplitude without witnessing runaway deformation (Seed, 1979).
12 LITERATURE REVIEW
This type of response was termed “cyclic mobility” by Castro (1975) or “initial liquefaction
with limited strain potential” by Seed et al. (1975).
Fig. 2.4 Stress parth and stress-strain curve for loose sand (Ishihara, 1985)
The differences between the response of loose sands and dense sands can be more clearly
observed in stress-strain hysteresis loops as suggested by Ishihara (1985). The results of
Ishihara (1985) are shown in Figure 2.4 and Figure 2.5. When excess pore pressure equals
the initial confining pressure, the deformation of loose sands develops abruptly, resulting in
the stress-strain loops being enlarged and flattened rapidly. It takes only a few cycles for
shear strain to become tremendous. For dense sands, the shear strain develops gradually and
the stress-strain loops also grow in such a manner.
2.1 Fundamentals of soil liquefaction 13
Fig. 2.5 Stress parth and stress-strain curve for dense sand (Ishihara, 1985)
While pattern of strain development differs for loose sands and dense sands, their response
in the stress space is rather similar. The cyclic behaviour of sands in terms of stress is closely
associated with the phase transformation states of sands. As defined in the preceding section,
phase transformation lines divide the stress space of sands into a contractive region and
a dilative region. Crossing the phase transformation line means the behaviour of sands
changes from contraction to dilation or vice versa. Before the stress path intersects the phase
transformation lines, positive excess pore water pressure accumulates steadily with cyclic
stress application, accompanied by a steady decrease in effective confining pressure.
14 LITERATURE REVIEW
Once the stress path intersects the phase transformation lines, the behaviour of sands will
change dramatically. During the loading phase of a cycle, sands exhibit generally contractive
behaviour prior to reaching the phase transformation line. Excess pore pressure increases
and shear strain develops rapidly without much change in shear stress (the section of the
stress-strain loops corresponding to this state is relatively flat). After the phase transformation
line is crossed, sands exhibit a dilative response, with excess pore pressure decreasing and
stress-strain curves becoming steep. The soil behaviour during the loading phase under cyclic
loading resembles the limited-liquefaction behaviour under monotonic shearing. During the
unloading phase, by contrast, the dilation state of sands is reversed. Excess pore pressure
increases with decreasing loading. Therefore, if a stress cycle involves crossing the phase
transformation line, the change of excess pore water pressure within this cycle should contain
two valleys (at the maximum shear stress points) and two peaks (at the phase transformation
points), as shown in Figure 2.6.
Fig. 2.6 Typical records of the measurements of shear stress and excess pore pressure (Ishihara
and Yasuda, 1975)
2.1.3 Interpretation of undrained monotonic behaviour of saturated
sand in Critical State Theory
It has been suggested by numerous researchers that initial void ratio and effective confining
pressure can exert an influence on the behaviour of soil. Roscoe et al. (1958) summarised the
2.1 Fundamentals of soil liquefaction 15
test results of Hvorslev and Taylor to develop Critical State theory. Readers are directed to
the paper of Roscoe et al. (1958) and the book of Schofield and Wroth (1968) and Schofield
(2005) for more details on this well-established subject. Debates on whether critical states
are the same as the steady states of Casagrande (1936) were settled by Been et al. (1991).
Jefferies and Been (2015) attributed the differences previously observed between the two
concepts to different testing methods rather than an intrinsic behaviour. For the sake of
simplicity and consistency, these two states are regarded as the same in this work.
Fig. 2.7 Schematic diagram of critical state theory (Yamamuro and Lade, 1998)
According to Critical State theory, initial void ratio and initial confining pressure col-
lectively determine the initial state of soil. If the initial state of sands is located in the ‘wet’
region (above the critical state line in e−σ ′ or e− logσ ′ plane as shown in Figure 2.7), sands
will show a contractive behaviour under monotonic shearing, which can lead to liquefaction.
On the contrary, if the initial state of sand is located on the ‘dry’ side, sand exhibits dilative
tendency, preventing liquefaction under monotonic shearing.
To illustrate the interpretation of undrained monotonic response of clean sands in the
Critical State framework, Figure 2.8 is presented here to give a typical example of undrained
monotonic triaxial test on sand specimens with the same initial void ratios, but different
initial confining pressure. For specimens with equal initial void ratios, higher confining
pressure leads to a more contractive behaviour because higher confining pressure pushes the
16 LITERATURE REVIEW
initial states towards the ‘wet’ side, whereas lower initial confining pressure leads to a more
dilative behaviour.
Fig. 2.8 Undrained monotonic behaviour of sands with different initial confining pressure
(Yamamuro and Covert, 2001)
Since Critical State Theory has been demonstrated as capable of explaining the instability
of saturated sands, whether it is effective in predicting liquefaction potential becomes an
intriguing topic. Based on Critical State theory, parameters which can reflect the relative
relationship between initial states and critical states can be used as the indicator of different
types of sand behaviour. Efforts have thus been made towards this subject, among which
the most well known is the state parameter ψ proposed by Been and Jefferies (1985). ψ is
defined as:
ψ = eλ − ecs (2.1)
where ecs is the void ratio given by critical state line at a designated confining stress, and
eλ is the void ratio determined from the one-dimentional isotropic consolidation line at the
same confining stress. The definition is illustrated in Figure 2.9. A positive state parameter
generally indicates sands have initial states on the ‘wet’ side and hence a potential for
liquefaction.
While this approach sounds reasonable, the state parameter was criticized by Ishihara
(1993) for its inability to give reliable analysis for loose sands under low confining pressure.
2.1 Fundamentals of soil liquefaction 17
Fig. 2.9 Defination of state parameter ψ (Been and Jefferies, 1985)
Other types of parameters were then proposed by researchers to provide alternatives to state
parameter. For example, Ishihara (1993) proposed a state index Is based on the void ratio of
sands at quasi-steady state. Verdugo and Ishihara (1996) proposed relative contractiveness Rc
which was defined in a similar way to relative density. Nevertheless, all these parameters have
their own limitations, hindering their application (Coelho, 2007). Furthermore, when fines
content is involved, all these state parameters seem to lose meaning. Thus, the state parameter
proposed by Been and Jefferies (1985) remains the most used parameter to characterise the
state of sands, based on which many well-known models, such as Nor-sand (Jefferies, 1993),
were established.
Although to interpret test results with state parameter sounds attractive, however, it is
out of the scope of this study. On the one hand, sands usually cannot reach critical state
before shear strain accumulates to a huge amount. The performance of available testing
apparatuses, however, become inevitably inadequate at such a large level of shear strain. It
is difficult to obtain consistent critical states, if at all, and the interpretation of test results
with state parameter may thus mislead the investigation of the primary subject of this study.
On the other hand, since the principal objective of this study is to investigate the effects
of loading paths on soil liquefaction, the variation of other factors including density and
confining pressure is restrained to a small range to minimise their interference. Thus, the
18 LITERATURE REVIEW
Table 2.1 Idealized in situ boundary conditions for earthquake-induced soil liquefaction
(After Kammerer (2002))
Simplification Idealized in situ Boundary condition
Constant vertical load Although there are cases where the vertical load (overbur-
den) is reduced, for a majority of cases the soil above the
liquefiable layer remains in place
No lateral strain It is assumed that the deposit is of large lateral extent such
that lateral strains can be neglected
Constant height Because the volume is constant and there are no lateral
strains, the height must also be constant
Constant volume No drainage or pore pressure redistribution during loading
effects of state parameter cannot be examined systematically in this study, though a study of
it is recommended for future work.
2.2 Element-level laboratory testing for soil liquefaction
To investigate the response of soil whose behaviour is largely dependent on the in situ
condition, the most important and essential consideration is certainly to reproduce realistic
boundary conditions as much as possible. In terms of earthquake-induced soil liquefaction,
the cyclic shear stress induced on soil elements is primarily due to the upward propagation
of horizontally polarised shear waves (Seed, 1979). The idealized in situ stress condition
experienced by a soil element under a level ground surface is presented in Figure 2.10, while
the idealized in situ boundary condition for soil liquefaction was summarized by Kammerer
(2002) in table 2.1. In situ testing would be the ideal way to study the accurate response of
soil. However, inevitable inconveniences arise from the lack of low-cost methods to apply
loading in field, measure soil response and achieve soil variability in a controlled manner,
which makes it unfeasible to conduct in situ research extensively, especially in the area of
earthquake-induced soil liquefaction.
Many laboratory experimental techniques were developed to provide alternative research
method to in situ testing. They can be divided into two primary categories: physical modelling
and element-level testing. Physical modelling uses devices including centrifuge, shaking
2.2 Element-level laboratory testing for soil liquefaction 19
Fig. 2.10 Idealized field loading condition (Seed, 1979)
table, etc. to model the soil and the structures at single or multiple gravity levels. Physical
modelling is suitable for research into failure mechanisms, settlement distribution and pore
water movement. It can replicate the in situ conditions to a large extent, which makes it
suitable for boundary value problems, but the accuracy of the results from such testing
still suffers from the disturbance caused by testing devices and the limitations in terms of
measurement techniques.
In contrast, element-level testing involves making uniform soil specimens from which the
behaviour of a soil “element” can be investigated. Due to the inability to simulate pore water
redistribution across soil layers, the liquefaction resistance evaluation from element-level
tests can be rather different from reality. However, fundamental trends of soil behaviour and
liquefaction resistance can be reflected by element-level test results, making it still one of the
most useful research tools in geotechnics. Although the ability to replicate in situ condition
is very limited and the effectiveness of element-level experiment results can be affected by
issues such as imperfect sample preparation and inhomogeneous specimen deformation, this
testing method is still the most convenient research path when the amount of variables is
large and required number of tests is high.
There are three most widely used element-level testing methods: triaxial testing, simple
shear testing and torsional testing. As they all have their own advantages and limitations, the
trade-offs must be evaluated before the decision is made on methodology. Since element-
level laboratory testing will be the main research method of this study, a review of different
element-level laboratory testing methods is presented in this section to clarify the research
methodology.
20 LITERATURE REVIEW
2.2.1 Triaxial testing
The earliest attempt to use triaxial testing to investigate the undrained behaviour of soil under
an earthquake scenario was made by Seed and Lee (1966). They carefully analysed the
stress conditions of the triaxial tests and suggested that earthquake loading can be simulated
through triaxial testing with necessary corrections on measured pore water pressure, as shown
in Figure 2.11. As triaxial testing devices had already been available and easily accessible
around the world by that time, their implication ignited the passion of researchers all over
the world, making triaxial testing a favourite in the field of soil liquefaction research.
Fig. 2.11 Stress conditions for triaxial test on saturated sand under simulated earthquake
loading conditions (Seed and Lee, 1966)
Triaxial tests can be divided into two categories according to the loading condition
applied. If the confining stress is smaller than the axial stress (case b in Figure 2.11), the
specimen will contract, and this condition is termed compression. On the contrary, if the
confining stress is larger than the axial stress (case c in Figure 2.11), the specimen will
extend in the axial direction, and this stress scenario is extension. The compression condition
and extension condition create shearing on two principal planes perpendicular to each other,
2.2 Element-level laboratory testing for soil liquefaction 21
which is equivalent to applying shearing on a single plane if the specimen is an idealized
soil element. However, realistic specimens are not an ideal soil element, and the shearing
applied on two perpendicular planes usually leads to different soil responses. In cyclic triaxial
tests which involve both of the two loading conditions, the principal stress axes rotate 90◦
instantaneously when the stress condition changes from compression to extension or vice
versa. They do not describe actual shear wave propagation in the soil with smooth rotation of
principal stresses.
In terms of boundary conditions encountered in soil liquefaction problem, actually none
of the conditions in table 2.1 can be maintained in triaxial testing. To start with, the vertical
load has to change since shearing is applied through axial loading, although according to Seed
and Lee (1966), the normal stress working on the shearing plane can be kept constant. As for
lateral strain, since the membrane used to isolate soil from cell water is not reinforced, triaxial
specimens usually experience some lateral strain (usually bulging) during shearing, unless
cell pressure is carefully controlled and adjusted with loading to counter lateral deformation.
Correspondingly, although the overall volume of specimen can be kept constant in undrained
triaxial tests, the local volume can change and pore water redistribution may occur during
shearing due to the deformation of soil samples (DeGroot et al., 1994; Escribano et al., 2018).
The difference between triaxial boundary conditions and in situ boundary conditions, in
addition to the abrupt stress rotation issue, makes triaxial testing a less desirable method
to investigate earthquake-induced liquefaction. Thus, if a database is to be established for
development of numerical modelling, triaxial tests should not be the preferred testing method.
2.2.2 Simple shear testing
Simple shear testing has been used in geotechnical research for many decades. Various
versions of simple shear devices have been developed by different researchers (Bjerrum and
Landva, 1966; Boulanger, 1990; Casagrande, 1976; DeGroot, 1992; Franke et al., 1979;
Ishihara and Yamazaki, 1980; Roscoe, 1953; Rutherford and Biscontin, 2013). Although
some discrepancies exist in the results obtained with different apparatuses, simple shear
testing techniques, as a whole, are still regarded as an effective element-level test method to
22 LITERATURE REVIEW
replicate in situ loading conditions shown in Figure 2.10. When it comes to the effects of
multi-directional loading, the advantages of simple shear test become even more compelling.
The earliest use of simple shear in the study of earthquake-induced soil liquefaction was
conducted by Peacock (1968) with the type of simple shear apparatus designed by Roscoe
(1953). They compared their test results with those obtained with triaxial testing from Seed
and Lee (1966) and found the liquefaction resistance obtained from cyclic simple shear test is
much lower than that from cyclic triaxial test, suggesting simple shear can give more reliable
results due to its ability to provide closer simulation of the stress conditions in the field.
The limitations of simple shear test, however, lie in three major aspects. The foremost
drawback is its inability to measure lateral stresses. In NGI-type simple shear devices which
use short and wide specimens, either a stack of rings or a wire-reinforced membrane is
used to confine the specimen and ensure zero lateral strain. As a result, the measurement
of lateral stress with conventional stress monitoring techniques becomes impossible. The
lack of information about lateral stress results in the inability to determine the stress state
accurately, which makes it sometimes necessary to designate a K0 value to conduct stress
state determination, increasing the inaccuracy of the analysis. No effective solution to this
problem has been proposed, and this disadvantage continues to be the weak point of this
testing technique. This inability is acceptable if it does not affect the accuracy of the test
results of interest, which, fortunately, is often the case in research of soil liquefaction.
The second significant drawback of simple shear test rests in the difficulty in controlling
the drainage condition. Since back pressure is not usually used in simple shear test, full
saturation is hard to maintain and direct measurement of pore water pressure cannot be made.
However, this problem has been overcome by the recent versions of simple shear devices,
such as the ones reported by Boulanger (1990) and Rutherford and Biscontin (2013). A cell
chamber similar to that used in triaxial devices was introduced, which allows the application
of back pressure to ensure a full saturation condition. Excess pore water pressure can be
measured directly using these apparatuses. These improvements have made the simple shear
test results much more accurate and make in-depth investigation of the behaviour of saturated
or partly saturated soil possible.
2.2 Element-level laboratory testing for soil liquefaction 23
The third major drawback of existing simple shear devices is the lack of applied comple-
mentary stresses to the vertical boundaries of the specimen. This discrepancy from the in
situ stress conditions leads to such a result that the moment created by the shear stresses on
horizontal planes cannot be countered or balanced. Soil specimens thus have a tendency to
rotate, resulting in stress concentrations and the change of local density at the portion where
stress concentrations occur (Casagrande, 1976). Nevertheless, the influence of this problem
can be minimized by using specimens having large diameter-to-height ratio. As suggested by
Franke et al. (1979), a diameter-to-height ratio of 3.75 and 7.5 does not give much difference
on undrained cyclic test results, indicating that a diameter-to-height ratio over 3.75 is large
enough to minimize the effects of this issue. This conclusion was also supported by Vucetic
and Lacasse (1982) and has been widely accepted.
In conclusion, most of the major drawbacks of simple shear test can be minimized or
overcome by advanced testing devices with careful considerations, except for the lack of
information about lateral stress. In this study, although the unknown lateral stress restricts the
ability to assess the variation of K0, it does not influence the investigation into the effects of
multi-directional loading in the τ−σ ′v space. Therefore, the simple shear test is a reasonable
choice for this research.
2.2.3 Torsional shear testing
Two primary types of torsional shear test were developed to provide alternatives to triaxial
test and simple shear test: hollow cylinder test and non-hollow torsional shear test. The
non-hollow torsional shear test which uses tall cylindrical sample is so similar to the triaxial
test in many aspects that it was called triaxial torsion shear test by Ishihara and Li (1972). By
contrast, the torsional ring shear uses short soil sample, which resembles the simple shear
test. The torsional ring shear test is one of the best element-level test methods to obtain
residual strength condition because it can achieve “an unlimited amount of continuous shear
displacement” (ASTM D6467-13, 2013), with which the residual shear strength of soil can
be adequately mobilised.
24 LITERATURE REVIEW
Hollow cylinder testing further increases the number of controllable stress components by
replacing the centre cylinder of the soil specimen with water, through which an interior cell
pressure can be applied and controlled in addition to the exterior cell pressure. The specimen
of a typical hollow cylinder test is shown in Figure 2.12. The advantage in achieving complex
stress paths and stress states and improving the non-uniform strain distribution along the
radius has made hollow cylinder test preferable to its non-hollow counterpart in the research
on soil liquefaction.
Fig. 2.12 A hollow cylinder specimen under torsional shear (Ampadu and Tatsuoka, 1993)
Compared with the triaxial test, the hollow torsional shear test can achieve continuous
rotation of principal stresses and controls more stress components. Compared with simple
shear, torsional shear allows measurement of the lateral stress and can shear the specimen
into the large strain range.
Nevertheless, the hollow cylinder test has limitations as well. Many drawbacks of the
triaxial test similarly exist in the hollow cylinder test. On the one hand, the measurement of
lateral stress is achieved at the price of the development of lateral strain and local volume
change, reducing the reliability in terms of the boundary conditions for earthquake-induced
liquefaction. On the other hand, general multi-directional loading paths cannot be achieved
conveniently in either the hollow cylinder torsional test or the torsional ring shear test. Since
the objective of this study is to investigate the effects of multi-directional loading on soil
2.3 Multi-directional loading 25
liquefaction, the torsional shear test is not the appropriate methodology choice, even though
it features the advantage of a known lateral stress.
2.3 Multi-directional loading
Earthquakes apply shear loading on more than one direction. As waves propagate through
layered geological materials of decreasing density, the direction of propagation becomes
more vertical, resulting in the general simplification of treating seismic waves as vertical
propagating and horizontally polarised. The earthquake, however, is not limited to one
direction on the horizontal plane. An example of seismometer traces recorded during
San Fernando Earthquake is shown in Figure 2.13. The multi-directional characteristic of
earthquake loading is evident.
Fig. 2.13 Seismometer traces recorded in Dry Canyon Dam during 1971 San Fernando
Earthquake (Pyke et al., 1975)
Multi-directional loading on the horizontal plane is widely recognised to have significant
influence on the behaviour of soil. Pyke et al. (1975) conducted shake table tests to study
the settlement of sands. A schematic of the gyratory loading path they used and the test
results from their study is presented in Figure 2.14. They compared the settlement induced
26 LITERATURE REVIEW
by uni-directional loading with that by gyratory loading path, and found the accumulation of
settlement under gyratory loading is approximately twice that in uni-directional counterpart.
Fig. 2.14 The schematic diagram of gyratory loading path and settlement of sands (Pyke
et al., 1975)
Seed et al. (1978) proposed a pore pressure development model and extended the research
carried by Pyke to investigate the effects of multi-directional loading on pore water pressure.
Their calculation results indicate pore pressure increases faster under multi-directional
shaking than under uni-directional loading as shown in Figure 2.15.
Fig. 2.15 Pore pressure development (Seed et al., 1978)
2.3 Multi-directional loading 27
Casagrande and Rendon (1978) developed a gyratory shear apparatus and tested Ottawa
"Banding sand" at various relative densities under uni-directional linear shearing and gyratory
shearing. They showed that the development of excess pore pressure under gyratory loading
condition is different. Firstly, it takes fewer cycles to reach the maximum pore pressure,
indicating liquefaction resistance reduces in gyratory shearing. Secondly, the maximum
excess pore pressure under gyratory loading cannot increase to the value of initial vertical
stress, which is an important feature of circular loading and was further supported by the
experimental evidences from later research. Thirdly, the cyclic pore pressure fluctuations
in gyratory loading are smaller than in uni-directional cyclic tests. All these observations
indicate that sand behaviours differ when multi-directional loading is involved.
Ishihara and Yamazaki (1980) developed one of the pioneering multi-directional simple
shear apparatus and conducted multi-directional cyclic shearing tests on saturated sands
to investigate the effects of changing loading directions on liquefaction resistance. They
applied elliptical and bi-directional linear loading paths to water-pluviated Fuji River Sand
specimens. Three important observations were obtained by their study. Firstly, excess pore
pressure is not able to reach the value of the initial confining pressure under elliptical and
circular loading paths. Therefore, the use of the liquefaction criterion which requires excess
pore pressure ratio (excess pore pressure over initial confining pressure) to reach unity may
be un-conservative in these cases. The authors attributed this behaviour to the initial shearing
which locks some dilative deformation in the specimen. When the stress is released back to
zero, the dilative deformation is unlocked and the excess pore pressure could increase to the
value of the initial confining pressure. The second conclusion obtained by the authors was
that multi-directional loading reduces liquefaction resistance compared with unidirectional
shearing. The maximum reduction of the shear stress ratio required to induce 3% single
amplitude shear strain within a certain number of cycles was approximately 30% for circular
paths and 25% for bi-directional linear paths in their study, much greater than 15% proposed
by Seed et al. (1978). Furthermore, it was noted that the negative effect of multi-directional
loading on liquefaction resistance is maximised when two perpendicular components of
loading have the same amplitude.
28 LITERATURE REVIEW
Fig. 2.16 Comparison between multidirectional and unidirectional shaking using the proposed
method (Tokimatsu and Yoshimi, 1982)
Based on the experimental results in the two studies above, Tokimatsu and Yoshimi
(1982) proposed a method to estimate the effects of multi-directional shaking. They assumed
that the generation of excess pore pressure in one direction is independent of the other, which
suggests that the influence of loading along two directions can be superposed, as shown in
Figure 2.16. It was concluded that the intensity of the effect of multi-directional shaking
is related to the values of two parameters, which reflect the initial generation rate of pore
water pressure and the gradient of logτ − logN curves under unidirectional loading. The
earlier the excess pore pressure starts to increase and the steeper the logτ− logN curves are
under unidirectional shaking, the more significant the effects of multi-directional loading
become. The initial generation rate of pore pressure was later demonstrated to be affected by
multi-directional loading paths, cyclic stress ratio and cementation (Clough et al., 1989).
Yamazaki et al. (1985) developed a numerical model to estimate the development of the
effective stress path and deformation of soil during earthquakes. They applied this model
to investigate the effects of bi-directional loading on liquefaction resistance of soil and
concluded that the liquefaction potential increases when the loading pattern changes from
unidirectional lines to ellipses, and further to circles. The results of their numerical analysis
showed that soil tends to become more prone to liquefaction under bi-directional loading
than under unidirectional loading.
Nagase and Ishihara (1987) also used the bi-directional simple shear apparatus developed
by Ishihara and Yamazaki (1980); however, in that study they investigated the effect of
irregular unidirectional loading, rather than multi-directional loading, on cyclic strength of
2.3 Multi-directional loading 29
water deposited Fuji River sands. Samples were prepared at three different relative densities
corresponding to loose, medium dense and dense states respectively, and all consolidated at a
vertical overburden pressure of 196 kPa. They selected six pairs of earthquake horizontal
acceleration records based on which the cyclic stress to be applied can be determined because
the time history of shear stress, as they claimed, should be analogical to that of accelerations
with merely a phase difference. According to the results obtained in the study, the ratio of
peak shear stress versus initial confining pressure required to induce a specified value of
maximum shear strain within 20 cycles under irregular loading condition is higher compared
to uniform loading. Nonetheless, this effect is dependent on the factors including relative
density, maximum shear strain and wave form of loading. They concluded that the effect of
irregular loading tends to decrease with increasing relative density. Denser sands produce
a smaller amount of progressive shear strain, whereas they are not affected appreciably by
maximum shear strain or wave form of loading.
Ishihara and Nagase (1988) conducted undrained simple shear tests with the bi-directional
simple shear apparatus to investigate the effect of bi-directional horizontal loading as a part
of extension work on their previous study to develop irregularity factors for Japanese bridge
design code (Nagase and Ishihara, 1987). The experiment settings in terms of material, sample
preparation method and relative density of specimens were generally the same in these two
studies. Through extracting the coefficients of irregularity from the total differences between
the stress ratios required to induce a certain amount of maximum strain by multi-directional
irregular loading paths and by unidirectional uniform loading, they decoupled the effect of
loading multi-directionality from that of irregularity, thereby making the conclusion that
multi-directional loading decreases liquefaction resistance of sands. The resistance reduction
caused by multi-directional loading is shown to be generally independent of relative density
though with a slight decline for dense sands. It should be noted that although it is the
combined influence of irregularity and multi-directionality that dominates the response of
soil, the research on regular loading paths is still necessary to develop a better understanding
of the soil behaviours under multi-directional shearing.
30 LITERATURE REVIEW
Fukutake and Matsuoka (1989) proposed that the volumetric strain under multi-directional
shearing is composed of cumulative strain and resultant strain. Cumulative strain is related
to the plastic volumetric deformation accumulated during shearing and thus is irreversible,
whereas resultant strain is the reversible strain which is dependent on the magnitude of strain
vector at a given time. They conducted drained multi-directional simple shear tests to verify
the model they developed. Although numerical modelling is not in the scope of this study,
their hypothesis on the mechanism of strain accumulation under drained multi-directional
simple shear tests is still inspiring because the generation of excess pore pressure under
undrained condition is considered closely related to the volumetric change of soil in drained
condition.
Clough et al. (1989) used a cubical shear box to conduct multi-directional cyclic shearing
tests to investigate the effects of multi-directional loading on liquefaction resistance of
cemented sands. They reported that multi-directional loading always leads to a lower
liquefaction resistance compared with unidirectional loading, and the reduction can be as
high as 15%-40%.
Reddy and Saxena (1992) summarized the results of previous studies to develop a model
to estimate the generation of excess pore pressure and liquefaction resistance of cemented
soil under multi-directional stress paths. The experimental results they used were obtained
through true triaxial tests and indicated that the liquefaction resistance could be decreased
by up to 50% for uncemented sands, which was again in accordance with the widely-made
observation of multi-directional loading reducing liquefaction resistance.
Boulanger and his colleagues (Boulanger, 1990; Boulanger et al., 1993; Boulanger and
Seed, 1995) developed the U.C. Berkeley Bi-directional Simple Shear Device (UCB-2D)
to study the effects of bi-directional loading on the strength of moist tamped modified
Sacramento River Sand. An important improvement of this device was a chamber that allows
the application of back pressure for saturation. The bi-directional loading here refers to the
category of paths in which an initial static driving shear is applied in dip direction before
cyclic shearing is applied in perpendicular direction (strike direction), which represents the
loading situation encountered in sloping ground. It was suggested by their study that soils’
2.3 Multi-directional loading 31
shear resistance becomes lower in the case where cyclic shearing is perpendicular to initial
static driving shear because this loading scenario allows larger amount of shear stress reversal
and larger extent of axis rotation. Furthermore, they noticed that there exists an inverse linear
correlation between the largest maximum and minimum excess pore pressure within a cycle
and shear stress amplitude.
Another type of multi-directional direct simple shear (MDSS) apparatus is the one
concurrently developed by DeGroot (1992); DeGroot et al. (1993); Degroot et al. (1996) at
the Massachusetts Institute of Technology. They studied the effects of different directions
of static shear loading and ice shear loading on the response of Boston Blue Clay. They
observed that the peak undrained resistance of tested clay is affected significantly by the
angle between the two shear forces, which is similar to the previous results for sands.
Meneses et al. (1998) used a bidirectional simple shear apparatus to investigate the
influence of cyclic shearing superimposed on monotonic shearing in an orthogonal manner,
which represents a situation of small seismic shearing continuing after the main shaking of an
earthquake. The previous observation that superimposing cyclic shearing onto the monotonic
loading will reduce the strength of sand is further confirmed. Furthermore, the magnitude
and frequency of post-main-shaking cyclic loading was shown to influence the behaviour of
sands with same initial relative density and confining pressure.
Matsuda et al. (2011, 2004) developed their multi-directional cyclic simple shear test
apparatus and conducted strain-controlled tests on Toyoura sand and granulated blast furnace
slag (GBFS). Two sinusoidal signals were input into the two perpendicular loading actuators
with a phase difference which is controlled and changed to produce different loading paths.
Their test results showed that the number of cycles to trigger liquefaction decreases with
increasing phase difference, indicating again that multi-directional shearing can reduce
liquefaction resistance.
Sako and Adachi (2004) investigated the generation of excess pore pressure of air
pluviated Toyoura Sand subjected to uni-directional and bi-directional loading. It was
shown that excess pore pressure increases the fastest under bi-directional loading condition.
The suggested bi-directional excess pore pressure development can be virtually simulated
32 LITERATURE REVIEW
through scaling the pore pressures induced by unidirectional loading inputs by a modification
coefficient. The modification coefficient obtained in this study ranges from 1.08 to 1.26,
demonstrating an increased liquefaction potential when shearing is applied in bi-directional
manner.
Fig. 2.17 Schematic illustration of idealized multi-directional loading (Kammerer et al.,
2004b)
A comprehensive investigation into the influence of multidirectional stress paths (both
magnitude and direction) and initial static shear stress on saturated sands was conducted by
Kammerer, Wu and colleagues (Kammerer et al., 2004a,b; Kammerer, 2002). Water pluviated
saturated Monterey #0/30 Sand was tested with the improved U.C. Berkeley Bi-directional
Simple Shear Device (UCB-2D). The stress paths performed can be categorized into three
groups: bi-directional linear path, circular or oval paths and figure-8 paths, as shown in
Figure 2.17. Their results confirmed that the rotation of stress direction in the horizontal
plane can result in faster generation of excess pore water pressure, which was explained by
the aiding effect of stress rotation on particle rearrangement and densification of soil.
The inverse linear correlation between limiting excess pore pressure ratio and shear
stress ratio reported by Boulanger (1990) was also observed by Kammerer (2002), as shown
in Figure 2.18. While the author attributed it to the dilation characteristics of soil, the
mechanisms behind it remain unclear.
Furthermore, Kammerer (2002) observed that the tests which have no stress reversal
because of extremely large static driving shear, always have less strain potential. The counter-
2.3 Multi-directional loading 33
Fig. 2.18 Relationship between shear stress ratio and limiting excess pore pressure ratio
(Kammerer, 2002)
intuitive findings suggested that the primary effect of the initial static driving shear is to
affect the degree of stress rotation rather than to change the magnitude of the shear stress.
Stress rotation exerts complex effects on the behaviour of soil and generation of pore pressure
through influencing particle rearrangement and softening response. The author thus proposed
that stress rotation is more important than cyclic stress amplitude in terms of pore pressure
generation and shear strain.
In order to compare loading paths in a consistent manner, Kammerer (2002) proposed to
use aperture ratio and reversal parameter as the parameters to characterise loading paths. It
was shown that increasing aperture ratio (from oval to circular for instance) could cause a
decline in liquefaction resistance. But this negative effect itself tends to reduce with higher
aperture ratios.
Ueng and colleagues (Ueng et al., 2005) conducted two-dimensional shaking table tests
on Vietnam Sand to examine the influence of multi-directional shear on the re-liquefaction
of sands. It was suggested by the authors that peak pore pressure becomes higher under
two-dimensional shaking compared with one-dimensional loading with the same amplitude.
There was not a prominent effect of two-dimensional shaking on the rate of increment of
34 LITERATURE REVIEW
pore pressure. It should be noted that their two-dimensional tests were conducted after
one-dimensional shaking tests, and the behaviour of sands prior to initial liquefaction can be
rather different from their observations.
Based on the U.C. Berkeley Bi-directional Simple Shear Apparatus (UCB-2D), a new
device named the Digitally Controlled Simple Shear apparatus (DC-SS) was developed by
Duku et al. (2007). The major advantage of this device compared with the Berkeley one rests
on its ability to apply earthquake broadband loading with a large frequency. This device was
later mainly used to investigate the volumetric strain of sands (Duku et al., 2008; Yee et al.,
2013). No published multidirectional loading tests, however, have been performed by this
apparatus except for the tests conducted to demonstrate the capability of the device.
Jin et al. (2008) performed stress-controlled universal triaxial and torsional shear tests
on air pluviated loose Fujian Standard Sand to investigate the effects of complex loading
paths on pore pressure generation and strain development. The authors found that neither the
amplitude of vertical stress nor that of horizontal stress affects the peak pore water pressure;
instead, the peak pore pressure is inversely correlated to initial static shear. Their findings are
consistent with the observations of Boulanger (1990) and Kammerer (2002) despite different
testing techniques.
Rutherford and Biscontin (2013) from Texas AM University developed another multi-
directional direct simple shear apparatus (TAMU-MDSS) and experimentally studied the
cyclic shear response of Gulf of Mexico Clay. Rutherford (2012) observed that the largest
excess pore pressure occurs under the stress paths where both the magnitude and direction of
stress changes most. Furthermore, she concluded that shear strain of clay always accumulates
towards the orientation of initial static shear (downhill) regardless of the direction of cyclic
loading, which is in agreement with the conclusions of Biscontin et al. (2004). Although
the material tested here was clay rather than sand, the observations are still inspiring since
similar tendency can also be found in the tests on sands such as those conducted by Boulanger
(1990) and Kammerer (2002). The tests reported in this dissertation were conducted with
the modified TAMU-MDSS. The modification as well as the redesigned top cap and sample
preparation procedures will be detailed in chapter 3.
2.3 Multi-directional loading 35
Rudolph et al. (2014), Li et al. (2016) and Mirbaha (2017) developed their multi-
directional simple shear testing database, respectively, with the variable direction dynamic
cyclic simple shear device (VDDCSS) of GDS Instruments Ltd. The reduction of liquefaction
resistance in multi-directional loading condition was also supported by these studies. Due
to the limitations of VDDCSS, they conducted equivalent undrained tests rather than actual
undrained tests. They adopted constant-height boundary condition and regarded the reduction
of vertical stress as the generation of excess pore pressure. Although this type of tests is
accepted as an effective approximation of undrained tests, the actual discrepancies between
them are difficult to estimate, especially when the degree of saturation exerts an influence.
In conclusion, despite all the efforts outlined above, multi-directional simple shear test
data are still scarce and there has not been any well-established theoretical framework to
account for the effects of multi-directional loading. Although the consensus seems to be
that multi-directional shearing can reduce the liquefaction resistance of sands, the degree of
reduction varies from one study to another. The mechanisms behind the extensively-reported
inverse correlation between limiting excess pore pressure and shear stress amplitude remains
unclear. There have been no conclusions about the effects of loading paths on the stress
paths and shear modulus degradation. Clearly, more efforts are required before the effects of
multi-directional shearing can be explained consistently from a theoretical point of view, and
this study aims to contribute new insights towards this goal.

Chapter 3
MULTI-DIRECTIONAL SIMPLE
SHEAR TESTING
3.1 Material tested
The sand used in this study is Hostun S28 Sand. It is poorly graded and quartz dominated
(Ezaoui and Benedetto, 2009). The particles of Hostun Sand are sub-round to sub-angular in
shape as shown in Figure 3.1. The maximum void ratio, minimum void ratio and specific
gravity measured by Mitrani (2006) are used in this study, as summarized in table 3.1.
The particle size distribution of the sand was investigated with the conventional sieve
analysis method, following the procedures prescribed by ASTM D6913. The particle size
distribution curve is presented in Figure 3.2. The median particle diameter D50 is 0.35 mm.
Table 3.1 Properties of Hostun S28 Sand (Mitrani, 2006)
Properties Values
emax 1.01
emin 0.555
Gs 2.65
38 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Fig. 3.1 Shape of Hostun Sand particles (Ezaoui and Benedetto, 2009)
3.2 Review on specimen preparation technique
If the element-level laboratory tests are expected to give results that can represent the in situ
soil behaviour and satisfy the standards of numerical modelling, the technique used to create
reconstituted soil specimen has to be carefully selected. The ideal method is supposed to
convincingly simulate the fabric of in situ soil and the process of deposition, and produce
uniform specimens, which is the fundamental requirement of element-level testing. It is thus
worthy to review existing specimen preparation methods for sands.
3.2.1 Moist Tamping
The oldest specimen preparation method used to create reasonable reconstituted specimen
is moist tamping, in which a series of layers are poured into the mold to designated height
with each layer surface being levelled and tamped before placing the next layer (Lambe,
1951). This method effectively mimics the soil fabric of rolled construction fills for which
the method was originally designed (Kuerbis and Vaid, 1988).
3.2 Review on specimen preparation technique 39
0.1 1
Particle diameter (mm)
0
10
20
30
40
50
60
70
80
90
100
Pe
rc
en
ta
ge
 p
as
sin
g 
(%
)
Fig. 3.2 Particle size distribution curve of tested Hostun Sands
The moist tamping method can produce specimens with relative density ranging from as
low as near zero to approximately unity. As loose sands and silty sands are considered most
susceptible to liquefaction, the extremely loose state which can be achieved through water
tension force herein has naturally been a source of the interest in using this technique, while
other specimen preparation methods can hardly create such loose specimens.
However, this apparent advantage also serves as the primary cause of criticism. Casagrande
(1976) suggested that as a result of capillary forces, the moist condition induces the gener-
ation of a “honeycomb” structure by which the structure of soil is supported. During the
saturation process, water tension is gradually removed and the honeycomb structure can
collapse. For some silty specimens which have finer particles and thus experience large
water tension forces, a large strain can be observed due to this (Casagrande, 1976; Chang
and Whitman, 1988; Sladen et al., 1985). As for those specimens which do not exhibit a
large strain during saturation, though there is a chance that they behave in an incompressible
manner during consolidation and remain at large void ratios, the extremely loose specimens
are usually metastable and thus have a lower liquefaction resistance (Kuerbis and Vaid,
40 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
1988), that is, the loose specimens obtained through moist tamping may be not reliable.
Another drawback concerning this specimen preparation technique rests in its relatively low
uniformity compared with other methods. Miura and Toki (1982) conducted miniature cone
penetration test and concluded that non-uniformity in moist tamped specimens is remarkably
higher than that of pluviated specimens.
It should be noted, however, that moist tamping has been applied in most of the triaxial
tests on extremely loose Hostun sands in literature. The results of these tests should be
treated and interpreted with caution.
3.2.2 Air Pluviation
Air pluviation uses a funnel to pluviate sand from a designated height with a desired rate
of deposition to control the relative density of specimens (Miura and Toki, 1982; Vaid and
Negussey, 1988). The higher the drop height is, the larger the input energy becomes, and
thus the denser the specimen is. The loosest state is obtained through keeping the nozzle
right above the surface of the pluviated soil while lifting the funnel. The relative density
range that has been successfully achieved with this method is not as wide as that of moist
tamping method, but is still satisfactory for most of the research on soil liquefaction.
The uniformity of air pluviated specimens is desirable (Miura and Toki, 1982; Mulilis
et al., 1978) when preparing specimen made of well-sorted sand or well-sorted silt because
it simulates directly the natural deposition process of aeolian deposits (Kuerbis and Vaid,
1988). However, when it comes to well-graded sand with fines content, the performance of
the air pluviation technique is no longer reliable due to particle-segregation issues. If the
drop height is strictly restricted to be rather small, the segregation during drop of particles
may not be severe, though the saturation process will still induce segregation by washing out
the fine particles (Kuerbis and Vaid, 1988).
In addition to segregation, bulking is also an issue encountered in specimens with fine
contents prepared by air pluviation. Kuerbis and Vaid (1988) reported air-pluviated silty
sands exhibited a metastable behaviour with large strain developing during saturation and
3.2 Review on specimen preparation technique 41
consolidation. Extremely loose specimens prepared by air pluviation also undergo a reduction
in liquefaction resistance, just as their moist tamped counterparts.
For medium to dense clean sands with minimal silt content, nevertheless, the segregation
and bulking problems are not severe. Samples prepared by air-pluviation can provide promis-
ing and consistent results. Furthermore, for the samples with low height-to-diameter ratios,
the control of nozzle and particle drop height is straightforward. This sample preparation
technique is hence an adequate method for simple shear testing.
3.2.3 Under-compaction Moist Tamping
To overcome the issue of segregation when fines content is involved, the under-compaction
moist tamping method was proposed by Ladd (1978). Generally, this method is a modified
version of moist tamping with the lower layers not fully compacted. When the upper layers
are added, the compaction of the upper layer can simultaneously compact the lower ones, so
that the specimen will become uniform as a whole.
Although the issue of segregation can be improved to some extent, other drawbacks of
the moist tamping method, especially those related to the unrealistic soil fabric, cannot be
overcome by the under-compaction technique. Therefore, the test results obtained with this
sample preparation technique should also be treated carefully.
3.2.4 Water Pluviation
Water pluviation has been widely used in many previous studies regarding liquefaction
resistance of sands, clean sands in particular. The essence of this reconstitution technique is
rather similar to air pluviation, except that sands are pluviated through water instead of air
(Lee and Fitton, 1968).
Considering that the falling velocity of particles in water is much lower than in air due to
the clearly different buoyancy and friction, the energy input, provided mainly by gravity, is
thus much lower for water pluviation than air pluviation. The maximum void ratio obtained
by water pluviation is usually larger than its air counterpart. It is worth noting that Vaid and
42 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Sivathayalan (2000) contended it is unlikely that water-deposited in situ sands exist in a state
looser than the loosest state achievable by water pluviation, indicating that the extremely
loose specimens of hydraulic sands, such as those produced by moist tamping method, are
not realistic.
Similar to air pluviation, unwanted segregation is also an issue encountered by this
method (Amini and Qi, 2000). The falling velocities of fine particles are generally slower
than those of coarse particles in water, just as in air, causing fines sitting on the top of
the coarse particles as long as they have the same drop height. The uniformity of water
pluviated specimens can be impaired severely due to the layering of different-diameter
particles. Although stratification is encountered in situ from time to time, it is not acceptable
in element level experiments. Thus, again, the use of water pluviation technique is limited to
research on sand without fines content. Actually, when the effect of fine contents is involved,
neither a pluviation method nor moist tamping is preferred.
However, the advantages of this technique are significant since it is known to be able to
mimic the natural fabric of hydraulic sediments (Miura and Toki, 1982; Oda et al., 1978;
Vaid and Sivathayalan, 2000). This desirable feature is highly valued by geotechnical
experimentalists partly because hydraulic deposit is a type of fill of particular interest in the
area of liquefaction research and partly because the more realistic the fabric is, the more
meaningful the test results are. Furthermore, in terms of uniformity, Vaid and Negussey
(1988) suggested that the specimens prepared by water pluviation have better uniformity
compared with moist tamped ones. This method has been adopted by numerous researchers
in the last a few decades and is still widely employed now. Høeg et al. (2000), based on
comparison of stress-strain-strength behaviour of reconstituted specimens with undisturbed
in situ specimens, recommended that the most promising reconstitution technique is water
pluviation combined with vibration as long as segregation issue is avoided for very well-
sorted sands.
Nevertheless, the use of this sample preparation technique is not preferable in this study.
When the membrane is pulled over the top cap, a volume of air is inevitably trapped between
the membrane and the sidewall of the top cap. In triaxial tests, the influence of this issue is
3.2 Review on specimen preparation technique 43
not pronounced partly because the radial stress acts inward and partly because the overall
volume of sample is much larger compared with the volume of trapped air. In simple shear
tests, however, resultant radial stress acts outward when positive excess pore pressure is
generated and the volumetric proportion of trapped air is significant. As a consequence, the
degree of saturation is lowered if the sample is reconstituted using water-pluviation. Some
pilot tests were conducted in this study with water-pluviated samples and the Skempton’s
B-value of these tests were never higher than 0.7. This technique was therefore not adopted
in the subsequent tests.
3.2.5 Slurry Deposition
Slurry deposition, initially developed by Kuerbis and Vaid (1988), is another widely-used
reconstitution method proposed to cope with segregation of the particles of different sizes
while inheriting the advantages of water pluviation. The initial version of this method used a
tube filled with de-aired water to fully mix fines slurry and sand slurry before transferring
the mixture from the tube to the triaxial cell. Segregation can be inhibited by mixing fines
and sand adequately. Various versions of slurry deposition technique were later developed by
researchers including Høeg et al. (2000), Salgado et al. (2000) and Carraro (2004). However,
no matter how the detailed procedures differ, the crucial consideration remains that the fines
and coarse particles should be mixed thoroughly and the mixture should deposit from the
suspend-in-water state.
Such procedures were demonstrated by many researchers (Kuerbis and Vaid, 1988)
to produce specimens with attractive properties such as full saturation and a soil fabric
similar to natural deposits, as well as excellent duplication and high uniformity regardless of
gradation. It is these appealing features that make the slurry deposition method a fascinating
reconstitution technique, especially for research on the effects of fines content and gradation.
In this study, however, slurry deposition suffers from the same drawback as water-
pluviation method. Air can be trapped inside the sample when the membrane is clamped to
the top cap, resulting in a low degree of saturation. This technique is thus not applicable to
undrained simple shear tests either.
44 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
3.2.6 Air pluviation with CO2 flushing
The in situ fabric of soil and natural deposition process should be simulated as closely as
possible by the selected specimen preparation technique. Given that the sands of interest are
generally fluvial and hydraulic-deposited, air pluviation or water pluviation should be the
preferred methods to prepare samples. However, a low degree of saturation can result from
the air trapped inside the samples when top caps are mounted and sealed. This issue cannot
be solved if sands are pluviated into water, but can be reduced with air-pluviation followed
by a carbon dioxide flush. After CO2 is flushed through air-pluviated samples, the trapped
air is replaced by soluble CO2 that will dissolve into water with the help of increased back
pressure in the saturation phase.
CO2 has been extensively used in both element-level laboratory tests and physical mod-
elling tests (Ishihara and Nagase, 1988; Stringer and Madabhushi, 2009).Takahashi et al.
(2006) compared the P-wave velocity in the physical models saturated with different tech-
niques and concluded that carbon dioxide percolation and vaccum technique can achieve
high degree of saturation.
However, such a procedure cannot be applied if the sample is filled with water which
blocks the movement of air and isolates it. Thus, air pluviation with CO2 flushing was
selected as the best sample preparation for the simple shear tests conducted in this study.
Only 4 out of the 79 tests had B-value lower than 0.9, with the lowest one at 0.85.
3.3 Inflation problem in undrained simple shear tests
When it comes to the differences between simple shear tests and triaxial tests, there is an
aspect that is not often discussed, that is, the different confining stress condition in undrained
tests. In triaxial tests, lateral confining stress σ3 can be controlled by the difference between
cell pressure and back pressure. In most cases, cell pressure is always larger than back
pressure to keep soil samples confined by positive σ3 unless liquefaction happens. Therefore,
the generation of positive excess pore pressure in triaxial tests will not result in the resultant
radial stress turning from inward to outward.
3.3 Inflation problem in undrained simple shear tests 45
Fig. 3.3 Inflation of the rubber membrane (Ishihara and Yamazaki, 1980)
By contrast, what is simulated by simple shear tests is zero lateral deformation condition
of soil. Lateral confining stress is equal to vertical overburden stress multiplied by a lateral
stress coefficient K0. If the variation of K0 with shearing were well established, the condition
of zero lateral deformation could be conveniently achieved by controlling the lateral stress,
just as it is done in triaxial tests. Unfortunately, the knowledge of the variation of K0
with shearing is still limited, which eliminates the possibility of achieving zero lateral
deformation in laboratory tests by controlling exclusively lateral stress. Instead, the most
widely accepted method to ensure zero lateral deformation in the simple shear tests is to
provide physical confinement, such as a reinforced membrane or stacked rings, with the price
that the information about K0 is sacrificed.
Under this circumstance, if the cell pressure is set higher than back pressure, an additional
lateral stress will be applied. If the additional lateral stress is smaller than σv ·K0, the stress
condition of soil will not be altered because the difference can be compensated by lateral
confinement. But when the additional lateral stress is larger than σv ·K0, the stress condition
of the soil will change and the in situ K0 condition cannot be reproduced. Therefore, a
convenient and safe choice is to require that the cell pressure be equal to the back pressure,
so that all lateral stress is provided exclusively by lateral confinement.
46 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Fig. 3.4 The photo of water pouch forming in a sample which has outward water pressure
gradient of 60 kPa
However, such a testing condition produces issues in undrained tests where positive
excess pore pressure is generated. Ishihara and Yamazaki (1980) reported an inflation issue
of the rubber membrane as shown in Figure 3.3. When excess pore pressure increases,
the actual back pressure becomes higher than the cell pressure, from which an outward
water pressure gradient arises. As a result, water can be expelled from the soil sample and
accumulates at the place where physical confinement ends, leading to the phenomenon in
Figure 3.3. The apparent consequence of water discharge is the reduction of sample height.
By holding constant vertical load, the height of soil sample should remain constant if there is
no volume change. Once water flows out from the sample, on the contrary, sample height
reduces dramatically (more than 10% recorded in pilot tests), which not only makes the
constant volume assumption of undrained testing invalid but also results in uncontrolled
compression of soil samples.
To reduce the effects of the outward water pressure gradient, cell pressure can be raised
but cannot exceed σv ·K0,min with K0,min representing the minimum value of K0 that can be
experienced during a test. Boulanger (1990) compared soil samples consolidated under a
series of lateral stresses and concluded that initial K0 is closest to 0.4. While the value of
3.3 Inflation problem in undrained simple shear tests 47
Fig. 3.5 The photo of reinforced membrane deformed by outward water pressure gradient
K0,min can vary with the degree of saturation and is difficult to determine, a general assumption
is to regard K0,min as equal to initial K0. With this assumption, the maximum allowable
difference between cell pressure and back pressure is 40 kPa for a sample consolidated
with vertical overburden stress of 100 kPa. When liquefaction takes place and excess pore
pressure increases to be close to vertical overburden stress, the resultant outward water
pressure gradient can be reduced from 100 kPa with equilibrated cell and back pressures
to 60 kPa, with 40 kPa positve cell pressure. Figure 3.4 presents the photo of a soil sample
whose outward water pressure gradient is 60 kPa. The startling water pouch indicates clearly
that the inflation issue still exists at this amount of outward water pressure gradient.
Another potential method to solve the problem is to apply a physical constraint to the
full height of the membrane, eliminating the space for the expelled water to accumulate. If
stacked rings are used, it can be achieved by clamping O-rings right on the top of stacked
rings. But when top cap is lowered to maintain constant vertical load, the clamps and O-rings
will contact and press the stacked rings, resulting in undesirable vertical stress condition.
This issue can be avoided if a reinforced membrane is used instead of stacked rings and
O-rings are clamped at the bottom of the top cap. Although the vertical stress condition can
48 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
be guaranteed, however, the high outward pressure can deform the membrane severely even
if it is reinforced, as shown by the photo in Figure 3.5.
As all the convenient options were not effective, the top cap and sample preparation
procedures were redesigned in this study to minimise the issue of inflation. Figure 3.6 shows
the schematics of a soil sample in preparation (a) and after preparation (b). The modified
sample preparation method allows physical confinement to be applied to the full height of
membrane. The modified specimen procedure will be detailed in the next section.
Fig. 3.6 The schematics of a soil sample in preparation and after preparation
3.4 Testing details 49
Table 3.2 The summary of transducer replacement
Replaced
transducers
Measurement/
Control
Current model
X Load cell Force in X direction Strainsert universal flat load cell
Model FL025U(C)2NSPKT.
Capacity: 250 lbf (1.11 kN).
Non-Linearity: ±0.10% F.S.
Z Load cell Force in Z direction
(vertical)
Interface Force universal load cell
Model 1010-AF-500. Capacity: 500
lbf (2.22 kN). Non-Linearity:
±0.04% F.S.
Pore pressure
transducers 1
Pore pressure at the
bottom of volume
change measurement
Festo pressure transducers Model
SPTW-P6R-G14-VD-M12
Air pressure
controller
Control the air pressure
inside chamber
SMC pressure controller Model
ITV2030-31F2BL3
Volume change
pressure controlled
Control the volume
change regulation air
pressure
SMC pressure controller Model
ITV2030-31F2BL3
3.4 Testing details
3.4.1 Multi-directional direct simple shear apparatus
The multi-directional direct simple shear apparatus developed by Rutherford and Biscontin
(2013) has been upgraded and used for the experimental program in this study. A photo of
the testing device is shown in Figure 3.7.
The apparatus is composed of four primary parts: loading chamber, data acquisition and
control system, hydraulic power system and back pressure system. The loading chamber
contains a vertical loading actuator and two horizontal loading actuators, as well as two arms
to uplift or lower the chamber. The power of the chamber lifting arms and load actuators
50 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
is provided by the hydraulic power system. The air pressure in the chamber and the water
pressure in soil sample are controlled through the back pressure system. Data acquisition
and loading programming is conducted through control software Dimension. The schematic
and more details about the device can be found in Rutherford and Biscontin (2013).
The load cells, hydraulic system and data acquisition system were modified to provide
better control of testing conditions. The load applied by vertical and horizontal actuators is
measured by load cells. The displacements are measured by LVDTs. Cell pressure and back
pressure is controlled by pressure controllers and measured by pressure transducers. Excess
pore pressure is measured directly by a differential pressure transducer. A summary of the
replacement of transducers and digital controllers and their function is tabulated in table 3.2.
The information of other transducers was reported by Rutherford and Biscontin (2013).
This apparatus has two primary features. Firstly, the testing system has two horizontal
tables that can move in perpendicular directions independently. The application of multi-
directional shearing is thus made possible. Secondly, a chamber encloses the specimen,
allowing air confining pressure to balance back pressure so that the measurement of excess
pore pressure is viable. The photo of the chamber with a mounted sample is presented in
Figure 3.8, with the components inside the loading chamber indicated.
As shown in Figure 3.8, soil samples are reconstituted between bottom cap and top cap,
and enclosed by an unreinforced latex membrane. A stack of lubricated thin steel rings
is employed to confine lateral deformations of the soil sample. The stacked rings are the
same as those used by Bernhardt (2013) with average inner diameter of 101.6 mm. The
stacked rings are lubricated regularly with Teflon spray to reduce friction between the rings.
Following Franke et al. (1979), the height of samples was ensured to be lower than 25 mm so
that the diameter-to-height ratios of all soil samples were always larger than 3.75 to reduce
the effects of the lack of complementary stresses.
3.4.2 Specimen preparation procedures
The schematics of samples in preparation and after preparation are shown in Figure 3.6 with
the essential components indicated. The procedures of specimen preparation are as follows:
3.4 Testing details 51
Fig. 3.7 A photo of the multi-directional direct simple shear testing apparatus
(1) Assembling the split ring stand. Latex membrane and split ring stand are assembled
on the bottom cap. The top surface of the ring stand needs to be aligned with that of
the bottom cap to ensure the stacked rings can move horizontally when the sample is
deformed. The latex membrane is enclosed by the split ring stand and the ring stand is
fastened with a clamp to provide a bottom seal for soil samples.
52 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Fig. 3.8 A photo of the testing device with a sample mounted
(2) Assembling the split mold. Steel rings are stacked on the ring stand followed by the
split mold assembled. The split mold and ring stand are aligned and clamped together
to hold the rings in the correct position. The membrane is rolled over the mold and
held temporarily by O-rings. A small vacuum is applied to the split mold to help the
membrane cling to the stacked rings.
(3) Air pluviation of sand and surface levelling. A funnel is used to pluviate sand into the
mold. The nozzle of the funnel is kept at the top of sand surface and moved in a spiral
path to produce samples in a loose state. Vibration is applied to the mold with a rubber
hammer to densify soil samples, if required. A leveller is used to flatten the top surface
of sands and remove redundant sand. The surface of the sand should be slightly lower
than the mold to leave the necessary space for mounting the top cap.
3.4 Testing details 53
(4) Mounting the top cap. The top cap is placed onto the sand surface. The membrane is
stretched and pulled over the sidewall of the top cap. Three O-rings are used to clamp the
membrane into the groove on the top cap. More steel rings are placed onto the existing
ones until they are stacked to the lower edge of the groove. A pair of split spacers is put
into the groove and fastened by a clamp.
(5) CO2 flushing. CO2 is flushed through the reconstituted sand sample via the inlets on the
bottom cap and top cap. The speed of CO2 flow needs to be controlled carefully to avoid
sample blow up. The three-way valves at the outer ends of the cap pipes are closed for
sample transfer when the CO2 flushing is finished.
(6) Transferring the sand sample to the testing device. The sand sample is placed into
the multi-directional direct simple shear testing apparatus and aligned with the vertical
loading ram. The vertical loading ram is lowered until a small amount of seating load
(around 5 kPa) is applied to the sample. The top cap and the vertical loading ram are
clamped together. The T-bar is then tightened to fix the sample assembly. This prevents
pre-shearing of the specimen.
(7) Water flushing. The two ports on the three-way valves of the top cap and bottom cap
are connected to the back pressure system and a water reservoir, respectively. A small
amount of vacuum (around 5 kPa) is applied to the water reservoir connected to the top
cap, while de-aired water is filled into the water reservoir connecting to the bottom cap.
The port to the top cap water reservoir is first opened, then the bottom cap one. De-aired
water in the bottom cap reservoir will be sucked into the sample and flow out into the top
cap reservoir when the sample is fully filled with de-aired water. The process is reversed
and repeated for several times until no visible air bubbles can be observed in the tubes.
The ports on the three-way valves, which connect to the back pressure system, are finally
opened.
A sample is successfully prepared by this point. With the newly-designed top cap,
physical constraints can be fully applied to the specimen. The bottom of the sample is sealed
by the clamped split ring stand. The top is fastened by the clamped split spacers and O-rings.
54 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Steel rings are stacked along the full height between the top and bottom. A water pouch
was not formed in the tests prepared with this method and the largest reduction of sample
height was 2.2%. B-values above 0.9 were obtained in 75 of the 79 tests in this study. The
B-values in this study were generally lower than the typical values (above 0.95) seen in
literature because the capacity of pore pressure transducer did not allow the application of
higher back pressure. But the test results were considered valid since a B-value of 0.9 still
gives a nearly complete saturation, with the saturation degree over 99.5% according to Black
and Lee (1973).
3.4.3 Simple shear testing procedures
Once water flushing is finished and the connections of the sample assembly are switched
to back pressure system, the chamber is closed while the sand specimen is still confined by
seating load. The testing procedures adopted in this study are as follows:
(1) Saturation. Cell pressure is raised in small increments to a target value (usually 270
kPa). Back pressure is increased simultaneously but always kept slightly lower than
cell pressure to ensure a small net compressive pressure. The sample is saturated with
the high cell pressure and back pressure for a period of time (usually 1 hour). Vertical
overburden stress remains at the seating load in this phase.
(2) B-value check. B-value check is conducted at the end of the saturation phase. The
valves are closed and the cell pressure is increased rapidly by a small amount. The
change of pore pressure inside soil sample is measured and the B-value is calculated.
Back pressure is then raised to a value slightly lower than cell pressure. Valves are
opened and pore pressure stabilises. These steps are repeated until a sufficient saturation
is reached.
(3) Consolidation. Once B-values reach satisfactory values, the vertical stress is ramped
up to the required consolidation stress while cell pressure and back pressure remain
constant. The consolidation phase ends when the volume change becomes insignificant
and excess pore pressure is dissipated.
3.5 Data correction 55
(4) Initial static shearing. If the loading path to be tested does not pass through the origin,
a ramp stress phase needs to be conducted to avoid abrupt increase of shear stress. Static
shearing is applied with drainage open. LVDT readings are zeroed when initial static
shear stress target is reached.
(5) Shear testing. Valves are closed to conduct undrained shear tests. Monotonic tests
are displacement-controlled with increasing displacement at a constant rate (5%/hour).
Cyclic tests are stress-controlled with required loading amplitude and waveform.
3.5 Data correction
Frictions and cross-coupling are two of the major issues concerning multi-directional simple
shear tests. The measurement of applied load can be biased by the frictions that exists in
loading tables and stacked rings or by the cross-coupling between the two horizontal strokes.
Some pilot tests were conducted to probe the effects of friction and cross-coupling and the
results of an example test are shown in Figure 3.9 and 3.10. This example test was conducted
with sample assembly (caps and stacked rings) but without soil. Cell pressure was increased
to 280 kPa to reproduce actual pressure situation. Sinusoidal displacement with amplitude of
1 mm was applied in X direction at a frequency of 0.1 Hz to simulate cyclic loading scenario.
Figure 3.9 presents the measurements of horizontal load and displacement in this test.
Synchronization is found between the variation of X-load and X-displacement. Since the
sample assembly had no sand in it, the load readings should come from either friction or
transducer noise. Load readings are thus filtered through a low pass Butterworth filter as
shown in Figure 3.10(a). The filtered loads are replotted versus X-displacement in Figure
3.10(b). The correlation between filtered X-load and X-displacement evidences the existence
of friction which can be determined from the angle of inclination at approximately 0.003
kN/mm, corresponding to a shear stress of 0.36 kPa/mm. It should be noted that the actual
load cell readings can drift away from the calculated friction because of device oscillations
and imbalanced motion of the load actuator. The correction for friction will not be consistent
and accurate under this circumstance. Fortunately, the effects of friction are not pronounced
56 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
0 50 100 150 200 250 300
Time (s)
-0.01
0
0.01
0.02
Lo
ad
 (k
N)
(a) Load mesurement
X Load
Y Load
0 50 100 150 200 250 300
Time (s)
-1
-0.5
0
0.5
1
LV
D
T 
(m
m)
(b) LVDT mesurement
X LVDT
Y LVDT
Fig. 3.9 The measurements of horizontal load and displacement load in a pilot test
in this study, considering the amplitude of shear stress applied in this study is always above
15 kPa and displacement becomes higher than 2 mm only when liquefaction takes place. The
friction can be further reduced if the horizontal strokes and steel rings are lubricated regularly
by oil and Teflon spray, respectively, as done in this study.
The variation of Y-load and Y-displacement with X-displacement, by contrast, is minimal.
It indicates that the effects of the cross-coupling between horizontal loading tables are not
significant. It should be noted, however, that the cross-coupling phenomenon can become
more discernible in the tests with actual soil samples, which can be observed clearly in the
results of bi-directional linear tests such as Figure 5.5b. It is impossible to determine whether
this is a result of device limitations or due to the anisotropy of the soil samples. Corrections
to account for the effects of cross-coupling are thus not applied in case inaccurate “correction”
misleads the understanding of test data.
3.6 Liquefaction criterion 57
0 50 100 150 200 250 300
Time (s)
-0.01
0
0.01
0.02
Lo
ad
 (k
N)
(a) X Load filtering
Unfiltered X Load
Filtered X Load
Unfiltered Y Load
Filtered Y Load
-1.5 -1 -0.5 0 0.5 1 1.5
X LVDT (mm)
0
2
4
6
Lo
ad
 (k
N)
10-3 (b) Filtered Load versus X LVDT
Filtered X Load
Filtered Y Load
Fig. 3.10 Filtered load versus X displacement
Other potential corrections including membrane penetration and inertial loading were
demonstrated by Boulanger (1990) and Wu (2002) to be insignificant for the grain size and
loading frequency used in this study. Thus, no corrections are made to the test data presented
in this study. All the test results shown in this dissertation are based on uncorrected data.
3.6 Liquefaction criterion
A consistent liquefaction criterion in simple shear tests is required to compare liquefaction
resistance and understand liquefaction phenomenon of sands. There are two widely accepted
categories of liquefaction criterion. The first category of liquefaction criterion relates lique-
faction to an excess pore pressure ratio of unity (Seed and Lee, 1966). Since early studies
considered the cause of liquefaction as the complete loss of effective confining pressure,
this criterion was generally effective, especially in the tests with uni-directional loading.
58 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
However, the existing results of multi-directional simple shear tests indicated that excess
pore pressure ratio does not always increase to unity in multi-directional loading conditions.
Thus, this criterion may not be applicable to the tests in this study. The other category of liq-
uefaction criterion relates liquefaction to a certain level of shear strain because a large strain
level indicates sand’s resistance to deformation has been reduced significantly. There are
various strain-based liquefaction criteria in the literature. For instance, Tatsuoka et al. (1982)
defined liquefaction as double amplitude shear strain of 15%, while Vaid and Sivathayalan
(2000) used single amplitude shear strain of 3.75% as the requirement of liquefaction. In
one of the earliest liquefaction studies, however, Seed and Lee (1966) indicated that sand
loses resistance to deformation after shear strain exceeds 3%. The 3% shear strain criterion
was subsequently used by Ishihara and Yamazaki (1980) in the earliest multi-directional
simple shear tests on saturated sands. Realising the shear strain in uni-directional cyclic
simple shear tests is not always symmetrical, Kammerer (2002) and Wu (2002) included a
double amplitude shear strain of 6% in their liquefaction criterion. The criterion selected
for uni-directional tests in this study follows this criterion to define liquefaction as either
single-amplitude shear strain over 3% or double-amplitude shear strain over 6%, whichever
happens first. A discussion on the liquefaction criterion for multi-directional loading will be
made in Chapter 7.
3.7 Parameter definition
Figure 3.11 presents a graphical illustration of a figure-8 loading path, which starts from the
point S. Some important parameters that will be used for the analysis in this dissertation are
illustrated on the graph. SSR is shear stress ratio, which is the ratio of shear stress over initial
effective vertical stress. CSR is cyclic stress ratio and equals half of the range of shear stress
ratio. Although Figure 3.11 is plotted in shear stress ratio coordinates, the definition of the
parameters can be transformed straightforwardly to shear stress and shear strain coordinates.
Dip direction (or downhill direction) is defined as the direction along which an initial
static shearing is applied. Strike direction is perpendicular to dip direction. It is worth to
3.7 Parameter definition 59
mention that the loading system has a different coordinate system, with two axes named as X
and Y. In most of the tests performed, initial static shearing is applied only in X direction,
that is, X direction of loading system is also the dip direction. But there are also some tests
where Y direction or other directions (such as Figure 3.11) of the loading system become the
dip direction. For simplicity, X and Y direction are preferred in this study to present test data
in an intuitive manner, while dip and strike direction are used to facilitate the calculation
of certain parameters and the analysis of certain loading paths (bi-directional loading in
particular).
Fig. 3.11 Graphical illustration of a figure-8 loading path and the definition of directions and
parameters
The subscript total is used to indicate the calculation is based on resultant shear stress or
strain and directionality is not taken into consideration. The subscript dir means a parameter
is defined with directionality considered. For instance, CSRdir is the maximum cyclic shear
stress ratio across all directions.
60 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Kammerer (2002) proposed to use aperture ratio (AR) and reversal parameter (RP) to
characterise a loading path, which are defined as follows:
AR =
CSRperp
CSRdir
(3.1)
RP =
CSRα −α
CSRdir
(3.2)
where CSRperp represents to the cyclic shear stress ratio in the direction perpendicular to the
orientation along which CSRdir is determined, α is the initial static shear stress ratio and
CSRα is the cyclic shear stress ratio in the direction of static shearing. Initial static shear
refers to the shearing condition at the geometric center of a loading path in this study, unless
indicated otherwise.
The definition of other parameters will be given when they are introduced in the following
chapters.
3.8 Summary of tests
The summary of the tests performed is tabulated in table 3.3. Four monotonic and seventy
five cyclic undrained simple shear tests were conducted on clean Hostun sands in this study.
The relative density of sand samples ranged from 30.3% to 77.8%, though 77 out of the 79
tests had relative density between 46.8% and 68.7%. All the tests started with initial vertical
overburden stress of 100 kPa except for test LM-4 which was consolidated under σv,0 of
20 kPa. Four tests had Skempton’s B-values lower than 0.9 (test LS8, C15-2, C20-2 and
E2010-2) with the lowest value at 0.854. All the other tests held B-values higher than 0.9
which corresponds to a degree of saturation over 99.5% according to Black and Lee (1973).
Constant vertical load was adopted as the boundary condition of all the tests. The largest
change of height was measured in test C20-4 where the height of the soil sample reduced
by 2.2% (from 16.785 mm to 16.415 mm) when the test was terminated, manifesting that
satisfactory undrained condition was maintained with the modified top cap and sample
preparation procedures.
3.8 Summary of tests 61
Cyclic loading was applied in a sinusoidal manner. Uni-directional and bi-directional
linear tests had sinusoidal shear waves in only one direction with a frequency of 0.1 Hz.
Circular, oval and figure-8 tests had shear waves in both X and Y direction with the same
master frequency of 0.05 Hz but different frequency multiplier (figure-8 tests), amplitude
and phase difference (circular and oval tests).
These loading paths were widely used in previous research and were thus selected to
provide a baseline for comparisons. The loading condition of these tests is summarized in
table 3.3. CSRX and CSRX represents the amplitude of cyclic shear stress ratio in the X and
Y direction of the loading system, while SSRX ,0 and SSRY,0 indicates the initial shear stress
ratios along X and Y direction. The direction of the resultant initial static shear force is the
dip direction (or downhill direction), which will be discussed when certain loading paths
(bi-directional loading in particular) are analysed. Again, the dip and strike directions are not
necessarily aligned to X or Y direction.
Some of the important results that will be used in the analysis in the following chapters
are also included in table 3.3. ru,max is the maximum excess pore pressure ratio recorded
in a test by the cycle where liquefaction takes place. ru,min refers to the minimum excess
pore pressure ratio in the liquefaction cycle. Kµ is the coefficient of multi-directional loading
based on directional cyclic shear stress ratio CSRdir, which will be clarified in chapter 7.
62 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Ta
bl
e
3.
3
Su
m
m
ar
y
of
th
e
co
nd
uc
te
d
te
st
s
Te
st
na
m
e
Te
st
ty
pe
D
r
(%
)
B
-v
al
ue
N
L
σ v
,0
SS
R
X
,0
C
SR
X
SS
R Y
,0
C
SR
Y
r u
,m
ax
r u
,m
in
K
µ
L
M
-1
U
ni
-d
ir
ec
tio
na
lm
on
ot
on
ic
61
.1
0.
94
4
0
10
0
L
M
-2
U
ni
-d
ir
ec
tio
na
lm
on
ot
on
ic
59
.7
0.
92
0
10
0
L
M
-3
U
ni
-d
ir
ec
tio
na
lm
on
ot
on
ic
57
.4
0.
92
2
0
10
0
L
M
-4
U
ni
-d
ir
ec
tio
na
lm
on
ot
on
ic
57
.4
0.
92
2
0
20
L
C
1
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
55
.8
0.
97
86
10
0
0
0.
14
9
0
0
0.
95
0.
54
L
C
2
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
56
.1
0.
96
13
8
10
0
0
0.
14
8
0
0
1.
01
0.
78
L
C
3
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
50
.5
0.
91
5
39
10
0
0
0.
15
2
0
0
0.
99
0.
78
L
C
4
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
51
.7
0.
93
2
83
10
0
0
0.
15
2
0
0
0.
97
0.
69
L
C
5
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
53
.9
0.
97
44
10
0
0
0.
19
5
0
0
0.
92
0.
69
L
C
6
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
57
.4
0.
95
27
10
0
0
0.
19
5
0
0
0.
95
0.
64
L
C
7
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
60
.7
0.
97
62
10
0
0
0.
19
7
0
0
0.
93
0.
65
L
C
8
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
65
0.
94
3
13
2
10
0
0
0.
19
9
0
0
0.
99
0.
68
L
C
9
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
58
.5
0.
94
4
41
10
0
0
0.
19
5
0
0
0.
96
0.
69
L
C
10
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
61
.9
0.
93
19
10
0
0
0.
24
4
0
0
1.
02
0.
69
L
C
11
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
60
0.
95
9
10
10
0
0
0.
24
6
0
0
0.
90
0.
54
L
C
12
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
67
0.
95
3
51
10
0
0
0.
30
1
0
0
0.
96
0.
56
L
C
13
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
58
.2
0.
96
9
67
10
0
0
0.
16
4
0
0
0.
87
0.
49
3.8 Summary of tests 63
Ta
bl
e
3.
3
co
nt
in
ue
d
fr
om
pr
ev
io
us
pa
ge
Te
st
na
m
e
Te
st
ty
pe
D
R
(%
)
B
-v
al
ue
N
L
σ v
,0
SS
R
X
,0
C
SR
X
SS
R Y
,0
C
SR
Y
r u
,m
ax
r u
,m
in
K
µ
L
C
14
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
58
.4
0.
92
28
12
1
10
0
0
0.
19
5
0
0
0.
93
0.
69
L
C
15
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
57
.5
0.
93
1
2
10
0
0
0.
25
9
0
0
0.
97
0.
66
L
C
16
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
77
.8
0.
92
1
90
10
0
0
0.
29
8
0
0
0.
94
0.
44
L
C
17
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
59
.9
0.
94
5
4
10
0
0
0.
24
6
0
0
0.
92
0.
54
L
C
18
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
57
.7
0.
93
1
6
10
0
0
0.
29
5
0
0
1.
00
0.
49
L
C
19
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
67
.8
0.
93
7
12
10
0
0
0.
34
4
0
0
0.
89
0.
41
L
C
20
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
50
.9
0.
95
5
10
0
0
0.
19
4
0
0
0.
89
0.
53
L
C
21
U
ni
-d
ir
ec
tio
na
ll
in
ea
r
61
.1
0.
91
3
32
10
0
0
0.
19
4
0
0
0.
96
0.
66
L
S1
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
57
.6
0.
93
6
46
10
0
0.
03
3
0.
20
9
0
0
1.
01
0.
59
1.
11
L
S2
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
66
.7
0.
96
1
22
2
10
0
0.
04
8
0.
21
0
0
0
1.
02
0.
62
1.
04
L
S3
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
53
.4
0.
95
8
5
10
0
0.
05
8
0.
25
5
0
0
1.
00
0.
51
1.
16
L
S4
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
58
.4
0.
94
1
74
10
0
0.
06
1
0.
25
7
0
0
0.
95
0.
51
1.
42
L
S5
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
58
.6
0.
94
8
31
10
0
0.
07
6
0.
25
6
0
0
1.
00
0.
58
1.
26
L
S6
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
56
.6
0.
93
3
50
10
0
0.
05
9
0.
26
0
0
0
1.
01
0.
44
1.
43
L
S7
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
58
.2
0.
94
8
28
10
0
0.
12
2
0.
25
5
0
0
1.
01
0.
43
1.
25
L
S8
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
60
.5
0.
89
2
10
0
0.
07
4
0.
26
0
0
0
0.
97
0.
44
0.
81
L
S9
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
63
.5
0.
91
1
2
10
0
0.
15
8
0.
30
9
0
0
0.
99
0.
20
0.
70
64 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Ta
bl
e
3.
3
co
nt
in
ue
d
fr
om
pr
ev
io
us
pa
ge
Te
st
na
m
e
Te
st
ty
pe
D
R
(%
)
B
-v
al
ue
N
L
σ v
,0
SS
R
X
,0
C
SR
X
SS
R Y
,0
C
SR
Y
r u
,m
ax
r u
,m
in
K
µ
L
S1
0
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
55
.5
0.
94
5
10
0
0.
20
7
0.
25
4
0
0
0.
98
0.
25
1.
07
L
S1
1
U
ni
-l
in
ea
r/
st
at
ic
sh
ea
r
59
.9
0.
91
9
1
10
0
0.
29
2
0.
26
5
0
0
0.
58
0.
00
0.
80
B
ili
ne
ar
-1
B
i-
di
re
ct
io
na
ll
in
ea
r
55
.8
0.
95
4
40
10
0
0.
10
4
0
0
0.
25
1
0.
90
0.
61
1.
37
B
ili
ne
ar
-2
B
i-
di
re
ct
io
na
ll
in
ea
r
56
.7
0.
94
1
20
10
0
0
0.
24
8
0.
06
7
0
0.
97
0.
62
1.
21
B
ili
ne
ar
-3
B
i-
di
re
ct
io
na
ll
in
ea
r
61
.5
0.
92
4
2
10
0
0
0.
24
8
0.
07
6
0
0.
93
0.
50
0.
69
B
ili
ne
ar
-4
B
i-
di
re
ct
io
na
ll
in
ea
r
55
.7
0.
96
7
3
10
0
0.
07
2
0
0
0.
30
2
0.
95
0.
50
1.
19
B
ili
ne
ar
-5
B
i-
di
re
ct
io
na
ll
in
ea
r
61
.2
0.
93
1
2
10
0
0.
11
9
0
0
0.
31
3
0.
86
0.
47
0.
89
B
ili
ne
ar
-6
B
i-
di
re
ct
io
na
ll
in
ea
r
30
.3
0.
92
6
1
10
0
0.
15
2
0
0
0.
20
1
0.
78
0.
58
B
ili
ne
ar
-7
B
i-
di
re
ct
io
na
ll
in
ea
r
49
.9
0.
94
9
3
10
0
0.
15
2
0
0
0.
14
7
0.
85
0.
75
0.
73
B
ili
ne
ar
-8
B
i-
di
re
ct
io
na
ll
in
ea
r
61
.1
0.
92
2
4
10
0
0.
15
3
0
0
0.
30
5
0.
81
0.
46
0.
98
B
ili
ne
ar
-9
B
i-
lin
ea
r/
st
at
ic
sh
ea
r
55
.9
0.
93
9
1
10
0
0.
15
3
0
0.
11
1
0.
09
2
0.
82
0.
67
0.
31
B
ili
ne
ar
-1
1
B
i-
lin
ea
r/
st
at
ic
sh
ea
r
56
.9
0.
90
1
3
10
0
0.
15
4
0
0.
14
5
0.
16
9
0.
76
0.
47
0.
64
B
ili
ne
ar
-1
2
B
i-
di
re
ct
io
na
ll
in
ea
r
48
.7
0.
95
5
2
10
0
0.
30
4
0
0
0.
17
6
0.
55
0.
48
0.
84
B
ili
ne
ar
-1
3
B
i-
di
re
ct
io
na
ll
in
ea
r
59
.9
0.
90
2
3
10
0
0.
29
6
0
0
0.
29
5
0.
53
0.
34
1.
03
C
15
-1
C
ir
cu
la
r
56
0.
95
2
6
10
0
0
0.
15
0
0
0.
14
9
0.
84
0.
76
0.
64
C
15
-2
C
ir
cu
la
r
57
0.
88
2
58
10
0
0
0.
15
4
0
0.
15
1
0.
78
0.
73
0.
85
C
15
-3
C
ir
cu
la
r
57
.6
0.
95
1
40
10
0
0
0.
15
1
0
0.
15
0
0.
79
0.
73
0.
78
3.8 Summary of tests 65
Ta
bl
e
3.
3
co
nt
in
ue
d
fr
om
pr
ev
io
us
pa
ge
Te
st
na
m
e
Te
st
ty
pe
D
R
(%
)
B
-v
al
ue
N
L
σ v
,0
SS
R
X
,0
C
SR
X
SS
R Y
,0
C
SR
Y
r u
,m
ax
r u
,m
in
K
µ
C
15
-4
C
ir
cu
la
r
58
.9
0.
96
1
17
10
0
0
0.
15
1
0
0.
15
1
0.
88
0.
80
0.
68
C
15
-5
C
ir
cu
la
r
59
.7
0.
95
4
63
10
0
0
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4
0
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15
1
0.
80
0.
73
0.
81
C
20
-1
C
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cu
la
r
59
.6
0.
97
26
10
0
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20
3
0
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20
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0.
73
0.
66
0.
95
C
20
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C
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cu
la
r
60
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2
2
10
0
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20
1
0
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0
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71
0.
64
0.
62
C
20
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C
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la
r
63
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0.
95
6
19
10
0
0
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20
1
0
0.
20
1
0.
77
0.
72
0.
70
C
20
-4
C
ir
cu
la
r
63
.3
0.
96
1
6
10
0
0
0.
20
2
0
0.
19
9
0.
81
-0
.0
1
0.
57
C
20
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C
ir
cu
la
r
49
.2
0.
97
5
20
10
0
0
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20
1
0
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19
9
0.
71
0.
68
1.
19
C
25
-1
C
ir
cu
la
r
60
.9
0.
96
6
2
10
0
0
0.
24
5
0
0.
24
5
0.
72
0.
65
0.
73
C
15
-X
7.
5-
1
C
ir
cu
la
r/
st
at
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sh
ea
r
55
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0.
90
1
33
10
0
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07
2
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15
2
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3
0.
94
0.
69
0.
82
C
15
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15
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C
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cu
la
r/
st
at
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ea
r
56
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96
5
89
10
0
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14
7
0.
15
2
0
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15
0
0.
84
0.
55
0.
90
C
20
-X
05
-1
C
ir
cu
la
r/
st
at
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sh
ea
r
53
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97
1
8
10
0
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2
0
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81
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62
0.
98
C
20
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C
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cu
la
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at
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ea
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60
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0.
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92
10
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9
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80
0.
64
1.
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C
20
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10
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la
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st
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C
20
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la
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9
0.
86
0.
58
0.
63
C
20
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15
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C
ir
cu
la
r/
st
at
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sh
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62
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97
3
12
10
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8
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94
0.
55
0.
69
C
20
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20
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la
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st
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sh
ea
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62
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5
20
10
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1
0
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98
0.
42
0.
74
E
17
13
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va
l
58
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96
9
26
10
0
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0
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13
0
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79
0.
74
0.
73
66 MULTI-DIRECTIONAL SIMPLE SHEAR TESTING
Ta
bl
e
3.
3
co
nt
in
ue
d
fr
om
pr
ev
io
us
pa
ge
Te
st
na
m
e
Te
st
ty
pe
D
R
(%
)
B
-v
al
ue
N
L
σ v
,0
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R
X
,0
C
SR
X
SS
R Y
,0
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SR
Y
r u
,m
ax
r u
,m
in
K
µ
E
20
10
-1
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l
54
0.
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4
19
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1.
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30
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69
1.
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20
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8
60
10
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0.
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0.
89
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20
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89
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76
1.
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*
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Y
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of
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.
Chapter 4
UNI-DIRECTIONAL SIMPLE SHEAR
TEST RESULTS
4.1 Uni-directional linear tests
Uni-directional loading is the most commonly tested category of loading paths that have
been used in existing research regarding the cyclic behavior of soil, with liquefaction-related
research being no exception. A group of uni-directional tests were thus conducted in this
study, including both monotonic tests and cyclic tests, with or without initial static shearing
Together, these tests illustrate the sand behaviour under uni-directional loading scenarios and
provide the baseline for the comparisons with the multi-directional tests.
4.1.1 Uni-directional tests without static shearing
Monotonic Tests
Four uni-directional linear monotonic tests were conducted to investigate the fundamental
soil behaviour and characteristics of clean Hostun sand. The angle of the failure line and
phase transformation line for the sand used in this study can be determined through these
tests. The relative density of the sample after consolidation varies from 57.4% to 61.1%. Test
68 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
LM-1, LM-2 and LM-3 were consolidated with an initial vertical stress at 100 kPa while test
LM-4 was consolidated to 20 kPa.
0 20 40 60 80 100 120 140 160 180 200
Effective vertical stress 
v
' (kPa)
0
20
40
60
80
100
120
140
160
180
Sh
ea
r s
tre
ss
 
 
(kP
a)
Fai
lure
 Lin
e
Phase
-trans
forma
tion L
ine
LM-1
LM-2
LM-3
LM-4
Fig. 4.1 The stress paths of the four monotonic uni-directional tests
Figure 4.1 shows the stress paths of all the four monotonic tests. The solid black line in
the graph represents the failure line and the dashed black line is the phase transformation line.
The angle of the failure line is 33.4◦, and the angle of phase transformation line is 19.3◦. It
is found that the angle of failure of the four tests did not converge to a single value. This is
attributed partly to the inability of the tests to reach steady state since they were all stopped
at a shear strain smaller than 20%, and partly to the friction in testing system. Judgement
was thus made in determining the value. The determined angle of 33.4◦ is comparable to the
value of 33◦ by Mitrani (2006) and 33.7◦ by Doanh et al. (1997).
All the samples exhibit typical dilative behaviour as described by Castro (1969). While
shear stress increases, the effective vertical stress decreases slightly first because of the
generation of positive excess pore pressure. As the stress paths approach and hit the phase
4.1 Uni-directional linear tests 69
transformation line, however, the effective vertical stress is gradually reversed due to the
mobilised dilative behaviour and the decrease of excess pore pressure thus induced. The
stress paths then develop towards the failure line until they eventually reach it and develop
along the critical state line. The development of excess pore pressure in the four tests is
presented in Figure 4.2. The initial increase of pore pressure, followed by a turn, is clearly
shown in this figure.
0 2 4 6 8 10 12 14 16 18
Shear strain  (%)
-120
-100
-80
-60
-40
-20
0
20
Ex
ce
ss
 p
or
e 
pr
es
su
re
 
u
 (k
Pa
)
LM-1
LM-2
LM-3
LM-4
Fig. 4.2 The development of excess pore pressure in the four monotonic uni-directional tests
The dilative behaviour of the samples can also be seen from its stress-strain response, as
presented in Figure 4.3. The shear stress climbs all the way up with increasing shear strain.
No local peaks of shear stress are encountered and no flow-type development of shear strain
is observed, confirming the dilative response of the Hostun Sand of the density range tested.
The critical state point where the increment rate of shear stress becomes and remains zero is
not reached before these tests were terminated.
70 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 2 4 6 8 10 12 14 16 18
Shear strain  (%)
0
20
40
60
80
100
120
Sh
ea
r s
tre
ss
 
 
(kP
a)
LM-1
LM-2
LM-3
LM-4
Fig. 4.3 The stress-strain relationships of the four monotonic uni-directional tests
Cyclic Tests
For a soil element underneath level ground, the earthquake loading scenarios are convention-
ally stimulated by uni-directional cyclic tests without static shearing (ground-level tests). As
discussed earlier, the occurrence of soil liquefaction is determined when double-amplitude
shear strain exceeds 6% or single-amplitude shear strain reaches 3%.
The results of an example test (LC10) are shown here to illustrate the characteristics of
this category of tests. For cyclic tests, shear stress is more commonly reflected by cyclic shear
stress ratio (SSR) rather than the true value of deviator stress. In simple shear tests, shear
stress ratio is defined as the ratio of shear stress over initial vertical stress. This convention
will be followed in this study. The example test has a relative density of 61.9% and a B value
of 0.93, with an actual initial vertical stress at 98.7 kPa. The sample liquefies at the 19th
cycle, sheared by a cyclic shear stress of 25 kPa (CSR=0.25) along the X direction.
4.1 Uni-directional linear tests 71
Fig. 4.4 The development of shear strain and excess pore pressure of test LC10
The development of shear strain and excess pore pressure as the shear stress is applied is
presented in Figure 4.4. Excess pore pressure accumulates with shearing cycles from the very
beginning of the test. Both the maximum excess pore pressure and the residual excess pore
pressure of each cycle increases in general. By contrast, the shear strain does not change a
lot initially, maintaining a double amplitude of around 2%, until the last few cycles where the
excess pore pressure is high enough and the residual strength of the sample is much reduced
due to the lowered effective vertical stress.
The stress path of the test is shown in Figure 4.5. With the generation of the excess pore
pressure, the effective vertical stress declines gradually, making the stress path move towards
the origin. A relatively large increase of excess pore pressure is witnessed in the first cycle,
as commonly observed. In the next tens of cycles when the excess pore pressure remains
low and the stress path is far away from the phase transformation line, the sample exhibits
a dilative response. During the two loading phases of each cycle, the shearing, together
with the dilation tendency of sand, results in a small drop in the effective vertical stress,
72 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
10 100
v
' (kPa)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R X
SSRX- v'
Fai
lure
 Lin
e
PT Li
ne
Fig. 4.5 The stress path of test LC10
if at all, or even increases with increasing SSR. In the unloading stages, by contrast, the
dilation mobilised in sand is released and the effective vertical stress reduces. The excess
pore pressure increases twice and decreases twice respectively within a single cycle, giving a
widely-recognised "double peak" feature in the pore pressure plot, which can be seen more
clearly in Figure 4.6, where the results of the 17th cycle of test LC10 are represented by solid
curves and those of the 18th cycle in dashed lines. There are pairs of peaks that seem to have
similar levels and are often misunderstood intuitively to be in the same cycle, but this may
not necessarily be the case. The final effective vertical stress of each cycle is always lower
than the initial one, since the residual excess pore pressure keeps increasing cycle by cycle,
as observed in Figure 4.4 and in Figure 4.6.
When the effective vertical stress approaches zero and liquefaction is about to be initiated,
the sand has softened dramatically and exhibits a contractive tendency. In the last cycle of
the test, the effective vertical stress decreases dramatically during the loading phases, as a
result of a contractive tendency. After the stress path intersects the phase transformation line,
the sample turns from being contractive to a dilative tendency, with the effective vertical
stress regained. During the unloading phase, the dilation is released again and excess pore
4.1 Uni-directional linear tests 73
0 10 20 30 40 50 60 70
Effective vertical stress (kPa)
-0.4
-0.2
0
0.2
0.4
SS
R X
30405060708090100
Excess pore pressure (kPa)
160
165
170
175
180
Ti
m
e 
(s)
-2 -1 0 1 2 3
Shear strain X (%)
-0.4
-0.2
0
0.2
0.4
SS
R X
-2 -1 0 1 2 3
Shear strain X (%)
160
165
170
175
180
Ti
m
e 
(s)
Fig. 4.6 The 4-way plot of the 17th and 18th cycle of test LC10 (the solid curves are the 17th
cycle and the dashed curves are the 18th cycle)
pressure keeps growing. In the second loading stage of the last testing cycle, the stress path
eventually reaches the failure line, thus the sample fails and liquefaction is triggered.
The stress strain response of test corresponds to the stress path, as plotted in Figure 4.7.
For the cycles with low excess pore pressure, the hysteresis loops remain small while there is
only limited shear strain developing with increasing shear stress. As the excess pore pressure
increases, the peak-to-peak strain within a single cycle increases gradually, as does the shear
strain that develops during the loading phases of each cycle, making flatter hysteresis loops.
There is not a section, even in the last cycle of the test, where the shear strain can accumulate
without the increase of shear stress (i.e. a flat period), again confirming that the sample
exhibits dilation-featured response (cyclic mobility).
74 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
Fig. 4.7 The stress-strain response of test LC10 (the first cycle is coloured red)
4.1.2 Uni-directional tests with static shearing
For a soil element underneath a slope, however, a static shearing force determined by the
inclination angle is applied constantly to the soil element in the geological dip direction.
If the cyclic loading caused by earthquakes is in the direction parallel to the dip of the
slope, the soil response in this loading scenario can be investigated by cyclic uni-directional
simple shear tests with static shearing (also termed "dip-shaking slope tests" in this study).
After saturation, the soil samples are sheared monotonically to the target shear stress with
drainage open and are consolidated in a Kα consolidation condition, before an undrained
cyclic shearing is applied in line with the static shearing direction.
The discussions regarding this category of tests usually involve whether shear stress
reversal exists. The importance of the stress reversal rests in the widely-made observations
that it determines if excess pore pressure can be fully developed or not, that is, if the stress
path can reach the origin in stress space or not. When the cyclic shearing is at a phase where
its shear force is in the opposite direction to the static shear force and counterbalances the
4.1 Uni-directional linear tests 75
latter, the resultant shear force may be either positive or negative, depending on the relative
amount of the two kinds of shear stresses. If the amplitude of cyclic shear stress is larger than
the static shear stress, stress reversal will happen; otherwise, there will be no stress reversal.
The results of a test (LS7) which has stress reversal and liquefies at the 28th cycle are
presented here. The sand sample of this example test has a relative density of 58.2% and a B
value of 0.95, consolidated to an initial vertical stress of 99.3 kPa. The amplitude of cyclic
shear stress is 25 kPa (CSR=0.25) and the static shear stress is 12.5 kPa (SSR=0.125). The
positive direction of shear stress and strain is equivalent to the downhill direction of a real
slope.
-0.1
0
0.1
0.2
0.3
0.4
SS
R
x
0
2
4
6
8
x
 
(%
)
0 50 100 150 200 250
time (s)
0
10
20
30
40
50
60
70
80
90
100
Ex
ce
ss
 P
or
e 
Pr
es
su
re
 (k
Pa
)
Fig. 4.8 The development of shear strain and excess pore pressure of test LS7
As shown in Figure 4.8, there is an obvious flaw in this test. The excess pore pressure
temporarily decreases after the 14th cycle and climbs back afterwards. The shear strain, as
well as the vertical stress and vertical displacement (Figure 4.9), nonetheless, do not have
76 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
such an abrupt change. Together with the fact that the sample is able to liquefy eventually,
this eliminates the possibility of significant leakage. The most reasonable explanation left
seems to be that a sudden dislocation happens in the stack of steel rings, resulting in a part of
the rubber membrane deforming abruptly or even being penetrated by sand grains, which
brings about an increase in the volume of the sample.
90
95
100
105
110
v
er
tic
al
 st
re
ss
 (k
Pa
)
0 50 100 150 200 250
time (s)
41.7
41.75
41.8
41.85
41.9
41.95
Z 
LV
D
T 
(m
m)
Fig. 4.9 The variation of vertical stress and vertical LVDT with time in test LS7
Considering the undesired variation is not observed in any of the other parameters,
this test is reckoned to be valid and the analysis based on it is still tenable, except for the
number of cycles to liquefaction. As the excess pore pressure had an unexpected decrease
in the middle of the test, the number of cycles to liquefaction should have become higher
correspondingly, compared with the case where no such decrease occurs. The “true” number
of cycles to liquefaction should fall in a range whose upper limit is the recorded number and
the lower limit is the cycle number where the abrupt change happens. To be more specific,
if a cycle is considered “recovered” when both the minimum and maximum excess pore
pressure in this cycle are higher than those in the cycle right before the prompt decrease
4.1 Uni-directional linear tests 77
takes place, the test can be assumed to have proceeded directly to the first “recovered” cycle
from the affected cycle if the problem did not happen, without going through the cycles in
between. Along with this assumption, the excess pore pressure recovers at the 19th cycle in
this test and the 15-18th cycles would not be there if the unanticipated drop had not occurred.
The number of cycles to liquefaction is hence adjusted from 28 to 24, which means the
true pre-liquefaction cycle number is very likely to fall somewhere around 24. While this
adjustment is rather arbitrary, it does provide useful information for the analysis in terms
of liquefaction resistance. It will be marked and clarified in the later sections whenever this
adjustment is applied.
Similar to the tests without static shearing reported in the preceding session, the first cycle
gives a larger increase of excess pore pressure. The so-called “double peak” phenomenon in
the development of excess pore pressure can also be observed in this test. But different from
the level-ground tests, the valleys between the pairs of peaks are not pronounced until the
last a few cycles. This can be explained conveniently by the uneven resultant amplitudes of
shear stresses, as shown in Figure 4.10. In the early cycles, except for the first cycle, the sand
sample at a density similar to that of LC10 exhibits a dilative response as expected, with the
excess pore pressure decreasing during loading and increasing during unloading. However,
the amplitude of negative shear stress ratio is only one third of its positive counterpart, as a
result of which the dilative response cannot be mobilised by negative shear stress to the same
extent as by positive shearing. That is, the relatively small negative shear loading cannot
counteract the increasing tendency to develop positive excess pore pressure, hence making
the valleys less prominent. The valleys do not appear until the sand sample is softened so
much that the inconsiderable loading phase in the negative side can also lead to a significant
change of excess pore pressure from increasing to decreasing.
This can be examined more carefully in Figure 4.11, where the 4-way plots for the 25th
cycle (solid) and the 26th cycle (dashed) are presented. The increase in negative shear stress
in the 25th cycle fails to reach the phase transformation line or accomplish the contraction-
to-dilation transition of the sand sample. The corresponding development of excess pore
pressure versus time shows a temporary plateau, rather than a valley, at the maximum negative
78 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
10 100
v
' (kPa)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
SS
R X
SSRX- v'
Fai
lure
 Lin
e
PT Li
ne
Fig. 4.10 The stress path of test LS7
shear stress, as shown in the bottom left plot. In the 26th cycle, however, the effective vertical
stress has been reduced to a point where even the relatively small negative shear loading
can still push the stress path to cross the phase transformation line, resulting in a significant
decrease of excess pore pressure.
In addition to the discrepancy in the valleys in the excess pore pressure plots, the with-
static-shearing tests also differ partly from the no-static-shearing ones in terms of strain
development, as illustrated in Figure 4.12. While the stress-strain loops become flatter and
flatter, indicating the same kind of strain softening behaviour as in level-ground tests, the
shear strain accumulates towards the positive direction progressively, exhibiting a clear
downhill preference that is not seen in level-ground tests. This is not surprising since the
downhill shear stress is more dominant in this test. Wu (2002) and Biscontin et al. (2004)
observed the same phenomenon on sand and clay, respectively. Furthermore, it is worth
noting that prior to the cycle where the double amplitude shear strain exceeds 6%, the
maximum cumulative shear strain has reached as much as 6.7%.
In stark contrast, the soil response becomes more unusual if there is no stress reversal.
The test results of a test (LS11) in this category are shown in Figure 4.13 and 4.14. The
sample has a relative density of 59.9% and a B value of 0.92, consolidated to an initial
4.1 Uni-directional linear tests 79
0 10 20 30 40 50 60 70
Effective vertical stress (kPa)
-0.2
0
0.2
0.4
SS
R X
30405060708090100
Excess pore pressure (kPa)
240
245
250
255
260
Ti
m
e 
(s)
2 3 4 5 6 7
Shear strain X (%)
-0.2
0
0.2
0.4
SS
R X
2 3 4 5 6 7
Shear strain X (%)
240
245
250
255
260
Ti
m
e 
(s)
Fig. 4.11 The 4-way plot of the 25th and 26th cycle of test LS7 (the solid curves are the 25th
cycle and the dashed curves are the 26th cycle)
vertical stress of 98.7 kPa. As test LC10 and test LS7, the amplitude of cyclic shear stress is
25 kPa (CSR=0.25). But this test has a high static shear stress at 30 kPa (SSR=0.3), higher
than the cyclic shear stress amplitude. The positive direction of shear stress and strain is still
equivalent to the downhill direction.
The sand sample fails in the first cycle due to the devastating single amplitude shear
strain within this cycle, but the test results of the four subsequent cycles are still presented
here to demonstrate the peculiar soil response in this test. The most remarkable feature
rests in the inability of stress path to reach the origin, that is, the inability of excess pore
pressure to develop to 100% of the vertical stress. Figure 4.13 gives the stress path of this
test. During the first loading period, the stress path initially develops towards the origin
with the generation of positive excess pore pressure, but soon curves back away from the
origin, indicating a phase transformation resulting in the decrease of excess pore pressure.
However, the exact position of phase transformation state is difficult to locate because the
adaptive loading system tends to overreact at the initial stage of loading and induce significant
80 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
-1 0 1 2 3 4 5 6 7 8 9
X (%)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R
X
SSR
X
- 
X
Fig. 4.12 The stress-strain response of test LS7 (the first cycle is coloured red)
oscillations of stresses. The oscillations of loading system, together with system friction,
may cause the phase transformation state to deviate from the PT line. Considering this effect
becomes particularly obvious when a test starts from a large initial static shear force, no
conclusions on the phase transformation states were made for these tests.
As the test continues, the stress path becomes oblique, taking the weaving-shuttle loop
shape. The stress path can reach the failure line, but can never reach the origin, with the
minimum effective vertical stress still larger than 40 kPa. To be more specific, differently
from the tests with stress reversal, only a small portion of the stress path of the failure
cycles in this test is really located on the failure line, while the majority is not. For a typical
liquefaction failure cycle that shows rather contractive response in test LC10 and test LS7,
the increase of shear stress should generate huge excess pore pressure very rapidly and soon
force the stress path to reach the failure line and stay there until unloading. By contrast,
it seems the sand sample in this test cannot be softened enough by the non-reversal cyclic
4.1 Uni-directional linear tests 81
10 100
v
' (kPa)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
SS
R X
SSRX- v'
Fai
lure
 Lin
e
PT L
ine
Fig. 4.13 The stress path of test LS11
shear stress before the test is terminated, so that it exhibits prominently a dilative type stress
response, with the excess pore pressure decreasing with the increase of shear stress.
This peculiar dilative behaviour can be observed in the stress-strain plot as well. In Figure
4.14, the loading period in the first cycle of the test results in a shear strain over 9%, causing
the sample to fail according to the single amplitude shear strain liquefaction criterion. In
contrast, the subsequent cycles have only moderate double amplitude shear strains (less than
2%). The hysteresis loops of these cycles are not flat at all, also demonstrating dilative soil
behaviour. While it is reasonable to attribute the phenomenon to the constantly positive
resultant shear stress, the actual mechanism driving is unknow.
A fundamental question arises from these observations: is the sample in this test really
liquefied? From the point of view of the shear strain liquefaction criterion, the answer
is positive: the sample does liquefy in the first cycle with a large shear strain. From the
perspective of hysteresis characteristics or mechanical failure, the answer is doubtful because
the sample still exhibits dilative response. The results of a test with marginal stress reversal
may help to further investigate this problem.
82 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 2 4 6 8 10 12 14
X (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
SS
R
X
SSR
X
- 
X
Fig. 4.14 The stress-strain response of test LS11 (the first cycle is coloured red)
Test LS10 is a test with relative density at 55.5%, a bit looser than test LS11. The
Skempton’s B value is 0.94. The sample is consolidated with a vertical stress at 100 kPa and
an initial shear stress towards positive X direction (dip direction) at 20 kPa, before a cyclic
shear stress of 25 kPa is applied. From Figure 4.16, the shear strain accumulated during
the loading phase of the first cycle is up to almost 8%, therefore the sample liquefies within
this cycle based on the shear strain liquefaction criterion. Similar to test LS11, however,
the subsequent cycles have only moderate double amplitude shear strains and stiff-featured
hysteresis loops. Correspondingly, the stress path of these cycles, as shown in Figure 4.15, is
in the shape of a weaving shuttle.
In contrast, the typical behaviour of softened sands appears again in the last two cycles.
The stress path of these cycles reaches the failure line at low stress level and develops along
it. The minimum vertical stress is very close to zero, indicating a significant generation of
excess pore pressure. The double amplitude shear strain exceeds 4% and becomes larger and
4.1 Uni-directional linear tests 83
larger while the hysteresis loops become notably flatter. It seems the sample liquefies again.
But differently from the first cycle, this time, the failure is caused by the stress path reaching
the failure line, accompanied by large excess pore pressure, large double amplitude shear
strain, and flatter hysteresis loop. From the mechanical point of view, this is more closely
resembling what we expect failure to look like.
A conclusion can hence be made here: test LS11 is not liquefied from the point of view
of the criterion requiring ru = 1. Accordingly, a potential explanation for the peculiar soil
behaviour in test LS11 is that the non-reversal shear stress hinders or prohibits a sample from
developing ru = 1. The soil behaviour observed is thus what it should be when the sample is
not fully softened.
10 100
v
' (kPa)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
SS
R X
SSRX- v'
Fai
lure
 Lin
e
PT Li
ne
Fig. 4.15 The stress path of test LS10
It is evident that large excess pore pressure ratio (close to unity) is not achieved at the time
when the shear strain exceeds criterion limit (6% double-amplitude strain) in test LS11 and
test LS10. Therefore, the liquefaction resistance analysis based on shear strain criterion may
be viable for practical use, but not fully useful when it comes to the mechanical assessment
of soil. More detailed discussion about the definition of liquefaction and its implications will
be made in chapter 7, together with the thoughts derived from multi-directional tests.
84 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 2 4 6 8 10 12 14
X (%)
-0.1
0
0.1
0.2
0.3
0.4
0.5
SS
R
X
SSR
X
- 
X
Fig. 4.16 The stress-strain response of test LS10 (the first cycle is coloured red)
4.2 Liquefaction resistance of uni-directional tests
One method to estimate the liquefaction resistance of sands is to determine the relationship
between the number of cycles needed for liquefaction triggering and the cyclic shear stress
ratio applied. The relationship developed from laboratory tests can only be used as a baseline
for this estimate; instead, the data from field tests or actual earthquake records weigh much
more in determining liquefaction resistance. The discrepancy between the liquefaction
resistance derived from laboratory and that from field is large enough that it cannot be
ignored. Considering the significant difference in terms of loading uniformity, loading
multi-directionality, soil uniformity, density measurement, saturation conditions, etc., it is
not difficult to understand why the liquefaction resistance attained in the laboratory is so
different from case histories.
Therefore, there is no intention in this study to modify or develop practical charts for
liquefaction hazard assessment. The reason why liquefaction resistance is still analysed here
4.2 Liquefaction resistance of uni-directional tests 85
1 10 100 1000
Number of cycles to liquefaction
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Cy
cl
ic
 sh
ea
r s
tre
ss
 ra
tio
Dr=50%
Dr=55%
Dr=60%
Dr=65%
Dr=78%
65%
60%
50%
55%
Wu(2002)
Dr=80% 
Wu(2002)
Dr=60% Wu(2002)
Dr=45% 
Fig. 4.17 Number of cycles to liquefaction versus cyclic stress ratio in uni-directional cyclic
tests with no static shearing
is, on the one hand, to provide a benchmark against which the effects of multi-directional
loading can be compared and clarified more conveniently, and on the other hand, to prepare
data and implications for potential liquefaction resistance research in the future.
Figure 4.17 presents the number of cycles to liquefaction versus cyclic shear stress ratio
of all the uni-directional cyclic tests without static shearing (level ground tests) in this study.
The tests are divided into four groups according to relative density. A trend line that fits a
power function is obtained for each group with least square minimisation, in the form of:
CSR = aNbL (4.1)
with the values of the two parameters, a and b, tabulated in table 4.1.
The scatter of the data points is obvious, but it is not unreasonable given the fact that
liquefaction resistance results are rather sensitive to sample density and loading application
86 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
Table 4.1 Fitting parameters of NL−CSR curves
Relative density Curve-fitting parameters
a b
50% 0.225 -0.097
55% 0.287 -0.127
60% 0.331 -0.132
65% 0.572 -0.193
as a rule, and the fact that it is not very likely to have perfect control of these factors in
simple shear tests. Especially for the parts where the curves become flatter, liquefaction
resistance results can drift away from the trend line due to a small error of mass measurement
or loading measurement, not to mention the inevitable sample non-uniformity or imperfect
control of loading conditions, even though much effort has been made to minimise their
influence. In addition, the samples categorized into the same density group do not always
have the designated density, which also aggravates the scatter.
Although scattered, however, the data in this study show the trend that is congruous with
general expectations. The decrease of cyclic shear stress ratio results in the rapid increase
of the number of cycles required to trigger liquefaction. The curves are steeper in the high
CSR parts and flatter in the low CSR parts. A higher density leads to a higher liquefaction
resistance and a steeper curve, indicating that, compared with dense samples, the liquefaction
resistance of loose samples is more sensitive to the change of CSR.
As Monterey 0/30 sand has been used in some of the seminal work on multi-directional
simple shear testing (Boulanger, 1990; Kammerer, 2002) and will provide a baseline for
comparison in the following sections, the uni-directional test results of Monterey 0/30 sand
are also included here. Wu (2002) performed uni-directional cyclic simple shear tests on
Monterey 0/30 sand, which is a sub-rounded beach sand with a D50 of 0.36 mm. The samples
were prepared by the wet-pluviation method. The correlation curves from one of his test
series with initial vertical consolidation stress of 80 kPa are also plotted in Figure 4.17, as
dashed curves. Wu’s correlations were not obtained with a best fit method; instead, Wu
determined these curves by assigning weights to the data. Samples whose density is closer to
4.2 Liquefaction resistance of uni-directional tests 87
the nominal density of the group are assigned more weight. This adjustment is not followed
in the current study because the weight allocation process is believed not to be tenable when
strict and consistent statistic assessment is not conducted. Nevertheless, from a qualitative
point of view and for comparison purpose, Wu’s curves can still be used as a reference to
validate the data of the current study.
The trend lines of this study are generally consistent with the trends found in Wu’s tests,
but the liquefaction resistance of the latter has a narrower variation range. His correlation
curve for loose samples (Dr=45%) is situated at the position of the curves for denser samples
(Dr=50-55%) in this study, whereas the liquefaction resistance of a dense sample (Dr=78%)
in this study is higher than his results for samples with relative density of around 80%.
Furthermore, the liquefaction resistance obtained from this study at large CSR is lower than
Wu’s, while that at small or moderate CSR range is higher than Wu’s. Considering that Wu
used a different sand, sample preparation technique, consolidation stress and curve fitting
method, this discrepancy is reasonable.
As for tests with static shear stress, the effects of initial static shearing on liquefaction
resistance has been a widely discussed issue, however, the effects observed by different
research groups differ significantly. Factors including density, confining pressure, sand type,
particle shape and anisotropic fabric, as well as the response type (cyclic mobility or strain
softening) that is determined potentially by them, exert together an influence on the effects of
static driving shear force. The research conducted by Seed and Harder (1990) and Harder et al.
(1997) showed that static shearing always lowers the liquefaction resistance of loose sands,
and can either increase liquefaction resistance or have no effects for medium-dense to dense
sands, depending on the relative density. In contrast, Vaid et al. (2001) found in their tests that
the increase of static shearing always results in an increase of liquefaction resistance, even
for loose sands, but this effect can reverse with static shear force increasing further in loose
sands. Ishihara (1985) and Iai et al. (1992) integrated the effects of anisotropic consolidation
into the effects of mean principal stress. Vaid and Chern (1985), however, concluded that
the increase of static shearing reduces liquefaction resistance if the sand can develop flow
liquefaction, while the effect is reversed if the sand develops cyclic mobility, which indicates
88 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
the response of sands also affects the influence of static shearing, in addition to density and
confining pressure. Sivathayalan and Ha (2011) observed that the effects of static shearing
are opposite on subrounded silica sands and on semi-angular Fraser River sand, as the former
strain-softens while the latter strain-hardens over a large density range. The investigation
into the effects of static shearing as well as the effects of anisotropic consolidation is also
attempted with discrete element modelling method, from a micro mechanical point of view
(Georgiannou and Konstadinou, 2014; Lashkari et al., 2017), but there is still no consistent
conclusion that can be made on this issue.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Initial static shear stress ratio 
0
0.5
1
1.5
K
Dr=55%
Dr=60%
Dr=65%
Fig. 4.18 The correction factor Kα versus initial static shear stress ratio α
The test results of the 11 uni-drectional tests conducted in this study with initial static
shearing are shown in Figure 4.18. The initial static shear stress ratio α is defined as the
ratio of initial static shear stress over the vertical overburden stress. The correction factor
Kα equals to the ratio of the cyclic stress ratio with static shearing over that without static
shearing, as proposed by Seed et al. (1983) to assess quantitatively the influence of static
shear on liquefaction resistance.
4.3 Excess pore pressure generation in uni-directional tests 89
α =
τstatic
σv,i
(4.2)
Kα =
CSRα=α
CSRα=0
(4.3)
Although no definitive conclusion can be made regarding how density and confining
pressure affects the influence of static shearing due to the scatter and limited number of tests
that have been done, a general trend can still be found. On the one hand, the increase of static
shearing up to a moderate level can increase liquefaction resistance, but this positive effect
no longer exists if static shearing is too large. On the other hand, the four tests with Kα < 1
all fail within one or two cycles due to the rising single-amplitude shear strain, without the
stress path reaching failure line, as discussed in the preceding session. That is, the Kα in
these four tests are obtained with strain-based failure criterion while the Kα in other tests are
attained with ru based failure. Evidently, the failure criterion has also an influence on the
assessment of the effects of static shearing, which can worsen the discrepancy in terms of
the effects of static shearing. In general, the existence of initial static shearing is beneficial
for liquefaction resistance for the sand at the density range and the vertical consolidation
pressure tested in this study. But when the maximum shear stress exceeds a certain level,
sands may experience an extremely large shear strain during the first a few cycles by this
overwhelming shear stress, triggering strain-based failure. Practical engineering design and
hazard assessment should take account of both of these two types of failure, especially when
the effects of them are contradictory.
4.3 Excess pore pressure generation in uni-directional tests
The development of excess pore pressure ratio in all the ground-level tests (solid) is presented
in Figure 4.19, along with the dip-shaking slope tests (dashed) which reach mechanical
failure rather than failing due to overlarge single amplitude shear strain. The cycle numbers
are normalized by the number of cycles required to trigger liquefaction in that test. The data
90 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
presented here are not adjusted and the influence of the drop of pore pressure artefact in some
of the tests is retained.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/NL
0
0.2
0.4
0.6
0.8
1
1.2
r u
Unidirectional tests without static shearing
Unidirectional tests with static shearing
Fig. 4.19 The development of excess pore pressure ratio with normalized cycle number
The tests can be divided into two categories according to the rate of pore pressure
ratio increase, as shown in Figure 4.20. Group I involves the tests that are very prone to
liquefaction, in which liquefaction can be triggered usually with only a small number of
shearing cycles. The excess pore pressure ratio in these tests keeps rising from the very
beginning and there is no slowing-down until liquefaction is about to be initiated.
In contrast, group II, which consists of the tests that need more cycles to reach liquefaction,
features a three-stage development pattern of pore pressure ratio. Excess pore pressure ratio
has a jump in the first stage, followed by a period of steady increase with a smaller increment
rate. As excess pore pressure is accumulated, its increment rate becomes higher and higher
due to the softening of sand. When the excess pore pressure reaches a certain level and
the sand is softened, a large pore pressure increase takes place and liquefaction ensues.
This three-stage pore pressure development pattern is consistent with the strain-softening
behaviour of sands as discussed earlier.
4.3 Excess pore pressure generation in uni-directional tests 91
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/NL
0
0.2
0.4
0.6
0.8
1
1.2
r u
Group I
Group II
Fig. 4.20 The development of excess pore pressure ratio with normalized cycle number, with
two groups recognised
Comparing Figure 4.19 and Figure 4.20, it can be observed that the pattern of excess pore
pressure accumulation is not altered much by the existence of initial static shearing. The
curves of the tests with static shearing fit well into the band of their ground-level counterparts
and show the same characteristics of three-stage development.
Following Oda et al. (2001), the number of cycles to liquefaction is correlated to the
excess pore pressure ratio of the first cycle. Qualitatively, if the excess pore pressure ratio
generated in the first cycle is high, the sample should be softened more rapidly and thus
initiate liquefaction with fewer cycles. Wu (2002) demonstrated that the excess pore pressure
ratio in the first cycle is not sensitive to varying density and confining pressure, further
validating the potential use of this relationship in liquefaction assessment. The data of all the
uni-directional tests in the current study are also compatible with the correlation, as shown in
Figure 4.21, with the fitted relationship:
ru,1 = N
−(0.67+
1.53
NL
)
L (4.4)
92 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
where ru,1 is the excess pore pressure ratio generated in the first loading cycle and NL is the
number of cycles to liquefaction.
The data of level-ground tests are coloured blue, with red markers for slope tests. It can
be noticed that the existence of static shearing does not alter this correlation, if only the
failure line can be reached.
100 101 102
Number of cycles to liquefaction NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
1
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Fig. 4.21 The relationship between the excess pore pressure ratio in the first cycle and the
number of cycles to liquefaction in uni-directional tests
4.4 Shear modulus degradation in uni-directional tests
The shear modulus degradation of all the uni-directional tests with and without static shearing
is shown in Figure 4.22. G refers to the secant shear modulus calculated for each cycle, and it
is normalized by the small-strain shear modulus Gmax following Hardin and Drnevich (1972):
Gmax = 1230 · (2.973− e)
2
1+ e
· (σ ′i,0)0.5 (4.5)
4.4 Shear modulus degradation in uni-directional tests 93
where e is void ratio and σ ′i,0 is the initial effective vertical stress in pounds per square inch.
The cyan dotted curves in Figure 4.22 represent the predicted shear modulus degradation
curves following the method of Hardin and Drnevich (1972):
γr =
τmax
Gmax
(4.6a)
G
Gmax
=
1
1+
γ
γr
(4.6b)
As clarified in chapter 3, the cyclic shear stresses used in this study are moderate to
high in order to liquefy the medium-dense Hostun sand samples. As a result, the double
amplitude shear strain induced is larger than 1% even in the first cycle, corresponding to
a G/Gmax ratio below 0.04. There is thus not much information about small strain shear
modulus degradation that can be extracted, though the comparison of test results with Hardin
and Drnevich’s prediction can still be made.
The shear modulus degradation curves are enlarged and plotted on the same graph for
a clearer view. It can be observed that the shear modulus of the test data in this study is in
agreement with the prediction at the strain range obtained. Hardin and Drnevich’s correlation
which was derived based on hyperbolic stress-strain curve works effectively at the large shear
strain range tested in this study.
The existence of static shearing is not seen to bring about a notable difference. The shear
modulus of the tests with static shearing degrades along the predicted curve, as does that of
ground-level tests.
In addition to shear strain, the relationship between shear modulus and excess pore
pressure ratio can also have implications since the development of excess pore pressure is
recognised to soften sands and hence reduce shear modulus. The shear modulus degradation
versus excess pore pressure ratios is shown in Figure 4.23. The shear modulus is not
normalized by Gmax. Gmax is related to the elastic shear strain, which can serve as a reference
for comparison in terms of shear strain, but there is no evidence that it is still relevant when
it comes to excess pore pressure.
94 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
Fig. 4.22 The shear modulus degradation with double-amplitude shear strain (cyan dotted
curves are the predictions following Hardin and Drnevich (1972); blue and red curves present
measured data)
Instead, the shear modulus is normalized by the secant shear modulus of the second cycle
in this study. The reason why the shear modulus of the second cycle is used rather than that
of the first cycle rests in the overshooting issues of the testing device, that is, the adaptive
loading system tends to overreact at the initial stage of loading. The loading control system
takes time to adapt itself to the stiffness of sand samples, which results in overshooting in the
beginning cycle and a first-cycle shear strain that is larger than the following cycles. This is
reflected by the increase of shear modulus ratio in the initial stage and the fact that G1/G2 is
usually lower than unity. Since the overshooting problem cannot be controlled or eliminated,
the shear modulus in the first cycle is not reliable. By contrast, the normalization against the
shear modulus of the second cycle is more consistent and less scattered. The tests which
4.4 Shear modulus degradation in uni-directional tests 95
0 0.2 0.4 0.6 0.8 1 1.2
r
u
0
0.2
0.4
0.6
0.8
1
1.2
G
/G
2
Unidirectional tests without static shearing
Unidirectional tests with static shearing
Fig. 4.23 The degradation of shear modulus with excess pore pressure ratio
fail within two cycles are removed because the G/G2 normalization does not have much
meaning for them.
The shear modulus degradation curves of all the other tests, no matter whether there is
static shearing, fall into a reasonably narrow band and exhibit the same trend. The degradation
process can be divided qualitatively into two phases according to the rate of the decrease of
shear modulus. When excess pore pressure ratio remains low and the sand samples are not
softened significantly, shear modulus degrades at a low and nearly constant rate. However,
when excess pore pressure ratio exceeds a certain level around 0.3-0.4, the decrease of
shear modulus accelerates. Shear modulus generally decreases to less than 40% of G2 when
liquefaction is initiated. The correlation between G/G2 and excess pore pressure ratio can be
described with a power function as follows:
G
G2
= 1.05−0.88ru2.5 (4.7)
It should be noted that the curves for the tests with static shearing also fall into the
same band as ground-level tests, demonstrating the effects of initial static shearing are not
96 UNI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
significant. The degradation of G/G2 is therefore also a type of characteristics shared by all
the uni-directional tests, which is not affected by static shearing and thus holds a potential
for modelling work.
4.5 Discussion on uni-directional tests
Uni-directional cyclic loading is the most examined loading condition in seismic geotechnical
study, and soil liquefaction research is no exception. A soil element under level ground
is sheared by seismic loading without static shearing; by contrast, a soil element under
sloping ground has a downhill static driving shear force. If the cyclic shear induced by the
earthquake is along the dip direction, in an ideal situation, the soil element still experiences a
uni-directional loading scenario, but with static shearing.
21 uni-directional level-ground tests and 11 uni-directional slope tests were conducted in
this study to validate the ability of the testing device and form a benchmark against which
multi-directional tests can be compared and studied.
An important difference between the tests with and without static shearing rests in what
type of failure actually happens when the strain-based liquefaction criterion is met. For level
ground tests and slope tests that have small or moderate static shearing, the strain-based
failure is compatible with mechanical failure, that is, when liquefaction happens from a strain
perspective, the stress path also reaches the failure line and excess pore pressure increases
to a high value close to the vertical overburden stress. However, this is not necessarily true
for tests holding large static shearing. In this case, the failure line may not be reached in
stress space, though liquefaction is considered to have occurred because of an overlarge
single-amplitude shear strain. Since the importance of serviceability is widely recognised
and large strain can result in undesirable loading situation for the structures built on soil, the
overlarge shear strain caused by large static shearing is still dangerous even if the sands do
not technically liquefy and the excess pore pressure develops only to a moderate level.
The analysis of liquefaction resistance in this study shows that small or moderate static
shearing is beneficial, while large static shearing has the opposite effect. An observation from
4.5 Discussion on uni-directional tests 97
this is static shearing up to a certain level can help resist mechanical failure in medium-dense
Hostun sands vertically consolidated at 100 kPa. But this observation should be treated
cautiously because the effects of static shearing can be influenced by factors including sand
type, relative density, confining stress and so forth (Harder et al., 1997; Ishihara, 1985;
Seed and Harder, 1990; Vaid et al., 2001). Considering liquefaction hazard assessment in
engineering practice is more often based on actual case histories and in situ testing rather
than laboratory testing, there is no intention to provide new charts or correction factors in
this study; instead, the analysis regarding liquefaction resistance in this study will be used to
investigate the effects of static shearing and multi-directional loading qualitatively.
In terms of excess pore pressure development and shear modulus degradation, however,
the existence of static shearing does not result in any notable difference under the test
conditions in this study, if only the sand samples are failed mechanically. Excess pore
pressure shows three-stage development pattern as widely reported in strain-softening soil
behaviour. Shear modulus decreases as sands are softened. The theoretical small-strain
shear modulus Gmax and G/Gmax degradation curve proposed by Hardin and Drnevich (1972)
works effectively for the tests in this study. Shear modulus degrades at a nearly steady rate
when excess pore pressure remains low. However, after excess pore pressure ratio exceeds
the level at approximately 0.3-0.4, the degradation of shear modulus accelerates until the
secant shear modulus drops eventually to less than 40% of G2.
In summary, the uni-directional tests conducted in this study show that the clean Hostun
sands with a relative density between 50% and 65% exhibit typical cyclic-mobility behaviour
under undrained uni-directional cyclic shear. The existence of static shearing along the cyclic
loading direction does not alter its characteristics in terms of mechanical failure state, excess
pore pressure development, and shear modulus degradation. Compared with uni-directional
tests, multi-directional tests show significant differences, which will be illustrated in the next
chapter.

Chapter 5
MULTI-DIRECTIONAL SIMPLE
SHEAR TEST RESULTS
5.1 Bilinear tests
The effects of initial static shearing become more complex and far less understood when the
direction of static shearing is perpendicular to that of cyclic shearing. A fundamental case of
such a type of loading condition refers to a soil element again under sloping ground, but with
earthquake-induced shaking occurring along the strike direction rather than dip direction.
Under this circumstance, the static shear stress which points downhill and the cyclic shear
stress along the strike direction become orthogonal.
13 bi-directional linear tests were conducted in this study to investigate the behaviour of
Hostun sands under this loading condition. After sand samples are consolidated vertically
with the target vertical overburden stress, the samples are sheared with a ramp loading, with
drainage open, to achieve the designated static shear stress. The ramping rate is set to 0.01
kN/min, a rate that is slow enough to allow any induced excess pore pressure to dissipate
immediately but not too slow in case large strain develops prior to cyclic shearing. Right
after the static shear loading is increased to the desired value, the drainage valves are closed
and cyclic shearing along the perpendicular direction follows. The information of all the
100 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
bi-directional linear tests, including relative density, B-value, loading conditions and the
number of cycles to initiate liquefaction, are summarized in table 3.3.
The liquefaction criterion becomes problematic in a multi-directional loading scenario.
Excess pore pressure does not always increase to unity when liquefaction happens; instead,
excess pore pressure is bound not to be able to reach unity in bi-directional linear tests, which
will be elaborated in the subsequent section. Moreover, the strain-based criterion encounters
challenges partly from the fact that there is no evidence that a certain level of shear strain in
one direction can be correlated to liquefaction, and partly from the difficulties in defining a
shear strain that can take directionality into account effectively.
From a point of view of serviceability, resultant shear strain which is defined as γtotal =√
γdip2 + γstrike2 may serve as a better indicator because an over-large resultant strain, no
matter what direction it is in, can always threaten the serviceability of the structures built on
the soil. But from a stress perspective, whether a certain resultant shear strain level can be
associated reliably with stress failure states is yet to be confirmed. A more detailed discussion
regarding liquefaction criterion will be made in Chapter 7.
Considering this, liquefaction is defined directly based on stress states for multi-directional
tests in this study in order to develop a better and more comprehensive understanding of soil
behaviour under multi-directional loading scenario. Liquefaction is not considered to occur
in bi-directional linear tests until the stress path in either strike direction or dip direction
reaches the failure line, whichever takes place first. The stress-related graphs presented in
this section are all plotted with the data up to the cycle where the the failure line is reached,
regardless of shear strain level.
The results of three examples tests (Bilinear-3, Bilinear-5, Bilinear-8) will be discussed
and compared. The three tests have almost the same relative density at 61% and Skempton’s
B-value at 0.92, as well as the same vertical consolidation stress of 100 kPa. The target
cyclic shear stress ratios applied were 0.25, 0.3 and 0.3. The primary difference among the
three tests is the amount of static shear stress in the dip direction, whose effects can thus be
studied. Test Bilinear-3 is consolidated with a target static shear stress ratio of 0.0625, and
5.1 Bilinear tests 101
failed mechanically at the 2nd cycle. Liquefaction is initiated also at the 2nd cycle in test
Bilinear-5 with a target α of 0.12, and the 4th cycle in test Bilinear-8 with α of 0.15.
0 5 10 15 20 25
dip (%)
-6
-4
-2
0
2
4
6
8
10
12
st
rik
e
 
(%
)
Bilinear-3, D
r
=61.5%, CSR=0.25, =0.0625
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
Fig. 5.1 The plan view of shear strain development of bi-directional test Bilinear-3, Bilinear-5
and Bilinear-8
The bi-directional shear strain development of the three tests is presented in Figure 5.1.
One more cycle is shown for test Bilinear-3 and Bilinear-5 to illustrate post-liquefaction
strain development. This is not done to test Bilinear-5 because that test was stopped when
mechanical failure happens.
The shear strain of the three tests all develops in a zigzag shape, with the strain amplitude
in the two directions both increasing gradually as the sand strain-softens. However, while
the shear strain in the strike direction exhibits a cyclic pattern, the dip shear strain keeps
increasing without ever reversing. Clearly the pattern of strain development in each direction
is controlled by the type of loading that is applied in that direction: cyclic shear leads to
cyclic shear strain accumulation and static shear brings monotonic increase of shear strain.
102 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
It is worth noting that the dip shear strain can develop to extremely high values (18% in
test Bilinear-8 when the failure line is reached), greatly exceeding the limiting amount of
shear strain accumulated in strike direction, though the latter can also reach as high as more
than 6%. The dominating role of the downhill strain becomes clearer in the post-liquefaction
cycles of Bilinear-3 and Bilinear-5. While the increase of dip shear strain is not necessarily
high within a single cycle, the constant accumulation towards the same direction results
in eventually a vast residual downhill shear strain. It is evident that static shearing in dip
direction can become as dangerous as, or even more dangerous than, the cyclic shearing in
strike direction. This is in agreement with the conclusions drawn for sands by Boulanger
(1990) and Kammerer (2002).
Another observation from the shear strain plan view is regarding the amplitude of cyclic
strain in strike direction. The double-amplitude strike shear strain in the three tests is all 10%
in the cycle where liquefaction happens. But the post-liquefaction development of strike
shear strain differs. Test Bilinear-3 holds the smallest α (0.06) among the three tests but its
γstrike,DA increases to 13% at one cycle after liquefaction. Sheared with α = 0.12, γstrike,DA in
test Bilinear-5 increases to 11.5% within one post-liquefaction cycle. In contrast, γstrike,DA in
test Bilinear-8 has stopped increasing when liquefaction happens, levelling out at 10%. The
existence of static shearing in one direction appears to inhibit the potential of cyclic shear
strain development in the orthogonal direction. This inhibition effect is made more clear with
that fact that the test having the largest cyclic strain (Bilinear-3) was actually sheared with a
lower cyclic stress ratio (0.25) compared with the other two tests (0.3).
This indicates that the existence of dip static driving shear force can reduce the degree
of softening when strike cyclic shearing results in liquefaction. The stress-strain curves in
strike direction of test Bilinear-3, Bilinear-5 and Bilinear-8 demonstrate the same tendency,
as presented in Figure 5.2. As expected, the stress-strain hysteresis loops in the three tests all
become larger and flatter with the cyclic shearing. But the final loop in test Bilinear-3 has
less significant strain-hardening features with a period of the curve lying almost flat, while
the final loops in the other two tests look still relatively stiff, evidencing the effects of dip
static shear on orthogonal seismic softening of sands.
5.1 Bilinear tests 103
-6 -4 -2 0 2 4 6 8
strike (%)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R
st
rik
e
Bilinear-3, D
r
=61.5%, CSR=0.25, =0.0625
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
Fig. 5.2 Strike-direction stress-strain curves of test Bilinear-3, Bilinear-5 and Bilinear-8
A reasonable explanation for this counter-intuitive phenomenon can be found in the
excess pore pressure developed at liquefaction in each test, as shown in Figure 5.3. The initial
rate of excess pore pressure generation in test Bilinear-3 is lower than that in the other two
tests because of the smaller cyclic stress, which is also suggested by the relatively small strain
level at this stage. However, the maximum excess pore pressure ratio shows a different trend.
With the increase of α , the maximum excess pore pressure ratio at liquefaction decreases,
hence reducing the degree of softening.
Correspondingly, the stress paths show the same tendency. The stress paths of the three
tests are presented in Figure 5.4, with the top graph showing the stress paths in strike direction
and the bottom showing dip. The stress paths in strike direction exhibit the similar behaviour
as uni-directional tests, featuring butterfly-shaped paths when it approaches failure, which is
also reflected in the “double-peak” phenomenon in excess pore pressure generation. None of
the stress paths in strike direction manages to reach the failure line when the dip stress paths
104 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 5 10 15 20 25 30
Time (s)
-0.2
0
0.2
0.4
0.6
0.8
1
Ex
ce
ss
 p
or
e 
pr
es
su
re
 ra
tio
 r u
Bilinear-3, D
r
=61.5%, CSR=0.25, =0.0625
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
Fig. 5.3 The generation of excess pore pressure in test Bilinear-3, Bilinear-5 and Bilinear-8
arrive there. And the lower the α , the closer to the origin the stress paths in strike direction
can reach.
Aiming to clarify further the effects of dip static shear on strike cyclic shear, the stress
paths of test Bilinear-6, Bilinear-7 and Bilinear-8 are shown in Figure 5.5. The relative
densities and cyclic stress ratio in strike direction differ hugely among the three tests, but they
all share the same amount of α . An impressive but anticipated finding is that the ultimate
effective vertical stress when liquefaction occurs in the three tests, or the maximum excess
pore pressure at that point, are almost the same.
A more careful comparison of Figures 5.4 and 5.5 leads to the conclusion that the
maximum excess pore pressure is controlled by the static shear rather than the cyclic shear in
the perpendicular direction. It is not difficult to understand this because it is always the dip
static shear stress paths, instead of the strike cyclic stress paths, that reach the failure line
first and trigger liquefaction. When a stress path in the dip direction arrives at the failure line,
5.1 Bilinear tests 105
60 80 90 100 110
v
' (kPa)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R s
tr
ik
e
Fai
lure
 Lin
e
PT L
ine
Bilinear-3, D
r
=61.5%, CSR=0.25, =0.0625
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
(a) Stress in strike direction
10 20 30 40 50 60 70 80 90 100 110
v
' (kPa)
-0.35
-0.25
-0.15
-0.05
0.05
0.15
0.25
0.35
SS
R d
ip
Fai
lure
 Lin
e
PT L
ine
Bilinear-3, D
r
=61.5%, CSR=0.25, =0.0625
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
(b) Stress in dip direction
Fig. 5.4 Stress paths of test Bilinear-3, Bilinear-5 and Bilinear-8
106 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
10 60 70 80 90 100 110
v
' (kPa)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R s
tr
ik
e
Fai
lure
 Lin
e
PT L
ine
Bilinear-6, D
r
=30.3%, CSR=0.2, =0.15
Bilinear-7, D
r
=49.9%, CSR=0.15, =0.15
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
(a) Stress in strike direction
10 20 30 40 50 60 70 80 90 100 110
v
' (kPa)
-0.35
-0.25
-0.15
-0.05
0.05
0.15
0.25
0.35
SS
R d
ip
Fai
lure
 Lin
e
PT L
ine
Bilinear-6, D
r
=30.3%, CSR=0.2, =0.15
Bilinear-7, D
r
=49.9%, CSR=0.15, =0.15
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
(b) Stress in dip direction
Fig. 5.5 Stress paths of test Bilinear-6, Bilinear-7 and Bilinear-8
5.1 Bilinear tests 107
the excess pore pressure at that crossing point becomes the maximum excess pore pressure
that can be generated in the corresponding bi-directional linear loading scenario. Based on
this consideration, a correlation between the static shear stress ratio and the maximum pore
pressure ratio can be derived with the straightforward geometry shown in the subplot in
Figure 5.6, as follows:
ru,max = 1− αtanφFL (5.1)
where φFL is the angle of failure line (33.4◦ in this study). This is congruous with the
relationship proposed by Boulanger (1990).
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
m
a
x
v
' (kPa)
SS
R d
ip Fai
lure
 Lin
e
stress path
Fig. 5.6 The relationship between the maximum excess pore pressure ratio and static shear
stress ratio for bi-directional linear tests, with the schematic for deriving equation 5.1
Also shown in Figure 5.6 is the maximum excess pore pressure ratio of the bi-directional
linear tests conducted in this study plotted versus the dip static shear stress ratio α . The solid
108 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
line in the main graph shows equation 5.1, with the circle-marked points representing the
data from tests. It can be said with a high degree of confidence that the derived correlation
fits well with the measured data, validating the hypothesis.
The effects of dip static shear stress ratio on soil liquefaction resistance are presented in
Figure 5.7. The solid data points are those from uni-directional tests and the unfilled ones
represent those from bi-directional linear tests. No unique relationship can be developed
with the large scatter, but a qualitative trend can be found. The increase of static shear stress
generally results in the decrease of liquefaction resistance for the relative density range and
confining pressure tested in this study. A low α is potentially beneficial for liquefaction
resistance (Kα > 1) while the effect is reversed when α enters a moderate range.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Initial static shear stress ratio 
0
0.5
1
1.5
K
Dr=50%
Dr=55%
Dr=55%
Dr=60%
Dr=60%
Dr=65%
Fig. 5.7 The effects of static shear stress ratio α on Kα in uni-directional tests and bi-
directional linear tests
Given the influence of dip static shearing on liquefaction resistance and excess pore
pressure generation, the correlation between the number of cycles to liquefaction and the
5.1 Bilinear tests 109
excess pore pressure generated in the first cycle needs to be re-examined to see if it is
applicable to bi-directional linear tests. Presented in Figure 5.8, the solid curve is the same
one as that in Figure 4.21, following equation 4.4. The general trend is similar but the scatter
at small liquefaction resistance becomes prominent, suggesting that such a correlation needs
to be treated cautiously in multi-directional loading conditions. More information from other
categories of multi-directional tests is needed to check how effective this correlation is.
100 101 102
Number of cycles to liquefaction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
1
Fig. 5.8 The relationship between the number of cycles to liquefaction and the generated
excess pore pressure ratio in the first shearing cycle
The G/Gmax degradation versus shear strain is analysed and presented in Figure 5.9, with
the dotted curves indicating the predictions proposed by Hardin and Drnevich (1972) and the
data points representing test data from this study. A reasonable match can be found between
the predicted values and test data, suggesting the shear modulus degradation curves can still
be used in the direction along which cyclic shearing is applied. However, it needs to be
pointed out that when loading paths become more complex, the definition of shear modulus
110 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
may become challenging. The conclusion from bi-directional linear tests may not hold true
for other multi-directional loading scenarios. A discussion on this issue will be given in
Chapter 7 by summarizing the results of all the tests with various loading paths.
Fig. 5.9 Strike-direction Shear modulus degradation curves in bi-directional linear tests
5.2 Multi-directional tests 111
5.2 Multi-directional tests
5.2.1 Tests with circular loading path
Realistic multi-directional loading paths can be far more complex than uni-directional or
bi-directional linear ones, partly because the interaction between the variation of the amount
of shear stress and the change of its direction can be more complicated and influence soil
behaviour significantly, and partly because the added dimension in loading paths raises
concerns about whether the conventional analysis framework is still effective. Circular
loading paths without initial static shearing is a simplified, fundamental and representative
type of loading condition of such kind. Compared with uni-directional or bi-directional linear
tests in which the amount of shear stress varies cyclically and its direction is constrained to
a single dimension, the shear stress under circular loading conditions rotates its direction
smoothly, with its magnitude, however, remaining unchanged.
19 undrained cyclic circular-path tests were conducted to develop insights into the soil
behaviour under this loading situation and investigate the effectiveness of the conventional
liquefaction analysis framework under this circumstance. 11 of the tests had path centre
located at the origin, while the other 8 tests had offset circular paths. Detailed testing
information is summarized in table 3.3. Along with bi-directional linear tests, liquefaction is
defined to take place when the stress states in either X or Y direction reach the failure line.
The results of test C20-3 are presented here as an example to illustrate the characteristics
of soil behaviour in this category of tests. This test had relative density of 63.1%, Skempton’s
B value of 0.956, and initial vertical consolidation stress at 100 kPa. The assigned amplitude
of circular shear stress ratio was 0.2. After vertical consolidation, ramp loading was applied
towards positive X direction, increasing the shear stress ratio in X direction to 0.2, to avoid
abruptly increasing the of shear stress if a circular loading path was applied directly. The
shear strain measurement was zeroed at this point, before the target circular loading started.
The plan view of measured shear stress is shown in the left graph in Figure 5.10, with
the round mark representing the starting point and the arrow indicating the anticlockwise
direction of loading path. As discussed in the preceding chapter, the initial part of the
112 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
-0.2 -0.1 0 0.1 0.2
SSRX
-0.2
-0.1
0
0.1
0.2
SS
R
Y
(a) Plan view of shear stress ratio
-6 -4 -2 0 2 4
X(%)
-8
-6
-4
-2
0
2
4
Y
(%
)
(b) Plan view of shear strain
Fig. 5.10 The plan view of shear stress and shear strain of test C20-3
first cycle veers away from the theoretical path because the device control system needs to
adjust its two loading strokes. The last few cycles also deviate from the theoretical circular
stress path because the stiffness of the sample is reduced significantly when liquefaction is
approached, making it difficult for the testing device to hold constant vertical stress and good
control of shear stress. Figure 5.10(b) presents the plan view of shear strain. The strain path
exhibits generally a circular shape, as expected, until the sample liquefies.
A four-way plot following Kammerer (2002) is shown in Figure 5.11. Total shear stress
ratio SSRtotal and total shear strain γtotal is used to provide overall characteristics in terms of
shear stress and strain, with their definitions as follows:
SSRtotal =
2
√
SSR2X +SSR
2
Y (5.2)
γtotal = 2
√
γ2X + γ2Y (5.3)
5.2 Multi-directional tests 113
The SSRtotal versus σ ′v and the SSRtotal versus γtotal curves are almost horizontal in the top
graphs, confirming the total shear stress ratio remained approximately constant and the
testing device performed satisfactorily during this circular-path test.
0 20 40 60 80 100
v
' (kPa)
0
0.05
0.1
0.15
0.2
0.25
0.3
To
ta
l S
SR
0 2 4 6 8
Total shear strain total (%)
0
0.05
0.1
0.15
0.2
0.25
0.3
To
ta
l S
SR
00.20.40.60.81
Excess pore pressure ratio r
u
0
10
20
30
N
um
be
r o
f c
yc
le
s
0 2 4 6 8
Total shear strain total (%)
0
10
20
30
N
um
be
r o
f c
yc
le
s
Last cycle
Other cycles
Fig. 5.11 The 4-way plot of test C20-3
Excess pore pressure in circular tests
The bottom left graph in Figure 5.11 shows the development of excess pore pressure ratio
with the number of cycles. The excess pore pressure accumulates gradually and the maximum
pore pressure ratio of 0.77 is reached at the 19th cycle where liquefaction is initiated. The
“double-peak” phenomenon seen in linear tests is not observed; instead, there is only one pore
pressure peak within a single cycle, demonstrating that the pattern of excess pore pressure
accumulation differs from that in linear tests.
114 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
Fig. 5.12 3D development of excess pore pressure ratio of test C20-3
Actually, the cyclicity itself is unexpected. As the total shear stress remains constant like
monotonic shearing, there is no distinguishable boundary between loading and unloading, and
the excess pore pressure is thus anticipated to develop monotonically rather than cyclically.
Figure 5.12 presents the excess pore pressure development with stress path in 3D in the
left graph. The right graph shows the same plot with the red dots representing the sections
where excess pore pressure increases the light blue dots presenting the portions where it
decreases. When the excess pore pressure is relatively low, it increases along half of the path
and decreases in the other half. In this test, excess pore pressure decreases in the semicircle
corresponding from 135◦ to 315◦, and increases in the other semicircle. In the last three
cycles where excess pore pressure ratio is high, however, the pattern changes again with
excess pore pressure decreasing only in the 90◦−150◦ section and increasing in the rest.
This can be seen more clearly in Figure 5.13 which illustrates the development of excess
pore pressure in a polar coordinate system. The polar angles in Figure 5.13 are the angle
of a point on the stress path relative to positive SSRX direction in Figure 5.10, measured
counterclockwise. The red parts indicate the increase of excess pore pressure while the blue
5.2 Multi-directional tests 115
0
30
60
90
120
150
180
210
240
270
300
330
0
0.2
0.4
0.6
0.8
(b) Excess pore pressure ratio
0
30
60
90
120
150
180
210
240
270
300
330
0
0.1
0.2
(a) Total shear stress ratio
Fig. 5.13 The shear stress and excess pore pressure ratio of test C20-3 in polar coordinate
system (the red parts in (b) represent increasing pore pressure and the blue parts indicate
decreasing pore pressure)
sections indicate a decrease. The variation of excess pore pressure ratio follows a pattern in
which the excess pore pressure grows in a semicircle and declines in the other, with the final
excess pore pressure ratio of a cycle always higher than the initial one in the same cycle. The
sections corresponding to pore pressure decrease shrink to 90◦−150◦ in the last three cycles,
only one sixth of the whole loading path, while the majority of the loading path witnesses
increase of pore pressure.
Although the other tests have the same testing procedure with this one, the portions of a
cycle in which excess pore pressure cycles from increasing to decreasing or the way round
is not always the same. Figure 5.14 presents the development of excess pore pressure of
circular-path test C20-1 and test C20-5, which have the same total shear stress ratio of 0.2
but are different in density. It is evident that the specific partition of the increase and decrease
of excess pore pressure can vary conspicuously. But it is also notable that there is a similarity
among all the pre-liquefaction and on-liquefaction cycles in these tests, that is, the sections
of pore pressure decrease always shrink to a small portion while pore pressure increase is
seen in greater parts.
116 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0
30
60
90
120
150
180
210
240
270
300
330
0
0.2
0.4
0.6
0.8
(a) Test C20-1
0
30
60
90
120
150
180
210
240
270
300
330
0
0.2
0.4
0.6
0.8
(b) Test C20-5
Fig. 5.14 The development of excess pore pressure ratio of test C20-1 and C20-5 in polar
coordinate system (the red parts represent increasing pore pressure and the blue parts indicate
decreasing pore pressure)
The mechanism behind this inconsistency is as yet unclear. On the one hand, the inevitable
experimental error such as loading asymmetry can contribute to this phenomenon since it
can cause latent subdivision of loading path into loading and unloading periods. But this
cannot explain why the subdivision is inhibited when excess pore pressure becomes high
and liquefaction is approached, since there is no evidence that the loading asymmetry is
reduced at this stage. On the other hand, the rotation of shear stress can also exert influences.
The anisotropic fabric in sand samples, caused by realistic but imperfect sample preparation
and consolidation, is very likely to contain an indiscernible directionality that can interact
with the stress rotation and thus distinguish the stress path into softer periods and stiffer
periods, from which the cycling of excess pore pressure can arise. If this hypothesis holds
true, the significant implications will be that, for one, the interaction between soil structure
and the directionality of loading path can have prominent effects on soil behaviours, and for
another, the directional non-uniformity of fabric anisotropy of sand can be either inhibited
or determined by stress path when sands approach failure state or critical state, so that the
cycling of excess pore pressure becomes inconspicuous at this stage. Although existing
modelling and experimental work on triaxial, biaxial and uni-directional simple shear tests
5.2 Multi-directional tests 117
have demonstrated, implicitly or explicitly, soil fabric anisotropies develop gradually towards
a critical state that is controlled by stress states (Li and Dafalias, 2011; Oda, 1972; Rothenburg
and Bathurst, 1989), its development under more complex loading scenario remains less well
understood. More specifically designed experiments are needed in the future to verify this
hypothesis.
While the pattern of the variation of excess pore pressure in early cycles varies, there
exists a more noteworthy and consistent phenomenon: the maximum excess pore pressure
ratios that can be generated in these tests are below unity. The excess pore pressure ratio
reaches only 0.77 when liquefaction is initiated. The clue to explain this phenomenon
can be found in the total shear stress path in the top-left graph in Figure 5.11. With the
generation of excess pore pressure, effective vertical stress decreases, pushing the stress path
leftward until it meets the failure line that is represented by the solid line. This crossing point
announces the constrained final stress state that can be achieved under this loading condition.
Correspondingly, the excess pore pressure at this point represents the maximum excess pore
pressure in this test.
A question that follows naturally is whether the situation will change if initial static
shearing is involved and total shear stress does not keep constant. Test C20-X20-1 is an
extreme example of such a category of tests, which has a static shear in X direction that is so
large that X shear stress is never below zero, as shown in Figure 5.15. The three-dimensional
development of excess pore pressure with shear stress ratios is presented in Figure 5.16.
Figure 5.17 shows the stress path and excess pore pressure ratio in polar coordinate system.
The polar angles in Figure 5.17 are the anticlockwise angle of a point on the stress path
relative to positive SSRX direction in Figure 5.15, measured counterclockwise.
Evidently, the excess pore pressure develops in a different pattern compared with test
C20-3. A distinct maximum can be seen in the last a few cycles, but the location of the
maximum varies with each cycle. The higher the excess pore pressure, the closer to the
SSRY = 0 axis the summit is situated. The reason why excess pore pressure develops in such
a way can be found again in the total shear stress path illustrated in the top left graph in
Figure 5.18 with the last cycle coloured red. Differently from circular tests without static
118 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 0.1 0.2 0.3 0.4
SSRX
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
SS
R
Y
(a) Plan view of shear stress ratio
0 5 10
X(%)
-6
-4
-2
0
Y
(%
)
(b) Plan view of shear strain
Fig. 5.15 The plan view of shear stress and shear strain of test C20-X20-1
Fig. 5.16 3D development of excess pore pressure ratio of test C20-X20-1
shearing, the total shear stress in this test is not constant, instead, it cycles, decreasing in the
first half of a loading cycle and increasing in the other half. In the last a few cycles when the
5.2 Multi-directional tests 119
0
30
60
90
120
150
180
210
240
270
300
330
0
0.2
0.4
0.6
0.8
1
(b) Excess pore pressure ratio
0
30
60
90
120
150
180
210
240
270
300
330
0
0.1
0.2
0.3
0.4
(a) Total shear stress ratio
Fig. 5.17 The shear stress and excess pore pressure ratio of test C20-X20-1 in polar coordinate
system (the changing angular coordinates of the points in (a) corresponding to transformation
states account for the varying position of pore pressure thresholds in (b))
test approaches failure, the increase of total shear stress results in the increase of excess pore
pressure initially, but after the stress path passes the phase transformation line (the dashed
line in Figure 5.18), the relationship is reversed, with the further increase of total shear stress
reducing excess pore pressure. The values of total shear stress at phase transformation state
varies from cycle to cycle, which results in the changing position of the excess pore pressure
peak in a loading cycle.
This can also be seen in polar coordinates in Figure 5.17. For instance, the value of
total shear stress at the phase transformation state in the final cycle is 0.027, corresponding
to the red circle in Figure 5.17(a). A ray can be drawn, coloured red in Figure 5.17(a),
originating from the pole and passing through the intersection point between the red circle
and the loading-phase section on the loading path. With the angular measurement of this ray,
a point can an be determined on the outermost ring in Figure 5.17(b). This determined point
is clearly where the variation of excess pore pressure turns from increase (red) to decrease
(blue) in this cycle.
Another difference caused by the cycling of total shear stress, compared with the constant
SSRtotal test C20-3, rests in the maximum excess pore pressure ratio that can develop in a
120 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 20 40 60 80 100
v
' (kPa)
0
0.1
0.2
0.3
0.4
To
ta
l S
SR
0 5 10 15
Total shear strain total (%)
0
0.1
0.2
0.3
0.4
To
ta
l S
SR
00.20.40.60.81
Excess pore pressure ratio r
u
0
10
20
30
40
N
um
be
r o
f c
yc
le
s
0 5 10 15
Total shear strain total (%)
0
10
20
30
40
N
um
be
r o
f c
yc
le
s
Last cycle (20th)
Other cycles
Fig. 5.18 The 4-way plot of test C20-X20-1
test. As discussed previously, the failure line constrains the ultimate stress states. In the
circular tests without static shearing, maximum excess pore pressure ratio can be determined
conveniently from the intersection point of total shear stress path and failure line. For the
tests with static shearing, however, the ultimate stress states form a section on the failure line,
which is controlled by the amplitude of total shear stress, and the lower bound of total shear
stress decides the minimum effective vertical stress, or the maximum excess pore pressure.
Therefore, it is rational to expect that maximum excess pore pressure ratio can be correlated
to the lower limit of the amplitude of total shear stress ratio.
Figure 5.19 presents the maximum excess pore pressure ratio with the minimum total
shear stress ratio in all the circular tests conducted in this study. The negative correlation
between maximum pore pressure ratio and minimum SSRtotal is in congruous with the reports
of Boulanger (1990); Kammerer (2002). The solid line follows:
5.2 Multi-directional tests 121
ru,max = 1− SSRtotal,mintanφFL (5.4)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Minimum SSRtotal
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
m
a
x
Circular tests without static shearing
Circular tests with static shearing
Fig. 5.19 Maximum excess pore pressure ratio versus minimum total shear shear stress in all
circular tests
The development of excess pore pressure ratio with normalised number of loading cycles
is shown in Figure 5.20. The three-stage increase of excess pore pressure ratio is also
observed in the circular tests, no matter whether there is static shear or not.
The relationship between the excess pore pressure ratio generated in the first cycle of a
test versus the number of cycles required to initiate liquefaction is shown in Figure 5.21, with
the correlation equation 4.4 plotted as the solid curve. Although derived from uni-directional
tests, equation 4.4 is also viable under circular loading conditions, at least qualitatively,
suggesting that this relationship is also not altered by loading type.
122 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
Circular tests without static shearing
Circular tests with static shearing
Fig. 5.20 The development of excess pore pressure with normalised number of cycles in all
circular tests
Shear strain and shear modulus in circular tests
The plan view of shear strain of test C20-3 and test C20-X20-1 is shown in Figures 5.10
and 5.15. Two primary observations can be made regarding the development of shear
strain. Firstly, the shear strain develops in a circular manner that is related to the circular
loading paths, as anticipated. Secondly, when initial static shearing exists, the shear strain
accumulated towards the static-shear direction is always larger than in the other directions.
This is congruous with the results of uni-directional and bi-directional linear tests: static
shearing always dominates the development of shear strain.
The situation is, nevertheless, not straightforward regarding stress-strain relationship.
On the one hand, there is no consistent way to describe the relationship between total shear
stress and total shear strain. For instance, the SSRtotal − γtotal curve is almost a straight
5.2 Multi-directional tests 123
100 101 102 103
Number of cycles to liquefaction NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
1
Circular tests without static shearing
Circular tests with static shearing
Fig. 5.21 The excess pore pressure ratio generated in the first cycle versus the number of
cycles required to trigger liquefaction in all circular tests
line in test C20-3 (Figure 5.11), but becomes more similar to a uni-directional test in test
C20-X20-1 (Figure 5.18). On the other hand, the actual stress-strain relationships should
be four-dimensional in circular tests. The stress-strain curve in a certain direction is just the
projection of the four-dimensional stress-strain relationship in that direction. Under such a
circumstance, projecting the stress-strain relationships into dip and strike direction, which is
the conventional method used in analysing uni-directional and bi-directional linear tests, is
not applicable.
The projections of shear stress path and stress-strain curves of test C20-3 and C20-X20-1
onto X and Y direction are presented in Figure 5.22 and 5.23. The S-shaped hysteresis loops
in uni-directional tests are not seen in the projections of circular tests. Instead, the projections
of stress-strain hysteresis loops in test C20-3 have a cocoon-like geometry, while test C20-
124 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
X20-1 resembles pulled springs. The softening of soil can be told with these projected loops
gradually becoming flatter, but shear modulus becomes an issue that needs reassessment
0 20 40 60 80 100
v
' (kPa)
-0.2
0
0.2
SS
R X
-5 0 5
X (%)
-0.2
0
0.2
SS
R X
0 20 40 60 80 100
v
' (kPa)
-0.2
0
0.2
SS
R Y
-5 0 5
Y (%)
-0.2
0
0.2
SS
R Y
Fig. 5.22 The projections of stress paths and stress-strain relationships of test C20-3 on X
and Y planes
If the directionality is not accounted for, the shear modulus calculated along uni-
directional definitions is meaningless. For example, the secant shear modulus calculated
using total shear stress and total shear strain, following uni-directional definition, is zero in
test C20-3, as illustrated in the top right graph in Figure 5.11.
Otherwise, if shear modulus is defined by the stress-strain relationships in a certain direc-
tion, it cannot give information regarding other directions or spatial situation. Furthermore,
defining a secant shear modulus in a single direction is also challenging. In direction Y in
test C20-X20-1 as presented in Figure 5.23, for example, secant shear modulus cannot be
determined easily because there are no distinct tips in the hysteresis loops.
5.2 Multi-directional tests 125
0 20 40 60 80 100
v
' (kPa)
-0.1
0
0.1
0.2
0.3
0.4
SS
R X
-5 0 5 10
X (%)
-0.1
0
0.1
0.2
0.3
0.4
SS
R X
0 20 40 60 80 100
v
' (kPa)
-0.2
0
0.2
SS
R Y
-5 0 5 10
Y (%)
-0.2
0
0.2
SS
R Y
Fig. 5.23 The projections of stress paths and stress-strain relationships of test C20-X20-1 on
X and Y planes
It is thus reasonable if shear modulus can be evaluated in a spatial way in the four-
dimensional stress-strain space, but there are no practical or convenient mathematical tools
available and the difficulties brought to engineering practice may prevent the use of such
a complicated concept. Clearly, barriers in terms of the definition and application of shear
modulus arise from multi-directional loading conditions, and the modelling work based on
shear modulus should be careful with this issue.
5.2.2 Tests with oval loading path
An oval loading path is a type of loading condition that is rather similar to a circular path.
The difference is that the shear stress not only rotates, but also changes in amount, even if
126 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
there is no initial static shearing. The comparisons between circular tests and oval tests can
shed some light on the interaction of directionality and amplitude of shear stress.
Six tests with oval loading paths were conducted in this study. Four of them had a
elliptical path without static shear, a shear stress ratio of 0.2 on the semi-major axis and 0.1
on the semi-minor axis, with the aim of providing comparisons against the circular tests with
shear stress ratio of 0.2. Liquefaction is defined consistently with circular tests as taking
place in the cycle where shear stress path on either direction reaches the failure line.
The results of an example test E2010-1 are shown here to illustrate the characteristics
of soil behaviour in this category of tests. This test has relative density of 54%, B value of
0.954, and was consolidated with a vertical stress at 100 kPa. Maximum excess pore pressure
ratio of 0.908 was achieved in the 19th cycle. The long axis and minor axis of this test does
not coincide with the X and Y axis of the loading system, with a 15◦ angular difference
between them. The cyclic loading was started after a ramp loading pushed the shear stress to
the vertex, and shear strain measurement was zeroed before the cyclic loading began.
-0.2 -0.1 0 0.1 0.2
SSRX
-0.1
-0.05
0
0.05
0.1
SS
R Y
(a) Plan view of shear stress ratio
-10 -5 0 5 10
X(%)
-10
-5
0
5
Y
(%
)
(b) Plan view of shear strain
Fig. 5.24 The plan view of shear stress and shear strain of test E2010-1
5.2 Multi-directional tests 127
The plan view of measured shear stress and shear strain is shown in Figure 5.24, and a
4-way plot is presented in Figure 5.25, with the round mark representing the starting point
and the arrow indicating the anticlockwise direction of the loading path in the left graph. As
in the circular tests, the initial part of the first cycle overshoots and veers away from the path
because the adaptive loading system requires some time to adjust. The last cycle also deviates
from the path as the stiffness of the sample is reduced significantly, making it difficult for the
testing system to maintain good control. The strain path on the plan view shows a general
oval shape, and the oval strain path enlarges as the test approaches liquefaction.
0 20 40 60 80 100
v
' (kPa)
0
0.05
0.1
0.15
0.2
0.25
0.3
To
ta
l S
SR
0 5 10 15
Total shear strain total (%)
0
0.05
0.1
0.15
0.2
0.25
0.3
To
ta
l S
SR
00.20.40.60.81
Excess pore pressure ratio r
u
0
10
20
30
N
um
be
r o
f c
yc
le
s
0 5 10 15
Total shear strain total (%)
0
10
20
30
N
um
be
r o
f c
yc
le
s
Last cycle
Other cycles
Fig. 5.25 The 4-way plot of test E2010-1
The shear stress path and stress-strain relationships on X and Y plane are presented
in Figure 5.26. The stress path in Y direction and the stress-strain hysteresis loops in X
direction exhibit a rather peculiar appearance. The SSRY versus σ ′v stress path has a crab-claw
shape that is unlikely to be interpreted with the uni-directional-based concept of failure line
128 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
or phase transformation line in their direction. The hysteresis loops in X direction when
liquefaction is imminent are in anomalous shape, from which secant shear modulus becomes
meaningless since the tips on the loops are multiple.
0 20 40 60 80 100
v
' (kPa)
-0.3
-0.2
-0.1
0
0.1
0.2
SS
R X
-10 -5 0 5 10
X (%)
-0.3
-0.2
-0.1
0
0.1
0.2
SS
R X
0 20 40 60 80 100
v
' (kPa)
-0.3
-0.2
-0.1
0
0.1
0.2
SS
R Y
-10 -5 0 5 10
Y (%)
-0.3
-0.2
-0.1
0
0.1
0.2
SS
R Y
Fig. 5.26 The projections of stress paths and stress-strain relationships of test E2010-1 on X
and Y planes
Viewing in a three-dimensional space, however, the peculiar stress path starts to make
sense. The development of excess pore pressure ratio with loading path is shown in three-
dimensional view in Figure 5.27. Since the excess pore pressure ratio can be regarded as the
inverse effective vertical stress ratio, this graph is actually equivalent to a three-dimensional
stress path. It is not difficult to notice that the 3D spatial structure of the excess pore pressure
development in this oval test looks rather similar to uni-directional test if viewing along
the minor axis. Viewing along the major axis, however, shows the “crab claw” seen in the
5.2 Multi-directional tests 129
shear stress path in Y direction, with the “claws” being in fact the side views of the peaks of
three-dimensional excess pore pressure development.
Fig. 5.27 3D development of excess pore pressure ratio of test E2010-1
The maximum excess pore pressure ratio of 0.908, with the minimum total shear stress
ratio of 0.07, further demonstrating the conclusion drawn in circular tests that the minimum
total shear stress dictates the maximum excess pore pressure that can develop in a test,
following equation 5.4.
The variation of excess pore pressure within a single cycle on the right graph in Figure
5.27 does not have a consistent pattern in the early cycles with relatively low or moderate
excess pore pressure, but in the cycle of liquefaction, a distinct pattern appears with the
vertices and co-vertices becoming the partition points between the increase and decrease of
excess pore pressure. The two peaks of excess pore pressure are at the co-vertices while
the two valleys appear at vertices. The reasons of this phenomenon rest in the loading and
unloading of shear stress. Different from circular tests, the amount of shear in oval tests
is not constant. The major axis has the largest amount of shear and the minor axis has
the least. Therefore, loading paths in oval tests contain distinguishable periods of loading
130 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
and unloading, with the former taking place when shearing proceeding from co-vertices to
vertices and the latter in reverse. It is evident that the three-dimensional spatial structure
of excess pore pressure development is influenced by the type of loading path, and the
conventional 2D analysis method may mislead the understanding of soil behaviour.
5.2.3 Test with figure-8 loading path
Figure-8 or butterfly loading path is another type of fundamental loading paths that have
both stress rotation and stress reversal. The most significant feature of it, compared with
circular and oval loading paths, is that shear stress can pass through origin accompanied by
continuous change of shear stress direction.
Seven tests with figure-8 or butterfly loading paths were conducted in this study, with
varying amplitude of shear stress and shape of loading path. The results of test F8-3 are
presented as an example to illustrate the soil behaviour under this type of loading condition.
This test has relative density of 46.8%, Skempton’s B value of 0.98, and vertical consolidation
stress of 100 kPa. Liquefaction took place in the 11th cycle. The plan view of shear stress
and shear strain is shown in Figure 5.28. The cyclic loading started from zero and proceeded
along the direction oriented by the arrow. The amplitudes of shear stress on X and Y axis
were both 0.15.
As circular and oval tests, the initial part of the first cycle has an issue of overshooting
due to imperfect synchronizing of the two loading strokes in the testing device, and the last
cycle deviates from the path as the reduced stiffness of the sample makes the testing system
lose good control. The loading path of the last cycle is coloured red or dark to indicate
whether the total shear stress is increasing (red) or decreasing (dark). The plan view of shear
strain has a butterfly shape that is narrower on the positive γY side and wider on the other.
The X-direction and Y-direction shear stress paths and stress-strain relationships are
presented in Figure 5.29. The stress path on X direction has a peculiar behaviour (top-left
graph in Figure 5.29). In the last loading cycle, after exceeding the phase transformation line
(the dashed lines), the decrease of SSRX can result in the increase of effective vertical stress.
This is in conflict with the well-established theory from uni-directional tests which states the
5.2 Multi-directional tests 131
-0.1 0 0.1
SSRX
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
SS
R Y
(a) Plan view of shear stress ratio
-5 0 5
X(%)
-4
-2
0
2
4
Y
(%
)
(b) Plan view of shear strain
Fig. 5.28 The plan view of shear stress and shear strain of test F8-3 (the loading path of the
last cycle in (a) is coloured, with red for increasing total shear stress and dark for decreasing
total shear stress)
decrease of post-phase-transformation shear stress always leads to the decrease of effective
mean stress. Furthermore, the stress-strain hysteresis loops in X direction are in a shape from
which secant shear modulus cannot be easily defined. It is highlighted by these findings that
caution should be exercised when applying uni-directional analysis to a specific projection
plane under a multi-directional loading scenario. Under extreme circumstances like in this
test, the conclusions drawn from a projection plane can be rather deceptive.
Viewing in three-dimensional space, the spatial structure of the shear stress path can
be restored and reassessed, as shown in Figure 5.30. The counter-intuitive soil behaviour
observed in X-direction stress path in Figure 5.29 is actually the projection of such a spatial
structure.
The four-way plot of test F8-3 is presented in Figure 5.31. The two minimum SSRtotal
points are the starting points of the two half of the last loading cycle (coloured red). Taking the
last half cycle as an example, the starting stress state is located below the phase transformation
line (dashed line), as a result of which the increase of SSRtotal leads to the increase of excess
pore pressure first. After the total shear stress path reaches the phase transformation line,
132 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 20 40 60 80 100
v
' (kPa)
-0.2
0
0.2
SS
R X
-5 0 5
X (%)
-0.2
0
0.2
SS
R X
0 20 40 60 80 100
v
' (kPa)
-0.2
0
0.2
SS
R Y
-5 0 5
Y (%)
-0.2
0
0.2
SS
R Y
Fig. 5.29 The projections of stress paths and stress-strain relationships of test F8-3 on X and
Y planes
however, the tendency is reversed, with further increase of SSRtotal reducing excess pore
pressure. This is why the initial part of the left half of the last cycle in Figure 5.30 has
excess pore pressure increasing first before decreasing, although the total shear stress in
this part keeps increasing. The total shear stress states in the subsequent periods remain in
post-phase-transformation, where all the increase of SSRtotal results in the decline of pore
pressure and all the decrease of SSRtotal increases pore pressure, which is compatible with
the results in Figure 5.30.
It should also be noted that the maximum excess pore pressure ratio generated in this
test is only 0.848, rather than the expected value of 0.95 calculated from equation 5.4 with
SSRtotal,min = 0.015. But this does not mean equation 5.4 is not applicalbe because this test
was stopped before the failure stress states can fully develop. If more shearing cycles had
5.2 Multi-directional tests 133
Fig. 5.30 3D development of excess pore pressure ratio of test F8-3
been applied, it was very likely that the final ru,max could advance towards 0.95 as the total
shear stress path approaches and stayed on the failure line.
5.2.4 Discussions on the interpretation of multi-directional tests
The results of circular, oval and figure-8 tests analyzed in this chapter contain significant
implications regarding the gaps in understanding the test results, which are worth of a
summary and further discussion.
Firstly, the maximum excess pore pressure that can be generated in a test is closely
associated with the type of loading path. On the one hand, the maximum excess pore pressure
ratio in many instances is below unity. The liquefaction criterion that requires excess pore
pressure ratio to reach unity as liquefaction takes place is not applicable in this situation.
On the other hand, there exists a linear correlation between the maximum excess pore
pressure ratio and the minimum total shear stress ratio, regardless of loading type, described
mathematically by equation 5.4.
134 MULTI-DIRECTIONAL SIMPLE SHEAR TEST RESULTS
0 20 40 60 80 100
v
' (kPa)
0
0.05
0.1
0.15
0.2
To
ta
l S
SR
0 2 4 6 8 10
Total shear strain total (%)
0
0.05
0.1
0.15
0.2
To
ta
l S
SR
00.20.40.60.81
Excess pore pressure ratio r
u
0
10
20
0
10
2
N
um
be
r o
f c
yc
le
s
0 2 4 6 8 10
Total shear strain total (%)
0
10
20
0
10
20
N
um
be
r o
f c
yc
le
s
Last cycle
Other cycles
Fig. 5.31 The four-way plot of test F8-3
The reason why such a correlation should exist rests in the failure states in terms of total
shear stress. SSRtotal versus σ ′v path reaches and stays on the failure line eventually, from
which the maximum pore pressure can be determined on the failure line with the minimum
SSRtotal . However, the concept of a failure line may not be fully applicable or descriptive.
More detailed discussions on this will be made in the next chapter, together with the concept
of conical failure surface.
Secondly, similar to the failure line, the concept of phase transformation line should also
be considered in a total shear stress manner. After SSRtotal versus σ ′v path passes through
phase transformation line, any increase of total shear stress will reduce excess pore pressure
while any decrease of total shear stress will result in the growth of excess pore pressure. In
the figure-8 test analysed, for instance, the lack of information about phase transformation
can mislead the understanding of pore pressure variation in the last loading cycle. However,
5.2 Multi-directional tests 135
it should be noted that the application of phase transformation line to total shear stress is
also an assumption. More comprehensive discussion regarding this will be given in the next
chapter.
Thirdly, the shear stress path or the equivalent development of excess pore pressure should
be three-dimensional under multi-directional loading conditions. The spatial structure of such
a path should not be interpreted in a projection plane without caution. The results of example
oval and figure-8 test demonstrates that some crucial information can be lost in the process of
projecting such a spatial structure onto a plane, and the observations made from a projection
can be deceptive or even erroneous. Evidently, the stress analysis of multi-directional tests
should be conducted in three-dimensional space and the spatial structure of a shear stress
path needs to be incorporated into the three-dimensional analysis framework.
Last but not least, stress-strain relationships become a four-dimensional issue under
multi-directional loading condition because both shear stress and shear strain become planar
problems. Correspondingly, the definition of shear modulus becomes challenging. The
meaning of the ratio of total shear stress over total shear strain is doubtful. On the other hand,
the projections of shear stress-strain relationships on a plane can be misleading. Furthermore,
although constructing the definition of the shear modulus in a multiple-dimensional space
is an attractive plan, there is not a straightforward or practical mathematical method to
achieve this objective. Therefore, deformation-oriented constitutive models based on the
shear modulus determined from hysteresis loops should be reassessed.
In conclusion, there are significant gaps in understanding the influence of multi-directional
loading paths on liquefaction in simple shear tests that need to be bridged. The conventional
analysis framework which is used in uni-directional tests is not necessarily effective under
multi-directional loading conditions. The 2D τ −σ ′v analysis should be reassessed and
extrapolated into 3D τ−σ ′v space, so that the influence of actual earthquake-induced multi-
directional loading paths can be better interpreted and perceived.

Chapter 6
FAILURE AND PHASE
TRANSFORMATION IN
THREE-DIMENSIONAL τ−σ ′v SPACE
6.1 Failure cone
6.1.1 The concept of failure cone
While the results of multi-directional simple shear tests were analysed and illustrated in
Chapter 5, the strain-based liquefaction criterion (6% double amplitude strain) was not
applied because the variation of shear strain happens both in its amount and direction,
which cannot be assessed as conveniently as in uni-directional loading scenarios. Instead,
liquefaction was enabled when the shear stress in any direction reaches the failure line.
However, this liquefaction criterion relies on the assumption that the position of the
failure line in any direction remains the same. If the failure of sand in simple shear tests were
a pure frictional issue and obeyed Coulomb’s friction law perfectly, the assumption could
become equivalent to a Friction Cone problem.
For an ideal case where a block slides on a non-smooth plane and the coefficient of
kinetic friction µs is constant in any position and direction on the plane, if the motion of a
138 THREE-DIMENSIONAL τ−σ ′v SPACE
Fig. 6.1 The schematic of friction cone
block is uni-directional, the frictional force between the body and the frictional surface can
be determined with Coulomb’s law in that specific direction, as represented in the left graph
in Figure 6.1. In contrast, if the block is allowed to move towards any direction in a plane,
the same Coulomb’s law should be established in all allowable directions, with its envelope
forming a cone whose semi vertex angle is tan−1µs, as sketched in the right part of Figure
6.1. The line representing uni-directional friction law becomes the generatrix and the N axis
serves as the axis of failure cone. It should be noted that the “cone” in friction cone theory
refers to a conical surface rather than a solid body, and the terms are used interchangeably in
this study.
The concept of friction cone has been widely adopted in research regarding rock me-
chanics since the 1950s (Goodman, 1989; John, 1968; Priest, 1985; Talobre, 1957) because
the discontinuities in rocks make it necessary to analyse the stability of rock slopes in a
three-dimensional manner. In the past a few decades, it was also given much attention to by
researchers in the areas of computer visualization and robotics, along with other theoretical
fundamentals for solving multi-face contact or multi-body motion problems (Renouf et al.,
2005; Sauer and Schömer, 1998; Wriggers and Krstulovic´-Opara, 2004).
6.1 Failure cone 139
In contrast, the application of this concept on soil mechanics has been rather limited. The
particles of soil are small enough to inhibit prominent discontinuities from happening in the
profile of a single soil layer. The macro mechanical behaviour of soil is usually analysed
with the framework of continuum mechanics. Furthermore, conventional laboratory testing
techniques, such as triaxial and uni-directional simple shear testing, cannot simulate the
multi-directional loading paths seen in real earthquakes, making it difficult to investigate
the effects of multi-directional loading. Dynamic loading induced by earthquakes, wind
or waves are hence simplified as uni-directional cyclic loading as a rule, and the effects of
mult-directional loading are represented by an adjustment factor in engineering practice.
The test results analysed in Chapter 5, nevertheless, show that the uni-directional sim-
plification fails to capture all the important characteristics of the generation of excess pore
pressure, accumulation of shear strain and stress response under multi-directional loading. To
develop a more in-depth understanding of how soil behaves under realistic complex loading
scenarios, the dimensions of analysis framework should be increased so that the directionality
of loading can be taken into account. The concept of a friction cone provides valuable
insights in achieving this objective.
The Mohr–Coulomb failure criterion is usually adopted for simple shear conditions:
τ = σ ′v tanφFL + c (6.1)
where σ ′v is the effective vertical stress and φFL is the angle of the failure line.
The use of the angle of internal friction φ ′ is avoided in this study. Due to the lack of
measurements of lateral stresses, the three principal stresses cannot be determined. The
angle of internal friction φ ′, which equals sin−1
(
σ1−σ3
σ1+σ3
)
, can not be calculated accurately.
Furthermore, internal friction angle is affected by intermediate principal stress (Broms and
Casbarian, 1965; Lade, 1973; Matsuoka and Nakai, 1974) but the latter cannot be determined
with available simple shear tests. Thus, the angle of the failure line φFL is used in this study,
which is not necessarily the same as internal friction angle φ ′.
140 THREE-DIMENSIONAL τ−σ ′v SPACE
The term c was regarded as true cohesion by Terzaghi but Schofield has claimed in many
of his work that c is the result of interlocking rather than cohesion. Readers are referred to
Disturbed Soil Properties and Geotechnical Design (Schofield, 2005) for comprehensive
discussions on this topic. In undrained tests on sands, however, c is usually considered as
zero in either of the definitions. The failure states thus fall into its simplest form in 2D τ−σ ′v
plane:
τ = σ ′v tanφFL (6.2)
which is congruous with Coulomb’s friction law with tanφFL serving as the equivalent
coefficient of friction.
If the directionality of shear stress τ is considered, the 2D τ −σ ′v space needs to be
expanded into a 3D space formed by τX , τY and σ ′v, where τX and τY are the shear stress
decomposed onto two arbitrarily-assigned orthogonal directions. Correspondingly, the
Mohr-coulomb failure line Equation 6.2 should be extrapolated into a failure cone having
semi-vertex angle of φFL, as defined by:
τX 2 + τY 2 = σ ′v
2 tan2(φFL) (6.3a)
√
τX 2 + τY 2 = σ ′v tanφFL (6.3b)
τtotal = σ ′v tanφFL (6.3c)
In liquefaction analysis, shear stress is more often normalised by initial vertical stress
as shear stress ratio SSR = τσv,i . A failure cone in 3D SSR−σ ′v space is illustrated in Figure
6.2, with the representative angle of failure φFL determined to be 33.4◦ for medium-dense
Hostun sands tested in this study.
If such a failure cone exists, it should envelop all possibilities of stress states of sands in
undrained simple shear tests. A shear stress path cannot run outside the cone, and flow-type
deformation or rapid accumulation of shear strain should take place when the failure cone is
reached.
6.1 Failure cone 141
Fig. 6.2 The schematic of failure cone in 3D SSR−σ ′v space
The existence of failure cone can be verified by examining if equation 6.3c describes
the undrained soil response. The angle of failure line φFL does not always hold the value of
33.4◦, partly because the density variation of sands leads to varying angles of failure line and
partly because the loading system in this study cannot guarantee good control after flow-type
deformation takes place. The angle of 33.4◦ is a representative average value for the range of
specimens tested. Therefore, to validate equation 6.3c is actually to prove that the failure
envelope for the total shear stress path is a straight line, whose inclination angle does not
have to be exactly 33.4◦ however.
The 4-way plot of test C20-X20-1 which was analysed as an example test in Chapter 5
(Figure 5.18) is reploted in Figure 6.3, with more cycles presented. It is evident that the stress
path tends to become stationary once it reaches the failure envelope, as seen more clearly in
the distance between the bottom tips of every two total shear stress cycles. The cycle which
has a bottom tip at σ ′v of 10 kPa (20th cycle) seems to be the turning point, before which
the distance between every two cycles keeps increasing, indicating the generation of excess
pore pressure is accelerating and the sample is approaching failure. After that cycle, however,
142 THREE-DIMENSIONAL τ−σ ′v SPACE
the distance between cycles gradually decreases. It appears that the total shear stress path
resists to move further leftwards. A reasonable guess is that the sample has failed since the
20th cycle with the development of total shear stress path constrained thereafter, but the
imperfect control of loading system results in it actually moving leftwards subsequently with
a limited degree. If the loading system had ideal response to the low stiffness of soil sample
at this point, it should be expected that the total shear stress path would stay on roughly
the same location since the 20th cycle and proceed on the same trace after that. The total
shear stress path of the 20th cycle is red in Figure 5.18, clearly overlapping equation 6.3c.
Furthermore, even with the nominal last cycle (23rd cycle) of the test, coloured red in Figure
6.3, the failure envelope in τtotal −σ ′v space seem to still have the shape of a straight line,
which validates equation 6.3c and the concept of the failure cone in this case.
0 20 40 60 80 100
v
' (kPa)
0
0.1
0.2
0.3
0.4
To
ta
l S
SR
0 5 10 15 20 25
Total shear strain total (%)
0
0.1
0.2
0.3
0.4
To
ta
l S
SR
00.20.40.60.81
Excess pore pressure ratio r
u
0
10
20
30
40
N
um
be
r o
f c
yc
le
s
0 5 10 15 20 25
Total shear strain total (%)
0
10
20
30
40
N
um
be
r o
f c
yc
le
s
Last cycle (23th)
Other cycles
Fig. 6.3 The 4-way plot of test C20-X20-1 (C.F. Figure 5.18)
6.1 Failure cone 143
The analysis of other multi-directional loading paths leads to similar results. Oval test
E2010-1 analysed in Chapter 5, for example, reaches failure at the 19th cycle, which is
reflected by the sudden jump of shear strain as shown in Figure 5.26. At the time when the
flow-type deformation occurs, the total shear stress path in Figure 5.25 coincides with the
failure line following equation 6.3c. Similarly, the distinct flow-type development of shear
strain happens in the 11th cycle in the example figure-8 test F8-3 as seen in Figure 5.31. The
corresponding total shear stress path of this cycle is also observed to reach the proposed
failure line.
The concept of failure cone can be validated against all the testing results in this study
and it is defensible to conclude that equation 6.3c is applicable to undrained simple shear
tests on medium-dense clean Hostun sands vertically consolidated to 100 kPa, no matter
what the shearing path is. In addition, it is possible to expect that the failure cone exists in
sand tested with other testing conditions, but the details of failure cone may vary. For drained
test, the failure cone needs to be modified to a piecewise-truncated cone to account for the
contribution of dilation. For soils with outstandingly anisotropic fabrics, the failure cone is
likely to become elliptical to take the anisotropy into account. While these assumptions need
verification by more test data in the future, it cannot be denied that a conical failure surface
can be established under the testing conditions in this study.
6.1.2 Failure states of bi-directional linear tests and multi-directional
tests
With the failure cone hypothesis, uni-directional linear tests represent the stress paths on the
unique axial section plane that cuts the cone through its vertex and along its axis. The outline
of the intersection of the cone and the plane is in the shape of two straight lines, which
form the uni-directional failure lines. Similarly, through analysing the relative locations of
the failure cone and loading paths, some peculiar phenomena in bi-directional and multi-
directional tests can be explained. In the first instance, the failure envelope of bi-directional
linear tests can be re-examined. As illustrated in Figure 5.4a and 5.5a, the failure states on the
144 THREE-DIMENSIONAL τ−σ ′v SPACE
cyclic loading direction can be altered by the static shearing on the perpendicular direction in
bi-directional linear tests. An interpretation of this phenomenon can be found in the conic
sections of failure cone.
Fig. 6.4 The schematic of the failure cone intersected by a plane
In contrast with uni-directional tests, the static shearing in bi-directional linear tests is
perpendicular to the direction of cyclic loading. The cyclic shearing effectively occurs on a
plane cutting across the failure cone parallel to its axis, forming a conic section (a branch of
and hyperbola) as illustrated in Figure 6.4. The hyperbolic outline can be described as:
tan2(φFL)
σ ′2v
τ2static
− τ
2
cyclic
τ2static
= 1 (6.4)
The shear stress paths of test Bilinear-8 in dip and strike direction are presented as an
example in Figure 6.5. The square marker indicates the end of the test and the circle marker
locates the point with minimum measured static shear stress ratio. The solid hyperbola
in Figure 6.5b represents the conic section calculated with τstatic of the end point (square-
marked point) at 0.149, while the dashed one is calculated with the measured minimum
6.1 Failure cone 145
τstatic (circle-marked point) at 0.122. It is clear that the hyperbola derived from failure cone
theory is effective in enveloping the failure states of bi-directional linear tests. A more
understandable graph of the same stress path is shown in 3D τ −σ ′v space in Figure 6.6,
which visualizes and validates the effects of failure cone on constraining potential failure
states under bi-directional linear loading paths.
Three primary conclusions can be made from equation 6.4. Firstly, the asymptote of the
hyperbola crosses zero, indicating that effective vertical stress σ ′v can never reach zero in
bi-directional linear tests. To put it in other word, the excess pore pressure ratio can reach
unity in linear tests only if orthogonal static shear is strictly zero (τstatic ̸= 0); otherwise,
imperfect uni-directional linear tests which have even marginal static shear force on the
orthogonal direction can still result in a maximum excess pore pressure less than unity.
Secondly, the higher the static shearing, the farther the ultimate cyclic shear stress path in the
perpendicular direction is away from origin. Thirdly, equation 6.4 can be transformed to give
the relationship between excess pore pressure ratio and shear stress ratios as follows:
ru = 1−
√
SSR2cyclic +α2
tanφFL
(6.5)
For cases where stress reversal is seen in the direction of cyclic loading, that is, SSRcyclic
can reach zero at times, the maximum excess pore pressure ratio can only be achieved when
SSRcyclic becomes zero. Correspondingly, the maximum excess pore pressure ratio can be
correlated exclusively with α as:
ru,max = 1− αtanφFL (6.6)
which is exactly the same as the correlation obtained from bi-directional linear testing data,
as introduced in equation 5.1.
In addition to bi-directional linear tests, the failure states of multi-directional tests are
also associated with the failure cone. The stress paths of circular test C20-3 and C20-X20-1,
oval test E2010-1 and figure-8 test F8-3 are shown in Figure 6.7- 6.10. The reason why the
stress path of test C20-3 cannot approach the origin is that it is constrained by the failure
146 THREE-DIMENSIONAL τ−σ ′v SPACE
10 20 30 40 50 60 70 80 90 100 110
v
' (kPa)
-0.1
0
0.1
0.2
0.3
SS
R X
(a) Stress in dip direction (static shearing)
10 60 70 80 90 100
v
' (kPa)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R Y
(b) Stress in strike direction (cyclic shearing)
Fig. 6.5 Stress paths of test Bilinear-8
6.1 Failure cone 147
Fig. 6.6 The stress path of test Bilinear-8 and hyperbolic conic section in 3D τ−σ ′v space
cone. When its stress path reaches the failure cone, the circular loading has no choice but to
stay on the conical surface and develops along a circular route. It is impossible to go beyond
that point with such a loading condition.
Test C20-X20-1, by contrast, has static shearing towards the positive X direction. The
part of the stress path that has high SSRX values and is closer to the conical failure surface
is restricted by the failure cone, while the other side of the stress path which has low SSRX
values has more freedom to develop towards the origin.
The stress paths of the oval test and the figure-8 test also support this conclusion. The
co-vertices of the oval loading path and the junctions in the figure-8 test obviously have lower
shear stress values. The failure states corresponding to these special points, therefore, always
develop higher excess pore pressure and are closer to the origin.
Recalling the projections of stress paths onto X and Y directions as presented in Figure
5.22, 5.23, 5.26, and 5.29, it should be emphasized again that the information regarding the
spatial structure of stress paths is lost in the process of projection, and the effects of the shape
of loading paths cannot be interpreted comprehensively from the projections. Stress-path
148 THREE-DIMENSIONAL τ−σ ′v SPACE
Fig. 6.7 The stress path of circular test C20-3 in 3D τ−σ ′v space
Fig. 6.8 The stress path of circular test C20-X20-1 in 3D τ−σ ′v space
6.1 Failure cone 149
Fig. 6.9 The stress path of circular test E2010-1 in 3D τ−σ ′v space
Fig. 6.10 The stress path of circular test F8-3 in 3D τ−σ ′v space
150 THREE-DIMENSIONAL τ−σ ′v SPACE
analysis in three-dimensional τ −σ ′v space should be preferred, with the projections as a
supplement to enhancing the understanding of spatial stress paths.
6.1.3 Determination of failure states with failure cone
Through the analysis of the failure states of linear tests, bi-directional linear tests and multi-
directional tests, a general method to determine the failure envelope can be established in
τ−σ ′v space under the framework of the failure cone.
The failure state envelope corresponding to a specific loading path can be predicted by
projecting the loading path in the τX − τY plane onto the conical failure surface, which can
be described by:
τX
2 + τY 2 = σ ′v
2 tan2φFL
g(τX ,τY ) = 0
(6.7)
where X and Y represent any two orthogonal directions arbitrarily assigned, and g(τX ,τY )
gives a cylinder or a plane intersecting the envelope of a certain loading path along the axis
of cone.
For uni-directional linear tests, a loading path is a line segment, from which a axial section
plane can be constructed as g(τX ,τY ) =
√
τ2X + τ2Y −τdir = 0 with the subscript dir indicating
the direction of loading. The failure envelope reduces to the uni-directional Mohr-Coulomb
failure criterion, τdir = σ ′v tanφFL.
In bi-directional linear tests, loading paths are also in the shape of a line segment, but
the plane constructed from it does not contain the axis of cone. To simplify the calculation,
the direction of cyclic loading is assigned as direction Y, and the orthogonal direction is
designated as X, so that the plane function has g(τX ,τY ) = τ2X − τ2static = 0 and the failure
envelope becomes the hyperbola expressed by equation 6.4.
For circular loading scenarios, the equation of a cylinder can be built from the loading path
following g(τX ,τY ) = (τX − τX0)2 +(τY − τY 0)2− r2 = 0, where r represents the radius of
the loading-path circles or the amplitude of applied shearing. Equation 6.7 needs to be solved
6.2 Phase transformation in 3D τ−σ ′v space 151
in polar coordinates for general circular loading paths as (τX0+r ·cos t)2+(τY 0+r ·sin t)2 =
σ ′v
2 tan2(φFL).
Parametrization needs to be conducted to attain solutions of equation 6.7 for more general
type of loading paths. Numerical methods are required when necessary. In general, equation
6.7 provides a fundamental method to locate the failure states of a loading path, with which a
more specific failure criterion can be established for multi-directional shearing.
6.2 Phase transformation in 3D τ−σ ′v space
Since the 1970s when Ishihara et al. (1975) observed the existence of phase transformation in
undrained tests on medium to dense sands, it has attracted intense interest among geotechnical
researchers and engineers. The phenomenon of phase transformation, which is commonly
described as the stress state where excess pore pressure turns from growing to declining
with the increase of shearing, is observed and reported in almost every experimental work
conducted on undrained sands. Recent development of discrete element modelling method
has also contributed to the understanding of this phenomenon. Although the mechanisms
behind it is still unclear, it is widely accepted nowadays that phase transformation states form
a line in 2D τ−σ ′v space, whose angle is lower than that of failure line and affected by the
state parameter and stress states of sands.
A natural inference arises from the similarity between phase transformation line and
failure line: since the failure line can be extrapolated to a failure cone in 3D τ−σ ′v space,
the same extrapolation should be applicable to phase transformation line as well. If such
a hypothesis can be proved, a phase transformation cone should exist in 3D τ −σ ′v space
following:
τtotal = σ ′v tanφPT (6.8)
where φPT is the angle of phase transformation line measured from uni-directional tests.
Sands should exhibit contractive tendenies when shear stress paths are inside this cone and
turn to show dilative tendencies after crossing outside the cone. The aim of this section is to
152 THREE-DIMENSIONAL τ−σ ′v SPACE
investigate the hypothesis of conical phase transformation surface with the testing data in
this study.
6.2.1 The validity of phase transformation cone in multi-directional
tests
When the data of circular, oval and figure-8 tests were analysed in Chapter 5, the concept
of phase transformation cone had already been applied explicitly and tentatively without
demonstration. In fact, the ability of the assumed phase transformation cone in explaining
effectively the soil behaviour under these multi-directional loading scenarios is an important
indicator of its validity.
In the first instance, the results of test C20-X20-1 are recalled: excess pore pressure
peaks vary significantly among cycles as shown in Figures 5.16 and 5.17. This intriguing
phenomenon was explained with a phase transformation line in SSRtotal−σ ′v plot in Figure
5.18. On the one hand, it is conspicuous that the suggested phase transformation line divides
the stress space into two regions. Before the phase transformation line is reached, the increase
of total shear stress decreases effective vertical stress as expected in typical contractive sand
behaviour. After it is passed through, the increase of total shear stress results in the increase of
effective vertical stress, which is consistent with the dilative tendency appearing due to phase
transformation. It is evident that this line holds the characteristics of phase transformation.
On the other hand, the total shear stress values of the assumed phase transformation line can
be pinned on the polar loading path as presented in Figure 5.17(a). The corresponding angular
coordinates obtained coincide perfectly with the locations of peak excess pore pressure in
Figure 5.17(b). It is demonstrated that the phase transformation hypothesis is capable of
providing spatial information about the peaks of excess pore pressure.
Although the existence of such a phase transformation line was in fact a hypothesis rather
than a validated theory, the results of test C20-X20-1 support its role. Considering that the
phase transformation line in Figure 5.18 is the graphical reflection of equation 6.8, it can be
concluded that a phase transformation cone exists, at least in test C20-X20-1.
6.2 Phase transformation in 3D τ−σ ′v space 153
The hypothesis can be examined further in 3D τtotal−σ ′v space. Figure 6.11a presents the
3D stress path of test C20-X20-1 together with the failure cone and the phase transformation
cone. The data points outside the phase transformation cone are marked with circles and
coloured by the variation of effective vertical stress, with blue indicating the effective vertical
stress of this point is smaller than that of the previous point and red otherwise. Loading
proceeds clockwise, which can be told from the initial part of the stress path that has the
highest σ ′v. When the stress path moves outside the phase transformation cone, the trend of
effective vertical stress is reversed, reflected in both the swerve of the stress path and the
change of data points’ colour from blue to red. At the maximum SSRX , shearing changes
from loading to unloading, resulting in another abrupt change in the development of effective
vertical stress.
Figure 6.11b shows the same stress path in a different way. All the data points after stress
path emerges from phase transformation cone for the first time are highlighted. While the
data points outside the phase transformation cone are still marked with circles, points inside
the cone are marked with diamonds. The colours have new meanings. For the points inside
the cone, if effective vertical stress of a point is lower than previous one, that is, excess pore
pressure keeps increasing, the point is in agreement with the typical behaviour inside the
phase transformation and coloured blue; otherwise, it is coloured red to indicate atypical
behaviour. For the points outside the phase transformation cone, however, if the increase
of total shear stress results in the increase of effective vertical stress, or the unloading of
shear reduces the effective vertical stress, the points are coloured blue to suggest that the
typical post-phase-transformation behaviour is occurring; the points become red if such a
behaviour is not observed. In other words, blue points support the phase transformation cone
hypothesis while red points do not.
Red points are dispersed in the early cycles where excess pore pressure is relatively low
or moderate. This is similar to the fluctuation of excess pore pressure seen in the initial part
of uni-directional simple shear tests. Since the oscillation is not steady or repeating with a
clear pattern, the odds are that it is caused by the imperfect control of the testing apparatus or
sensor noise instead of some law of soil mechanics. In the last few pre-liquefaction cycles,
154 THREE-DIMENSIONAL τ−σ ′v SPACE
however, red points become rare, indicating the phase transformation cone hypothesis is
satisfied almost everywhere on the loading path.
The test results of oval tests and figure-8 tests also support the hypothesis of a phase
transformation cone. The stress path of oval test E2010-1 is presented in Figure 6.12 in the
same manner as Figure 6.11. After stress path passes through the conical phase transformation
surface, the increase of shear loading bends the stress path away from origin. When checking
the discrepancy between the hypothesis and measured data in Figure 6.12b, test data agrees
closely with the hypothesis, demonstrating the existence of phase transformation cone in oval
tests. Shown in Figure 6.13 is the results of test F8-3 which has a figure-8 shaped loading
path. The visually indistinguishable division between loading and unloading periods of such
loading path hinders judgement-making based on the variation of effective vertical stress in
Figure 6.13a. But an inspection of Figure 6.13b reveals that the measured data from this test
show great consistency with the hypothesized phase transformation cone.
In conclusion, the existence of a phase transformation cone can be proved in multi-
directional shearing conditions with the testing data in this study. Inside the cone, sand
exhibits contractive tendencies, and both loading and unloading in terms of total shear stress
results in the increase of excess pore pressure. Outside the cone, by contrast, sand shows
dilative tendencies with further loading and excess pore pressure turns to decrease with
the increase of total shear stress; in unloading period, however, the dilation-like behaviour
is inhibited and excess pore pressure increases again. The analysis of undrained multi-
directional testing results should take both failure cone and phase transformation cone
into consideration so that a more comprehensive understanding of soil behaviour can be
developed.
6.2.2 The invalidity of phase transformation cone in bi-directional lin-
ear tests
Although the concept of phase transformation cone has been applied successfully in multi-
directional loading scenarios including circular, oval and figure-8 shaped loading paths, it is
6.2 Phase transformation in 3D τ−σ ′v space 155
(a) The variation of effective vertical stress of the points outside the hypothesised
phase-transformation cone
(b) The agreement between test data and the phase-transformation cone hypothesis
(for points inside the hypothesised PT cone, agreement is achieved if σ ′v decreases
with both loading and unloading; for points outside the PT cone, agreement is
achieved if σ ′v increases during loading or decreases during unloading)
Fig. 6.11 The verification of phase-transformation cone hypothesis with 3D stress path of
test C20-X20-1, with the outer cone indicating failure cone and the inner one representing
the hypothesised PT cone
156 THREE-DIMENSIONAL τ−σ ′v SPACE
(a) The variation of effective vertical stress of the points outside the hypothesised
phase-transformation cone
(b) The agreement between test data and the phase-transformation cone hypothesis
(for points inside the hypothesised PT cone, agreement is achieved if σ ′v decreases
with both loading and unloading; for points outside the PT cone, agreement is
achieved if σ ′v increases during loading or decreases during unloading)
Fig. 6.12 The verification of phase-transformation cone with 3D stress path of test E2010-1,
with the outer cone indicating failure cone and the inner one representing the hypothesised
PT cone
6.2 Phase transformation in 3D τ−σ ′v space 157
(a) The variation of effective vertical stress of the points outside the hypothesised
phase-transformation cone
(b) The agreement between test data and the phase-transformation cone hypothesis
(for points inside the hypothesised PT cone, agreement is achieved if σ ′v decreases
with both loading and unloading; for points outside the PT cone, agreement is
achieved if σ ′v increases during loading or decreases during unloading)
Fig. 6.13 The verification of phase-transformation cone with 3D stress path of test F8-3, with
the outer cone indicating failure cone and the inner one representing the hypothesised PT
cone
158 THREE-DIMENSIONAL τ−σ ′v SPACE
still imprudent to take it for granted that such a cone must exist in bi-directional linear tests.
In fact, a careful examination of bi-directional linear tests brings different results.
As elucidated in the preceding section, the 3D stress path of a bi-directional linear test
is contained in a plane whose distance from σ ′v axis or the axis of cone equals to static
shear stress. The 3D stress paths can thus be analysed in that plane through scrutinizing
strike-direction cyclic shearing. The intersection between this plane and a cone gives a
hyperbolic conic section. The conic section produced by the failure cone being intersected
forms the failure envelope on this plane, which can be derived from the geometry as equation
6.4. Similarly, should a phase transformation cone exist in bi-directional linear tests, a
hyperbolic phase transformation envelope could be delineated by:
tan2(φPT )
σ ′2v
τ2static
− τ
2
cyclic
τ2static
= 1 (6.9)
If a stress path escapes from this hyperbola and enters the space between it and the failure
envelope, sand is expected to exhibit dilative tendencies with increase of shear stress resulting
in the decrease of excess pore pressure.
The strike-direction stress paths of test Bilinear-6, Bilinear-7 and Bilinear-8, which were
shown in Figure 5.5a, are replotted in Figure 6.14. The failure envelopes of the tests are
calculated with measured minimum static shear stress in dip direction and are represented
by solid hyperbolas, while the hypothesized phase transformation envelopes are indicated
by dashed ones. Minimum static shear stress is used as the representative value in the
calculation to provide the largest hyperbolic conic section and account for the influence of
the fluctuations of static shearing. The hyperbolas are differentiated by grayscale. The higher
the number in test name, the darker the hyperbolas.
Since the three tests have the same target value of α , their stress-path planes are roughly
identical, as are their failure envelopes. The success of the failure envelopes in enclosing the
stress paths validates again the existence of a failure cone.
The assumed phase transformation envelopes, however, fail to envelop the turning points
on the stress paths. In test Bilinear-6 (blue curve), for instance, the loading period in negative
SSRstrike side clearly reduces the effective vertical stress, which is the typical soil behaviour
6.2 Phase transformation in 3D τ−σ ′v space 159
10 60 70 80 90 100
v
' (kPa)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R s
tr
ik
e
Bilinear-6, D
r
=30.3%, CSR=0.2, =0.15
Bilinear-7, D
r
=49.9%, CSR=0.15, =0.15
Bilinear-8, D
r
=61.1%, CSR=0.3, =0.15
Fig. 6.14 The strike-direction stress paths of test Bilinear-6, Bilinear-7 and Bilinear-8, with
hyperbolic failure envelopes and hypothesized hyperbolic phase-transformation envelopes
(the hyperbolas are differentiated by grayscale: the higher the number in test name, the darker
the hyperbola)
before the phase transformation is reached. Similar phenomenon is also observed in the other
two tests. The significant inconsistency impairs the effectiveness of phase transformation
cone, and the application of this concept in bi-directional linear tests becomes dubious.
The results of test Bilinear-4, Bilinear-5 and Bilinear-13 are presented in a similar manner
in Figure 6.15. The static shear stress ratios of the three are 0.075, 0.12 and 0.3, respectively.
The locations of the planes containing the stress paths of these tests are thus different, but such
a variation does not alter the observation that the phase transformation cone hypothesis cannot
be applied successfully in bi-directional linear tests. There is no recognised relationship
between the turning points on the stress paths and the hyperbolas.
The stress path of an example bi-directional linear test (ms50cyck) from Kammerer (2002)
are modified and presented in Figure 6.16. The values of tanφFL and tanφPT are determined
160 THREE-DIMENSIONAL τ−σ ′v SPACE
10 60 70 80 90 100
v
' (kPa)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
SS
R s
tr
ik
e
Bilinear-4, D
r
=55.7%, CSR=0.3, =0.075
Bilinear-5, D
r
=61.2%, CSR=0.3, =0.12
Bilinear-13, D
r
=59.9%, CSR=0.3, =0.3
Fig. 6.15 The strike-direction stress paths of test Bilinear-4, Bilinear-5 and Bilinear-13, with
hyperbolic failure envelopes and hypothesized hyperbolic phase-transformation envelopes
(the hyperbolas are differentiated by grayscale: the higher the number in test name, the darker
the hyperbola)
from the uni-directional tests conducted by Wu (2002), at 0.59 and 0.248, respectively. The
failure envelopes of this test do not coincide perfectly with the predicted hyperbola, but
this is acceptable considering Kammerer used different sample preparation techniques and
data processing methods. Like test Bilinear-13 in this study, test ms50cyck of Kammerer
(2002) also holds a large α that prohibits the development of high excess pore pressure.
The tendency of stress path stacks eventually along a hyperbolic boundary demonstrates
qualitatively that a failure cone must exist. As for phase transformation states, by contrast,
it is evident that the dashed PT hyperbola cannot locate the stress-state points where phase
transformation takes place.
6.2 Phase transformation in 3D τ−σ ′v space 161
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Normalised effective vertical stress
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
SS
R
st
rik
e
Kammerer (2002)
Test ms50cyck
Bi-directional linear test
Dr=59%
Static SSR=0.24
Fig. 6.16 The stress path of bi-directional linear test ms50cyck from Kammerer (2002),
modified to include hyperbolic failure envelope (solid), hypothesized phase transformation
hyperbola (dashed) and phase transformation lines (dashed).
In conclusion, test results suggest that the concept of phase transformation which was
applied successfully in multi-directional tests becomes ineffective under bi-directional linear
loading scenarios.
6.2.3 Discussions on the phase transformation in multi-directional load-
ing conditions
The examination of the test results in this study revealed a bimodal feature of the phase
transformation surfaces in 3D τ−σ ′v. In multi-directional loading condition, phase trans-
formation stress states form a conical surface that is analogous to the failure cone. Under
uni-directional or bi-directional linear loading scenarios, however, the hypothesis of phase
transformation cone is not valid.
162 THREE-DIMENSIONAL τ−σ ′v SPACE
Unfortunately, there has not been a model that can explain this phenomenon and match
existing test data. But the observations in this study still has some implications worthy of
discussions.
Firstly, it is confirmed that the multi-directionality of loading paths can affect the phase
transformation of sands. Conventional analysis methods that ignore the geometry of the
loading paths cannot provide thorough information about the response of soil under multi-
directional loading conditions. The interpretation of undrained soil behaviour should be
conducted with loading paths taken into consideration to avoid deceptive or misleading
conclusions.
Secondly, the bimodal phenomenon of phase transformation raises more questions about
the mechanisms of phase transformation. On the one hand, the existence phase transformation
cone in the circular, oval and figure-8 tests suggests that phase transformation is comparable
to failure in the sense that they both hold the characteristics of friction. On the other hand,
differently from the failure surface that is always a cone, phase transformation surface (or
surfaces) cannot be determined unless the loading path is known. Phase transformation
appears to be a type of soil behaviour that is similar to friction but not identical to it.
While the mechanics of phase transformation remains unclear, a potential explanation
is proposed here based on the observations from this study. Sands are a type of heteroge-
neous granular materials that exhibit pronounced elasto-plastic behaviours. Huge plastic
deformation can happen to accommodate the stress applied to it. But frictional behaviour is
not enabled unless localized deformation happens, that is, shear bands appear in the profile
of soil. Failure happens when the shear bands in soil develop to the state that a soil profile
is partitioned by it, with which unlimited deformation is allowed to accumulate. In drained
tests, the formation of shear bands results in localized increase of volume. In undrained tests,
it results in the dilative tendencies of phase-transformed soil behaviour. The formation of
shear bands is not immediate, instead, it takes time. If the end of this process is failure, the
start is likely to be phase transformation.
There is no intention to investigate in depth the details of shear bands or the mechanisms
of phase transformation in this work. But the existence of shear bands reminds us to reassess
6.2 Phase transformation in 3D τ−σ ′v space 163
the situation when loading is applied in a multi-directional manner. It is likely that shear
bands will not be plenary in multi-directional loading conditions, which, if verified, may
throw light on the discrepancies observed in this study. Further exploration can be conducted
to advance the understanding of phase transformation, especially under multi-directional
loading scenarios.

Chapter 7
THE EFFECTS OF
MULTI-DIRECTIONAL SHEARING
ON SOIL LIQUEFACTION
Multi-directional simple shear tests differ from uni-directional tests in various aspects, among
which one of the most prominent is seen in the pattern of the development of shear stress
and shear strain. The multi-directionality of loading paths clearly poses challenges to the
definition of shear stress and shear strain.
Shear stress and strain defined in a certain direction (denoted by subscript dir as in τdir
and γdir in this study) can help in examining the soil behaviour in a certain direction or
at a certain time point on a loading sequence. But the cost is that the information about
other directions or time points cannot be extracted and a comprehensive perspective is lost.
Although resultant shear stress and strain (represented by subscript total as in τtotal and γtotal)
takes directionality into account and provides a big picture of soil behaviour, it cannot give
information about how shear stress and strain are distributed physically.
The issues regarding the definition of shear stress and strain impairs significantly the effec-
tiveness of the conventional liquefaction analysis framework that has been widely accepted in
liquefaction research with uni-directional simple shear tests. The liquefaction criterion based
on double-amplitude shear strain needs modifications to account for the multi-directionality
166 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
of shear strain. Liquefaction resistance estimated from NL−CSR (number of cycles to
liquefaction versus cyclic shear stress ratio) curves becomes problematic, since cyclic shear
stress ratio cannot be defined consistently due to the existence of multi-directionality. The
pronounced non-coaxiality between shear stress and shear strain makes the problem even
more complex. Shear stress-strain hysteresis loops cannot be plotted conveniently because
the stress-strain relationships now become four-dimensional and, correspondingly, shear
modulus determined from stress-strain curves is no longer effective, which weakens the
effectiveness of the the models based exlusively on shear modulus.
To the best of the author’s knowledge, there have not been expedient analysis tools
or stress and strain definitions that can extrapolate the uni-directional analysis framework
to multi-directional scenarios in a compatible manner, partly because the data of multi-
directional tests are still too scarce to develop such tools and also partly because the effects
of multi-directional loading has not been extensively explored. With increasing efforts being
made to investigate the effects of multi-directional loading in recent years, these problems are
gradually becoming more prominent. Therefore, a discussion about these issues is covered in
this chapter with the objective of summarizing the insights from this study and providing
suggestions for research in the future.
7.1 Liquefaction criterion for multi-directional simple shear
tests
In chapter 4 and Chapter 5 where the test results in this study were presented, the occurrence
of liquefaction was defined as shear stress path in any direction reaching the failure line.
With the concept of failure cone introduced later, this criterion can be rephrased such that
liquefaction takes place when the conical failure surface is reached by multi-directional
shear stress paths. Such a liquefaction criterion is effective in analysing the soil behaviour
in stress space since it defines liquefaction directly as the failure envelope being reached.
But in practice, it may be more convenient to relate the implicit failure to other measurable
7.1 Liquefaction criterion for multi-directional simple shear tests 167
parameters such as excess pore pressure and shear strain, as conducted in the two most widely
accepted categories of liquefaction criterion.
The excess pore pressure ratio based liquefaction criterion defines the occurrence of
liquefaction as excess pore pressure ratio reaching unity. This criterion is extensively adopted
in liquefaction-related laboratory research including both element-level tests and physical
modelling tests. However, the test results of this study indicate that the application of such a
criterion is not as straightforward as expected. The maximum excess pore pressure is affected
by the multi-directional loading conditions because the conical failure surface constrains the
ultimate development of shear stress paths. An excess pore pressure ratio of unity is thus not
achievable in all categories of loading paths. In fact, whether this criterion is applicable in
uni-directional simple shear tests is also questionable even though it has been widely used
in existing research. If a testing apparatus cannot provide perfect loading control and have
shear stress in other directions, which is rather typical or just inevitable when the stiffness
of a sand sample is reduced dramatically, the loading scenario becomes bi-directional or
multi-directional and the maximum excess pore pressure ratio should not reach unity.
Since the maximum excess pore pressure ratio under a loading scenario can be estimated
from the failure cone, a potential modification of the ru-based liquefaction criterion is to
define liquefaction as excess pore pressure ratio reaching certain theoretical values. When
the maximum excess pore pressure ratio reaches the value given by equation 5.4, liquefaction
can be considered as occurring. However, the existence of static shearing also influences the
maximum excess pore pressure ratio generated. When static shearing is so high that stress
reversal is eliminated, maximum excess pore pressure can fall below the theoretical values,
with the mechanism still unclear. Therefore, the ru-based liquefaction criterion should be
addressed separately when a high level of static shearing is applied.
Alternatively, liquefaction can be correlated to a certain level of shear strain, as discussed
in Chapter 3. The strain-based liquefaction criterion also needs reassessment under multi-
directional loading scenarios because shear strain can be defined now in different ways. In the
first instance, the development of total shear strain γtotal with the number of cycles is presented
in a semilogarithmic manner in Figure 7.1. The γtotal here refers to the maximum total shear
168 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
strain in a cycle. In uni-directional tests, γtotal becomes equivalent to the maximum absolute
amount of shear strain achieved in a cycle. The blue dashed line represents γtotal = 3%, the
liquefaction criterion proposed by Ishihara and Yamazaki (1980). When γtotal exceeds 3%,
39 of the 75 cyclic tests reach liquefaction, as seen in most of the curves turning to proceed
vertically after passing through the blue dashed line. One test (F8-7) liquefies before γtotal
grows to 3%, but it is counted in the 39 tests because it is reasonable to expect γtotal in this
test to reach 3% with one more shearing cycle. The other 36 tests, however, cannot liquefy
with γtotal = 3%. The four dash-dotted tests need a large amount of shearing cycles to initiate
liquefaction after total shear strain exceeds 3%. Test LS6 needs up to 45 more cycles to
trigger liquefaction. It demonstrates the criterion of Ishihara and Yamazaki (1980) is not able
to envelope the liquefaction failure in all the tests in this study, and remarkable discrepancy
in terms of liquefaction resistance can arise from it.
An attempt to modify this criterion can be made through adjusting the total shear strain
level. The red dashed line in Figure 7.1 examples a modified criterion requiring γtotal > 6%.
Although the increase in required strain level makes it applicable to more tests (61 out of
75), there are still 14 tests that cannot reach liquefaction under this criterion, with the largest
discrepancy in terms of NL shrinking to 3 cycles. It appears that γtotal based criterion is
not particularly effective, if at all, in determining the occurrence of liquefaction. The tests
which do not fit the criterion are mainly those involving initial static shearing. Recalling
the results of test LS7, LS10 and C20-X20-1 presented in Figure 4.12, 4.16 and 5.18,
considerable shear strain (far more than 3%) can accumulate towards the direction of initial
static shearing without the stress path reaching the failure line. There is clearly a possibility
that the contribution of static-shear induced shear strain to γtotal is so large that a γtotal-based
liquefaction criterion is severely biased and misleading. Thus, liquefaction criterion is not
recommended to be established on γtotal .
To overcome the incompatibility associated with initial static shearing, a double amplitude
strain level can be used conveniently in uni-directional tests, but whether the concept of
double amplitude is applicable to multi-directional tests remains unclear. Two types of double
amplitude shear strain are constructed in this study and examined: γtotal,DA and γdir,DA.
7.1 Liquefaction criterion for multi-directional simple shear tests 169
100 101 102
Cycle number N
0
5
10
15
20
25
to
ta
l (%
)
Uni-directional linear loading paths
Bi-directional linear loading paths
Circular loading paths
Oval loading paths
Figure-8 loading paths
Fig. 7.1 The development of γtotal with cycle number in all tests in this study
γtotal,DA is the difference between the maximum γtotal and minimum γtotal in a cycle. It is
termed “double amplitude total shear strain” because it adopts a calculation method similar
to γDA in uni-directional tests, though it does not necessarily refer to the actual “double
amplitudes” of total shear strain. γtotal,DA is not equal to the increase of γtotal between cycles,
and the contribution of the strain components induced by static shearing and cyclic shearing
to γtotal cannot be distinguished. Qualitatively, when the excess pore pressure is relatively
low, both the static-shear induced strain increment and cyclic-shear induced strain increment
are roughly constant in a cycle, and γtotal,DA is supposed to remain small and change little.
When liquefaction is approached, static-shear induced strain and the cyclic-shear induced
strain will both increase rapidly, resulting in a swift growth of γtotal,DA. Thus, a potential
liquefaction criterion can be built based on γtotal,DA.
170 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
Figure 7.2 presents the development of γtotal,DA with cycle number for all the tests
conducted in this study. A red dashed line is plotted to represent γtotal,DA = 3%. The strain
level of 3% is selected to make this criterion equivalent to single amplitude shear strain of
3% in uni-directional loading scenarios. 7 tests liquefy before γtotal,DA reaches 3% but are
expected to meet the criterion with one more shearing cycle. 19 tests cannot liquefy when
γtotal,DA exceeds 3%, with the largest discrepancy in NL at 3 cycles (test LS2 and LS4). The
deviation is acceptable considering the numbers of cycles to liquefaction of test LS2 and LS4
are 222 and 74. γtotal,DA in 9 tests exceeds 3% from the initial loading cycle due to overlarge
initial static shearing, but liquefaction occurs in a few cycles later (up to 4 more cycles in test
LS10). Extra efforts are required to cope with the situation where γtotal,DA reaches 3% in the
initial shearing cycle.
Generally, the performance of γtotal,DA = 3% criterion in enveloping liquefaction is satis-
factory, but two pronounced drawbacks still remain. Firstly, the physical and mathematical
meaning of γtotal,DA is not straightforward and the relationship between it and the abrupt
increase of shear strain is not explicit, from which remarkable inconsistency in liquefaction
resistance assessment can arise. Secondly, the calculation of total shear strain can be altered
if the zero point of strain is changed. Shear strain measurements are always zeroed before
cyclic loading starts in this study, but a ramp loading stage is conducted prior to cyclic
loading if cyclic shear stress does not start from zero such as in circular and oval tests. If zero
strain is assigned to the point before the ramp loading is applied, γtotal,DA-based liquefaction
criterion is likely to give different results. The consistency of applying such a criterion in
determining liquefaction resistance is thus not guaranteed.
A convenient method to avoid the influences of the alteration of the zero strain point
is to define a strain parameter in a relative manner and γdir,DA is thus proposed to achieve
this objective. Subscript dir is used to differentiate it from γtotal,DA and double amplitude
uni-directional shear strain γDA. γdir,DA amounts to the maximum relative strain between any
two strain points within a cycle. Assuming two points γ i and γ j on a strain path in a loading
7.1 Liquefaction criterion for multi-directional simple shear tests 171
100 101 102
Cycle number N
0
2
4
6
8
10
12
14
16
to
ta
l,D
A
 
(%
)
Uni-directional linear loading paths
Bi-directional linear loading paths
Circular loading paths
Oval loading paths
Figure-8 loading paths
Fig. 7.2 The development of γtotal,DA with cycle number in all tests in this study
cycle that contains N points, γdir,DA can be defined as:
γdir,DA =
N
max
i, j=1
√
(γ iX − γ jX)2 +(γ iY − γ jY )2 (7.1)
where X and Y represent two arbitrarily assigned directions that have to be orthogonal, γ iX
and γ iY represent the projections of the shear strain of ith point in X and Y direction.
Figure 7.3 presents the γdir,DA in each loading cycle in all the tests conducted in this
study. The red dashed line gives γdir,DA = 6% which, in uni-directional tests, is equivalent
to the criterion requiring γDA = 6%. Seven out of the 75 tests reach liquefaction before the
red dashed line is passed through, but γdir,DA in these tests is expected to exceed 6% with
one more cycle since their curves have become almost vertical at liquefaction, suggesting
a rapid increase in shear strain after that. Ten tests cannot liquefy when γdir,DA = 6% is
172 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
satisfied; however, liquefaction is initiated with one or two more cycles in 9 of them. It can
be concluded that γdir,DA = 6% is effective in enveloping liquefaction in these tests with only
limited inaccuracy in terms of NL. Compared with the preceding two criteria, the performance
of this one is evidently better.
The application of such a criterion in three tests, nevertheless, is problematic. Test LS10
achieves a γdir,DA of 7.8% in the first cycle but liquefaction happens 4 cycles later. γdir,DA in
test Bilinear-11 and Bilinear-13 cannot reach 6% and does not increase rapidly at occurrence
of liquefaction as other tests. However, the three tests share the common characteristics of
γdir,DA decreasing significantly in the second loading cycle. Their first cycle witnesses the
accumulation of huge amount of shear strain due to the presence of initial static shearing,
but the development of shear strain in the next cycle becomes sluggish, as shown in Figure
4.16 for test LS10. Since such a pattern of shear strain accumulation threatens the use of
the γdir,DA-based liquefaction criterion, it should be addressed separately if the criterion is
applied. However, the effectiveness of this criterion is at least not worse than the widely-
accepted uni-directional liquefaction criterion γDA = 6%, given the fact that liquefaction in
uni-directional test LS10 cannot be captured by γDA = 6% either.
The comparison of the three examined strain-based liquefaction criteria in this section
can be summarised as follows:
(1) γtotal = 3% is the least effective liquefaction criterion. Almost half of the tests in this
study do not liquefy when this criterion is satisfied. Increasing the required level of
γtotal can improve the performance of γtotal based criterion, but there is no convincing
evidence that a certain level can be selected consistently to make this criterion applicable
to all categories of multi-directional tests. Furthermore, the alteration of zero-strain point
changes the calculation of γtotal , which aggravates the inconsistency of this criterion.
(2) Single amplitude shear strain of 3% in uni-directional tests is extended to γtotal,DA = 3%
for multi-directional tests. This criterion has a better ability in enveloping liquefaction,
but the application of it is constrained critically by its lack of physical and mathemat-
ical meaning as well as the issue regarding the definition of zero-strain point for the
calculation of γtotal .
7.1 Liquefaction criterion for multi-directional simple shear tests 173
100 101 102
Cycle number N
0
5
10
15
20
25
30
di
r,
D
A
 
(%
)
Uni-directional linear loading paths
Bi-directional linear loading paths
Circular loading paths
Oval loading paths
Figure-8 loading paths
Fig. 7.3 The development of γdir,DA with cycle number in all tests in this study
(3) Double amplitude shear strain of 6% γdir,DA = 6% in uni-directional tests is modified as
γdir,DA = 6% to be applied in multi-directional loading scenarios. γdir,DA is defined in a
relative manner as the maximum distance between two points on a multi-directional strain
path and is thus not altered by varying definition of zero-strain point. This criterion is
the most effective one among the three criteria, but the situations where γdir,DA decreases
need to be considered separately.
In summary, liquefaction criteria hinged on either excess pore pressure ratio or shear
strain need modifications before being used for multi-directional loading conditions. The
maximum excess pore pressure predicted by equation 5.4 can be used as the modified
ru-based liquefaction criterion. γdir,DA = 6% is recommended as the strain-based criterion
against which liquefaction resistance can be investigated and compared consistently. It should
174 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
be noted, however, that the essence of liquefaction rests virtually in the stress path reaching
the failure envelope. Thus, failure defined from stress-state perspective should be given
priority and considered as the liquefaction criterion whenever possible, before resorting to
ru-based or strain-based criteria.
7.2 The effects of multi-directional loading on liquefaction
resistance
Although the research regarding soil liquefaction has now a 50-year history,a reliable and
effective method to fully mitigate the effects of liquefaction on structures has not been
developed. The assessment of liquefaction resistance is of vital importance in engineering
practice to minimize or even eliminate the hazard of soil liquefaction.
Liquefaction assessment is typically based on case histories correlated to in situ tests.
Seed and Harder (1990) correlated standardised SPT blowcount (N)60 to the equivalent
uniform uni-directional cyclic shear stress ratio CSR f ield required to trigger liquefaction in
the soils consolidated under standard vertical stress (usually 1 ts f or 100 kPa) and sheared by
an earthquake with a magnitude M = 7.5. CSR f ield also needs to be standardised as follows:
CSR f ield =CSR ·CM ·Kσ ·Kα (7.2)
where CM is the coefficient to account for the magnitude of earthquakes, Kσ represents the
coefficient of consolidation stress, and Kα is the coefficient of static shearing. The design
codes nowadays generally adopt his recommendation with adjustments.
Research in the last a few decades demonstrated the effects of irregular loading and
multi-directional loading should also be accounted for, but no consistent conclusions could
be drawn regarding how to take them into consideration. The influence of irregular loading
is out of the scope of this study and will not be discussed here. This section presents insights
into the problem of the effects of multi-directional loading.
7.2 The effects of multi-directional loading on liquefaction resistance 175
An unavoidable issue in evaluating the effects of multi-directional loading on liquefaction
resistance is to select a parameter against which different types of loading paths can be
compared or standardised. If the tests are compared, for instance, based on cyclic shear
stress ratio in certain directions, the differences in terms of the stress states in 3D τ −σ ′v
space cannot be reflected. If the tests are compared by total shear stress ratio, however, the
effects of the geometric characteristics of the loading paths are not accounted for. A dilemma
arises from the trade-offs among different types of parameters, and unfortunately, a perfect
parameter that retains all crucial information of multi-directional loading paths has not been
found.
Following the prevalent liquefaction assessment method proposed by Seed and Harder
(1990), however, liquefaction resistance can be indicated by shear stress ratio versus the
number of cycles to initiate liquefaction, that is, NL−CSR curves. CSR is replaced by CSRdir
in mutli-directional test, which refers to the maximum directional cyclic shear stress ratio
throughout all directions, following:
τdir,DA =
N
max
i, j=1
√
(τ iX − τ jX)2 +(τ iY − τ jY )2 (7.3)
CSRdir =
τdir,DA
2σv,i
(7.4)
Figure 7.4 presents the liquefaction resistance of different categories of tests in the form
of NL−CSRdir. The curves in the figure are the ones fitted for uni-directional tests without
initial static shearing, which are the same as those in Figure 4.17. With the relative density
and the number of cycles to liquefaction of a test, an equivalent uni-directional CSRequivalent
can be determined on the curves fitted to uni-directional test data, and a coefficient of
multi-directional loading paths, denoted by Kµ in this study, can be obtained as:
Kµ =
CSR
CSRequivalent
(7.5)
Figure 7.5 plots Kµ versus CSRdir for the tests having no initial static shearing in this
study (derived from plot (c),(e) and (f) in Figure 7.4), together with the data presented by
176 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
1
10
10
0
0
0.
1
0.
2
0.
3
(a)
D
r=
50
%
D
r=
55
%
D
r=
60
%
D
r=
65
%
1
10
10
0 (b)
1
10
10
0
00.
1
0.
2
0.
3
(c)
1
10
10
0
0
0.
1
0.
2
0.
3
(d)
D
r=
50
%
D
r=
55
%
D
r=
60
%
D
r=
65
%
1
10
10
0(e)
1
10
10
0
00.
1
0.
2
0.
3
(f)
Maximum single-direction cyclic shear stress ratio CSR
dir
N
um
be
r o
f c
yc
le
s t
o 
liq
ue
fa
ct
io
n 
N
L
Fi
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7.
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fa
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io
n
re
si
st
an
ce
re
fle
ct
ed
in
th
e
fo
rm
of
N
L
−
C
SR
di
r:
(a
)U
ni
-d
ir
ec
tio
na
l
te
st
s
w
ith
st
at
ic
sh
ea
ri
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;
(b
)B
i-
di
re
ct
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na
l
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st
s;
(c
)C
ir
cu
la
rt
es
ts
w
ith
ou
ts
ta
tic
sh
ea
ri
ng
;(
d)
C
ir
cu
la
rt
es
ts
w
ith
st
at
ic
sh
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ri
ng
;(
e)
O
va
lt
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ts
;(
f)
Fi
gu
re
-8
te
st
s
7.2 The effects of multi-directional loading on liquefaction resistance 177
Kammerer (2002). Data points in this study are colour-filled, with the colours indicating
relative density, while the data from Kammerer (2002) are hollow. The categories of loading
paths are suggested by the marker types. Linear interpolation is conducted in-between
NL−CSR curves to account for the deviation of actual relative density from the nominal
densities.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
CSRdir
0
0.5
1
1.5
K
Dr=50%
Dr=55%
Dr=60%
Dr=65%
Circular
Oval
Figure-8
Fig. 7.5 The coefficient of multi-directional loading Kµ versus CSRdir
Kµ obtained in this study ranges from 0.57 to 1.19, with the majority below 1.0. It
suggests that from the point of view of required CSRdir to trigger liquefaction, the effects of
multi-directional loading on liquefaction resistance can be either positive or negative, but
negative in most cases. In extreme situations, CSRdir can be reduced by up to 43%. The
largest reduction of required CSRdir in multi-directional loading scenarios reported previously
was 30% by Ishihara and Yamazaki (1980), 50% by Reddy and Saxena (1992), 40% by
Kammerer (2002) and 33% by Mirbaha (2017). The smallest Kµ of 0.57 in this study is in
agreement with their findings. The scatter in the graph are remarkable but inevitable because
178 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
both the numerator and denominator in the calculation of Kµ are affected by experimental
uncertainties.
The fact that Kµ is below unity in most cases, regardless of relative density, suggests
that ignoring the reduction of liquefaction resistance in multi-directional loading condition
is un-conservative. Furthermore, there is a general tendency of Kµ to decrease more in
the tests with higher relative density. Dense sands, which are conventionally considered
resistant to liquefaction in uni-directional tests, may experience more dramatic reduction in
liquefaction resistance when it comes to multi-directional loading. Caution should thus be
taken in assessing the liquefaction resistance of dense sands.
Another observation from Figure 7.5 is that circular loading paths tend to result in lower
Kµ if CSRdir and relative density are around the same, compared with oval and figure-
8 loading paths. The effects of varying categories of loading paths have been studied.
Ishihara and Yamazaki (1980) reported that the most significant reduction of CSRdir occurs
when the amplitudes of the loading in two perpendicular directions are equal. But little
advancement has been seen in terms of the parametrization of the differences among loading
paths. Kammerer (2002) proposed to use aperture ratio (AR = CSRperpCSRdir ) and reversal parameter
(RP = CSRα−αCSRdir ) to indicate the geometrical characteristics of a loading path (illustrated in
section 3.7). The higher the aperture ratio, the lower the Kµ . Circular paths have the highest
aperture ratio and thus the lowest Kµ .
Similar trends are also observed in this study, but the effects of aperture ratio on Kµ
recorded in this study are not as pronounced as Kammerer suggested. Figure 7.6 presents the
relationship between Kµ and aperture ratio for the tests having reversal parameters of 0 and
0.5. The five figure-8 tests (blue diamond) in Figure 7.6(a) have the same relative density,
CSRdir and reversal parameter, but the decrease of Kµ with increasing AR in these tests is
only marginal.
The effects of reversal parameter on Kµ are also examined. Figure 7.7 shows the re-
lationship between Kµ versus absolute reversal parameter |RP| for the tests which have
aperture ratios of 0 and 1. |RP| = 0 means static shearing is not involved, while higher
values of |RP| indicate generally larger static shearing. The tests shown with AR = 0 are
7.2 The effects of multi-directional loading on liquefaction resistance 179
Aperture Ratio AR
0
0.5
1
1.5
K
(a) RP=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Aperture Ratio AR
0
0.5
1
1.5
K
(b) RP=0.5
Uni-directional linear
Circular
Oval
Figure-8
Dr=50%
Dr=55%
Dr=60%
Dr=65%
Fig. 7.6 The coefficient of multi-directional loading Kµ versus aperture ratio AR for tests
with reversal parameter at 0 and 0.5
uni-directional linear tests with static shearing and bi-directional linear tests. For the former,
the coefficient to account for the effects of loading paths is equivalent to the coefficient of
initial static shearing Kα for uni-directional tests as presented in Figure 4.18. As discussed
in section 4.2, the relationships between Kα and static shear stress ratio α are complex and
controversial since factors including relative density, stress reversal, and maximum shear
stress are all claimed to exert an influence. It is inferred that similar complexity should exist
in multi-directional tests.
Uni-directional and bi-directional tests exhibit the tendency of Kµ to grow with increasing
RP in each density group when |RP| is smaller than 0.5. That is, a moderate increase of
downhill static shearing may help increase liquefaction resistance. But the trend is reversed
with further growth of |RP|. Large static shearing can result in over-large total shear stress,
180 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
in which case liquefaction can happen in the first one or two loading cycles. However, such a
tendency is not seen in circular, oval and figure-8 tests. As seen in Figure 7.7(b), Kµ remains
relatively constant in these test regardless of |RP|.
Absolute reversal parameter |RP|
0
0.5
1
1.5
K
 
o
r 
K
(a) AR=0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Absolute reversal parameter |RP|
0
0.5
1
1.5
K
(b) AR=1
Uni-directional linear
Bi-directional linear
Circular
Oval
Figure-8
Dr=50%
Dr=55%
Dr=60%
Dr=65%
Fig. 7.7 The coefficient of multi-directional loading Kµ versus reversal parameter RP for
tests with aperture ratio at 0 and 1
The inconsistency and scatter in the relationships between Kµ and |RP| as well as AR
suggests that reversal parameter and aperture ratio may not be able to fully capture the features
of a loading path with static shearing. But the philosophy contained in this pioneering attempt
to characterise loading paths raises thought-provoking questions.
Firstly, should the influence of static shearing be decoupled from that of multi-directional,
or, is it possible? The conventional liquefaction assessment method following Seed and
Harder (1990) uses coefficient Kα to involve the influence of static shearing on liquefaction
resistance. Kα can be obtained conveniently in uni-directional tests because the reduction of
7.2 The effects of multi-directional loading on liquefaction resistance 181
CSR is caused exclusively by static shear. In multi-directional loading conditions, the reduc-
tion of CSR results from the combined effect of static shearing and the multi-directionality of
loading paths. If the effects of either of the two were unambiguous, they could be decoupled
and described by two coefficients; however, there is no consensus yet.
Furthermore, from the perspective of total shear stress and geometry, there is a possibility
that the existence of static shearing can be included in more fundamental characteristics
of loading paths. It is likely that Kα is just a “particular solution” of Kµ in uni-directional
loading condition, and that the controversy in terms of Kα is a consequence of the lack
of understanding of multi-directional loading. The aperture ratio and reversal parameter
proposed by Kammerer can be regarded as an example of the attempt towards a generalised
way of characterising loading paths. More effective parameters are needed in the future to
explore the essence of multi-directional loading.
Secondly, is it reasonable to correlate N to CSR and are there alternatives to it? The
convenience of relating liquefaction resistance to cyclic shear stress ratio rests in that CSR
indicates effectively the amplitude of shearing in uni-directional tests without static shearing.
When it comes to multi-directional loading or uni-directional loading with static shear,
the effectiveness of CSR becomes questionable. On the one hand, CSR cannot reflect the
maximum shear stress applied if static shearing is involved. On the other hand, the definition
of CSR in multi-directional tests requires a direction to be designated and the representative
cyclic shear stress ratio CSRdir is the maximum CSR across all directions. A natural defect
of CSRdir is that the information of shearing on other directions is lost, which is what the
aperture ratio aims to complement. In this case, to correlate liquefaction resistance to CSRdir
is still reasonable, but not necessarily optimal.
In addition, cyclic shear stress ratio CSR is actually equivalent to maximum total shear
stress ratio SSRtotal,max in uni-directional tests without static shearing, that is, NL−CSR
curves and NL− SSRtotal,max coincide under such loading condition. The coefficients in-
cluding CM and Kσ in equation 7.2 will remain the same in SSRtotal,max based liquefaction
resistance. The resemblance suggests a possibility that SSRtotal,max may have been misinter-
182 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
1
10
10
0
0
0.
1
0.
2
0.
3
0.
4
0.
5
(a)
1
10
10
0 (b)
D
r=
50
%
D
r=
55
%
D
r=
60
%
D
r=
65
%
1
10
10
0
00.
1
0.
2
0.
3
0.
4
0.
5
(c)
1
10
10
0
0
0.
1
0.
2
0.
3
0.
4
0.
5
(d)
1
10
10
0(e)
D
r=
50
%
D
r=
55
%
D
r=
60
%
D
r=
65
%
1
10
10
0
00.
1
0.
2
0.
3
0.
4
0.
5
(f)
Maximum total shear stress ratio SSR
total,max
N
um
be
r o
f c
yc
le
s t
o 
liq
ue
fa
ct
io
n 
N
L
Fi
g.
7.
8
L
iq
ue
fa
ct
io
n
re
si
st
an
ce
re
fle
ct
ed
in
th
e
fo
rm
of
N
L
−
SS
R t
ot
al
,m
ax
:(
a)
U
ni
-d
ir
ec
tio
na
lt
es
ts
w
ith
st
at
ic
sh
ea
ri
ng
;(
b)
B
i-
di
re
ct
io
na
lt
es
ts
;
(c
)C
ir
cu
la
rt
es
ts
w
ith
ou
ts
ta
tic
sh
ea
ri
ng
;(
d)
C
ir
cu
la
rt
es
ts
w
ith
st
at
ic
sh
ea
ri
ng
;(
e)
O
va
lt
es
ts
;(
f)
Fi
gu
re
-8
te
st
s
7.2 The effects of multi-directional loading on liquefaction resistance 183
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
CSRdir
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
K
Uni-directional with static shearing
Bi-directionial
Circular
Circular with static shearing
Oval
Figure-8
Dr=50%
Dr=55%
Dr=60%
Dr=65%
(a) NL−CSR
0 0.1 0.2 0.3 0.4 0.5 0.6
SSR
total,max
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
K
Uni-directional with static shearing
Bi-directionial
Circular
Circular with static shearing
Oval
Figure-8
Dr=50%
Dr=55%
Dr=60%
Dr=65%
(b) NL−SSRtotal,max
Fig. 7.9 Comparison of Kµ derived from NL−CSR and NL−SSRtotal,max relationships
184 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
preted as CSR and NL−SSRtotal,max is a potential alternative to the NL−CSR relationship in
the assessment of liquefaction resistance.
Figure 7.8 presents the NL−SSRtotal,max relationship for the tests in this study. The curves
are the ones fitted to uni-directional level-ground tests. They are slightly different from the
NL−CSR curves in Figure 7.4 because a small amount of shear stress exists perpendicular to
cyclic shearing, resulting in a slight difference between CSR and SSRtotal,max. Remarkable
differences can be found in uni-directional tests with static shearing, bi-directional tests,
circular tests with static shearing and figure-8 tests (comparing plot (a),(b),(d) and (f) in
Figures 7.4 and 7.8) because the maximum total shear stress in these tests differ from CSRdir
due to static shear or the special geometry of loading paths (such as non-standard figure-8
paths).
Based on NL − SSRtotal,max curves, a set of coefficients of multi-directional loading
can be derived. Figure 7.9 compares the Kµ derived from NL−CSR and NL−SSRtotal,max
relationships. Tests with static shear are also presented and Kα is not decoupled from Kµ .
Different from CSR-based Kµ which is below unity in most situations, half of the tests have
SSRtotal,max-based Kµ larger than unity. This is particularly true for tests with static shearing
(points marked by hollow triangle, hollow circle and solid pentagon), which suggests that the
effects of static shearing are positive in these cases from the perspective of SSRtotal,max-based
liquefaction resistance. The circular and oval tests without static shearing, by contrast,
have generally unchanged Kµ , which in most situations is below unity. Furthermore, the
SSRtotal,max-based Kµ exhibits a stronger tendency to increase with SSRtotal,max in each
category of tests of similar density. This indicates the increase of SSRtotal,max due to variation
of loading paths can be beneficial in a counter-intuitive manner.
There is no intention in this study to demonstrate NL−SSRtotal,max is a more effective
method to assess liquefaction resistance. Multi-directional shearing has only been investi-
gated experimentally in element-level tests and test data are far from sufficient to determine
if a new liquefaction assessment method is compatible with the field. SSRtotal,max also has
drawbacks, such as the lack of information about directionality of loading. A series of
parameters analogous to aperture ratio and reversal parameter need to be proposed and work
7.3 The development of excess pore pressure in multi-directional simple shear tests 185
in a consistent manner with NL−SSRtotal,max if such a method is to be validated. However,
the thoughts about NL−SSRtotal,max suggest that there are potential alternatives to NL−CSR
for the assessment of liquefaction, especially in multi-directional loading scenarios. The
coefficient of static shearing Kα and multi-directional loading Kµ and even the potential
coefficient of loading irregularity is not reliable without more comprehensive demonstration.
The effect of multi-directional loading on liquefaction resistance is an area worthy of more
examination in the future so that more effective and accurate methods can be developed for
liquefaction assessment in engineering practice.
7.3 The development of excess pore pressure in multi-directional
simple shear tests
The development of excess pore pressure has attracted extensive attention in the field of
liquefaction research because the occurrence of liquefaction is caused by the generation of
positive excess pore pressure. Great efforts have been made towards correlating excess pore
pressure to liquefaction resistance, shear modulus and deformation based on experimental
and numerical studies. However, the understanding of how excess pore pressure develops in
multi-directional loading scenarios is far from extensive. Boulanger (1990) and Kammerer
(2002) observed an inverse relationship between limited excess pore pressure ratio and shear
stress ratio, but the mechanisms were not investigated in depth. Little discussions have
taken place on whether the conclusions regarding the development of excess pore pressure
in uni-directional tests are also applicable to multi-directional conditions. This section will
summarize the test results of this study and examine the influence of multi-directional loading
on excess pore pressure generation with the newly-developed test database and the newly-
established concept of failure cone. The findings from this study can provide a foundation
for modelling work in the future.
186 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
7.3.1 Maximum excess pore pressure ratios in multi-directional simple
shear tests
The concept of failure cone has been introduced in Chapter 6 and will be considered as the
actual failure envelope for multi-directional simple shear tests. Important inferences can be
drawn from the theory of the failure cone regarding the limits of excess pore pressure ratios.
When liquefaction is initiated, the shear stress path should reach and develop along
the failure line in τtotal −σ ′v space during loading. The point on the failure line, which
has minimum total shear stress ratio, also has the minimum effective vertical stress, or the
maximum excess pore pressure ratio. In contrast, the point that holds the maximum total
shear stress ratio sets the limit for the maximum effective vertical stress in a loading cycle.
Figure 7.10 shows the relationships between limiting excess pore pressure ratios and total
shear stress ratios of all the tests in this study. The solid lines represent the correlation
following equation 6.5.
As shown in Figure 7.10a, the measured maximum excess pore pressure ratios are
scattered around equation 6.5. The maximum pore pressure ratios in 16 of the tests are higher
than the predicted values mainly because the loading system cannot sustain constant vertical
stress after the stiffness of soil sample becomes extremely low and thus causes unanticipated
increase of excess pore pressure. The other tests, however, all have ru,max lower than the
predicted values, which is attributed to two primary reasons. Firstly, the effects of relative
density on the angle of the failure line are not taken into consideration here. The actual value
of φFL is likely to be lower than 33.4◦ so that ru,max reduces correspondingly. Secondly,
extremely low effective vertical stress is difficult to achieve in simple shear tests because the
large deformation of the soil sample increases the discrepancies between realistic boundary
conditions and the ideal ones. The phenomenon that effective vertical stress cannot approach
zero can be observed in Figures 4.10, 4.15, 5.18, and 5.31. One point (test LS11) deviates
remarkably from the prediction line because the lack of stress reversal results in a different
pattern of stress development in this test, as reported in Chapter 4 and presented in Figure
4.13. With the scatter clarified, a strong inverse linear correlation can be identified between
7.3 The development of excess pore pressure in multi-directional simple shear tests 187
0 0.1 0.2 0.3 0.4 0.5 0.6
Minimum total shear stress ratio SSR
total,min
0
0.2
0.4
0.6
0.8
1
M
ax
im
um
 e
xc
es
s p
or
e 
pr
es
su
re
 ra
tio
 r u
,
m
a
x
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Bi-directional linear tests
Circular tests without static shearing
Circular tests with static shearing
Oval tests
Figure-8 tests
(a) Inverse correlation between maximum excess pore pressure ratio and minimum
total shear stress ratio
0 0.1 0.2 0.3 0.4 0.5 0.6
Maximum total shear stress ratio SSR
total,max
0
0.2
0.4
0.6
0.8
1
La
st
 c
yc
le
 m
in
im
um
 e
xc
es
s p
or
e 
pr
es
su
re
 ra
tio
 r u
,
m
in
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Bi-directional linear tests
Circular tests without static shearing
Circular tests with static shearing
Oval tests
Figure-8 tests
(b) Inverse correlation between the largest minimum excess pore pressure ratio
and maximum total shear stress ratio
Fig. 7.10 Relationships between limiting excess pore pressure ratios and total shear stress
ratios
188 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
ru,max and minimum total shear stress, validating the theory of failure cone in enveloping
stress states under multi-directional loading conditions.
Figure 7.10b presents maximum total shear stress ratio versus the largest single-cycle
minimum excess pore pressure ratio.The largest single-cycle minimum excess pore pressure
ratio is denoted as rr,min and is determined as the minimum excess pore pressure ratio in
the loading phase of the loading cycle where liquefaction is triggered. If the failure cone
exists and the shear stress paths stay on the conical failure surface after reaching it during
loading, the stress point with the largest SSRtotal should give rr,min in this cycle. Clearly,
the data points in Figure 7.10b show a strong inverse correlation. The reason why most of
the points are located above the prediction line is total shear stress paths tend to deviate
slightly backward from the failure line at the end of the loading phase, as seen in Figures
5.18 and 5.31. In general, the strong correlation between ru,min and SSRtotal,max demonstrates
the failure cone is effective in predicting the critical values of excess pore pressure ratios.
The test results of Kammerer (2002) and the correlations proposed by Boulanger (1990),
as well as the prediction line from this study are presented in Figure 7.11, which is modified
from figure 5.60 of Kammerer (2002). The red dashed line is calculated following equation
6.5 with φFL = 30.5◦ (after the data of Wu (2002)). The relationship proposed in this study
is capable of correlating Kammerer’s data with limited scatter. In fact, the degree of scatter is
acceptable given that different methods were adopted in processing test data and the effects of
sample density on φFL were not taken into consideration. The linear-regression correlations
proposed by Kammerer (2002) were actually statistical approximation of equation 6.5.
Evidently, the existence of an inverse relationship between limited excess pore pres-
sure ratios and critical total shear stress ratios has significance in soil mechanics. Such a
relationship is not a statistical coincidence, instead, it is predicted by the failure cone that
constrains the stress states at failure in multi-directional simple shear tests. This correlation
is an important feature of multi-directional simple shear tests and an essential theoretical
basis for interpreting simple shear test data. The generation of excess pore pressure in multi-
directional loading scenarios should be understood in 3D τ−σ ′v space, whenever possible,
to avoid misleading results.
7.3 The development of excess pore pressure in multi-directional simple shear tests 189
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SSR
total,limit
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
r u
,
lim
it
Correlation based on failure cone 
 (proposed in this study)        
Fig. 7.11 Correlation between limiting excess pore pressure ratios and total shear stress ratios
(modified from Kammerer (2002))
7.3.2 Development of excess pore pressure in multi-directional simple
shear tests
While the existence of a failure cone constrains the largest single-cycle maximum and
minimum excess pore pressures that can be generated in multi-directional simple shear tests,
the pattern of excess pore pressure development within a single loading cycle, on the other
hand, is affected by phase transformation states. With τtotal increasing in the loading phase
of a cycle, excess pore pressure increases first, but the tendency is reversed after phase
transformation states are reached. When the increase of τtotal reverses and the unloading
phase begins, excess pore pressure turns to grow again until another loading period starts.
The variation of excess pore pressure within an individual cycle should not be examined in an
190 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
isolated manner because the lack of information of the phase transformation states prevents
accurate interpretation. Instead, it must be understood with real-time stress states. Readers
are referred to Chapter 6 where the effects of the phase transformation states on the change
of excess pore pressure within a single shearing cycle were discussed.
The development of excess pore pressure between loading cycles, however, needs to
be addressed because it provides valuable insights for constitutive modelling. As reported
in Chapter 4 in Figures 4.19 and 4.19, the development of excess pore pressure shows a
three-stage pattern that is not altered by initial static shearing in uni-directional tests. It is
worth exploring if such a pattern holds true in multi-directional loading scenarios as well.
Figure 7.12 presents the development of normalised excess pore pressure ratio with
normalised cycle number for all the tests in this study. The format in Figure 4.19 that plots
the absolute values of excess pore pressure ratio is not used because maximum excess pore
pressure ratios vary from a test to another due to the influence of the failure cone, resulting
in the end points of curves being widely scattered, from which undesirable confusion arises.
Instead, a normalisation of excess pore pressure ratios by their maxima is applied to make
the curves reach the same final point. There is no clear influence of multi-directional loading
paths on the development of excess pore pressure, indicating that the pattern of excess pore
pressure generation is one of the characteristics of Hostun sands that is not altered by the
multi-directionality of loading paths, at least for the tested range of relative densities and
vertical consolidation stresses in this study.
The patterns of excess pore pressure development can be categorised into three groups, as
shown in Figure 7.13. Group I is composed of the tests with NL > 5. All the curves of these
tests fall into a band. The same pattern of excess pore pressure generation is shared among
these tests, which feature three stages: excess pore pressure ratio jumps in the initial phase
and enters subsequently a stable period where excess pore pressure accumulates gradually,
followed by a period of excess pore pressure ratio increasing rapidly to its maximum amount.
This is congruous with the typical strain-softening type of behaviour of sands observed in
uni-directional tests.
7.3 The development of excess pore pressure in multi-directional simple shear tests 191
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
/r u
,
m
a
x
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Bi-directional linear tests
Circular tests without static shearing
Circular tests with static shearing
Oval tests
Figure-8 tests
Fig. 7.12 The development of normalised excess pore pressure ratio with normalised cycle
number
Group II represents the tests that have NL ⩽ 5 and normalised excess pore pressure ratio
lower than 0.5 in the first cycle. Excess pore pressure in these tests keeps rising until the
maximum values are achieved, without distinguishable stages or distinct variations in the
increment rate of excess pore pressure ratio.
Group III, by contrast, contains the tests where normalised excess pore pressure ratio
jumps to over 0.5 in the first cycle, including the tests that liquefy with only one loading
192 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
cycle. The tests in group III manifest a two-stage increase of excess pore pressure, with a
considerable increase of excess pore pressure (more than half of ru,max) taking place in the
first cycle, followed by moderate increase in the subsequent cycles. The scatter of group III
results is much more significant than for the other two groups, suggesting that there may not
be a consistent way to describe the pattern in this group.
The examination of patterns of excess pore pressure development in the tests with low
liquefaction resistance reveals that the three-stage pattern of excess pore pressure increase
is applicable only in tests that have moderate to high liquefaction resistance. For the tests
that are prone to liquefaction, however, the development of excess pore pressure needs to
be considered individually, especially when the increase of pore pressure in the first loading
cycle is extraordinarily high. Although NL = 5 appears to work satisfactorily in dividing
the tests into a group which is prone to liquefaction and a group which is not, whether
this threshold is applicable to other types of sands or tests with different testing conditions
remains unconfirmed. Only the qualitative conclusions should be transferable.
Another relationship worthy of inspection is that between liquefaction resistance and the
excess pore pressure ratio developed in the first loading cycle of a test. Such a correlation was
proposed for uni-directional tests by Oda et al. (2001) and supported by Wu (2002) as shown
in Figure 4.21, but whether it is applicable to multi-directional loading scenarios needs to be
examined. Figure 7.14a plots the first-cycle excess pore pressure ratio ru,1 versus the number
of cycles to liquefaction NL for the tests in this study. The solid curve is the same one as
in Figure 4.21 following equation 4.4. It appears that the same correlation is applicable to
multi-directional loading scenarios in general. The number of cycles to liquefaction reduces
rapidly with increasing first-cycle excess pore pressure ratio. The role of ru,1 seems to be
equivalent to either a descriptor of the stiffness of sands of a certain density under a certain
consolidation pressure or an indicator of the detriments that sands have experienced during
the first loading cycle, but in either way it indicates the ability of sands to withstand shearing
and resist liquefaction.
But considering the influence of multi-directional loading on achievable maximum excess
pore pressure ratio, it is reasonable to consider excess pore pressure in a normalised manner.
7.3 The development of excess pore pressure in multi-directional simple shear tests 193
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
/r u
,
m
a
x
Group I
Group II
Group III
Fig. 7.13 The development of normalised excess pore pressure ratio with normalised cycle
number categorised into three groups
Figure 7.14b presents the normalised first-cycle excess pore ratio versus liquefaction resis-
tance. The same solid curve is seen to work adequately, though the degree of scatter becomes
dramatic for the tests with rather low liquefaction resistance. Inevitable inaccuracies exist
in the determination of liquefaction resistance because factors such as unconsidered change
of φFL with relative density, overshooting issues of the loading system, imperfections in
samples and inhomogeneity induced by realistic simple shear testing setup can all contribute
to the uncertainties in determining liquefaction resistance. The inaccuracies are exaggerated
when the number of cycles to liquefaction is small, resulting in the seemingly large scatter.
Considering this, the solid curve is deemed as capable of describing the relationship between
liquefaction resistance and normalised first-cycle excess pore pressure ratio. Equation 4.4
can be hence modified as follows:
194 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
ru,1
ru,max
= N
−(0.67+
1.53
NL
)
L (7.6)
It should be noted that 0.67 and 1.53 are the coefficients fitted from unnormalised excess
pore pressure ratios in uni-directional tests. They are not modified partly because they still
work satisfactorily from the statistical point of view and partly because their use can highlight
the similarities between equations 4.4 and 7.6. It is recommended to replace equation 4.4 with
7.6 so that the effects of varying maximum excess pore pressure ratios under multi-directional
loading conditions can be accounted for.
In summary, while the multi-directionality of loading paths can influence the upper limits
of excess pore pressure and the variation of pore pressure within a single loading cycle, it
does not influence the development of normalised excess pore pressure ratios as well as the
relationship between liquefaction resistance and normalised first-cycle excess pore pressure
ratio. The latter can be regarded as an intrinsic characteristic of sands that is independent of
loading paths.
7.4 The development of shear strain in multi-directional
simple shear tests
Although shear strain and deformation is one of the most intriguing topics in geotechnical
research because of its value in practical engineering design, the understanding of shear
strain development in multi-directional loading remains insufficient. As discussed in Chapter
5, the development of shear strain induced by multi-directional shearing also occurs in a
multi-directional manner. It can not only pose a challenge to current models for geotechnical
deformation prediction, but also bring about problems for the seismic design of structures.
It is very likely that multi-directional motion of soil, depending on the specific path, can
result in detrimental torque in construction and aggravate the whiplash effects on high-rise
buildings. The accumulation of shear strain under multi-directional shearing is worthy of
more investigation.
7.4 The development of shear strain in multi-directional simple shear tests 195
100 101 102
Number of cycles to liquefaction NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
1
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Bi-directional linear tests
Circular tests without static shearing
Circular tests with static shearing
Oval tests
Figure-8 tests
(a) The relationship between the excess pore pressure ratio in the first cycle and the number
of cycles to liquefaction (the solid curve gives equation 4.4)
100 101 102
Number of cycles to liquefaction NL
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r u
,
1/
r u
,
m
a
x
Uni-directional linear tests without static shearing
Uni-directional linear tests with static shearing
Bi-directional linear tests
Circular tests without static shearing
Circular tests with static shearing
Oval tests
Figure-8 tests
(b) The relationship between the normalised excess pore pressure ratio in the first cycle and
the number of cycles to liquefaction (the solid curve gives equation 7.6)
Fig. 7.14 The relationship between the excess pore pressure ratios in the first cycle and the
number of cycles to liquefaction in all tests in this study
196 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
This section summarizes the characteristics of multi-directional shear strain development
that are observed in the tests in this study. It needs to be emphasized again that the sudden
change of stress from zero to a point on the loading path was avoided in the tests. For loading
paths such as circular and oval ones, a ramp loading was conducted first in drained conditions
to drive the shear stress to a point on the loading paths before cyclic shearing is applied in
undrained condition. The “zero” strain point is defined as the point before cyclic loading
starts rather than the point prior to ramp loading. Different values of total shear strain were
obtained if zero-strain point is assigned differently.
First and foremost, the geometry of strain paths is analogous to that of stress paths. Figure
7.15 compares the plan-view shear stress and strain of uni-directional linear test LC-7, oval
test E2010-3 and circular test C20-3 (loading proceeded in the anticlockwise direction).
The three tests have the same CSRdir at 0.2 with the primary shear directions assigned to
the X axis, as shown in Figure 7.15a. Clearly, the strain paths of the these tests reflect the
geometrical features of their stress paths. The shear strain accumulates in a circular way in
the circular test. By contrast, the uni-directional test has minimal shear stress on the direction
perpendicular to cyclic shearing and thus has only marginal shear strain in that direction. The
major direction of the shear strain development in the oval test is generally in accordance
with the orientation of the major axis of its elliptical stress paths. Similar trends can also be
seen in other categories of loading paths. For instance, three figure-8 tests which have similar
relative density and the same major-direction shear stress ratio of 0.15 are compared in Figure
7.16. The plan-view shear strain of the three tests all show the butterfly-like geometry. As
test F8-6 has smaller shearing in the perpendicular direction (SSRY,max is 0.075 in test F8-6
and 0.15 in F8-2 and F8-5), the strain accumulated in Y direction in test F8-6 is limited
compared with the other two tests.
However, it is found that the strain paths of oval tests and figure-8 tests exhibit special
characteristics that are not seen explicitly in uni-directional and circular tests. The strain
paths in oval tests tend to rotate when the amplitude of shear strain increases dramatically and
liquefaction is approached. The explanation is related the variation of excess pore pressure
within a loading cycle.
7.4 The development of shear strain in multi-directional simple shear tests 197
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
SSRX
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
SS
R
Y
LC7 (Dr=60.7%)
C20-3 (Dr=63.1%)
E2010-3 (Dr=62.6%)
(a) Plan view of shear stress ratio
-8 -6 -4 -2 0 2 4 6 8
X
-8
-6
-4
-2
0
2
4
6
8
Y
LC7 (Dr=60.7%)
C20-3 (Dr=63.1%)
E2010-3 (Dr=62.6%)
(b) Plan view of shear strain
Fig. 7.15 Comparison of uni-directional linear, oval and circular tests with the same CSRdir
at 0.2
198 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
SSRX
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
SS
R
Y
F8-2 (Dr=55%)
F8-5 (Dr=53.4%)
F8-6 (Dr=52.5%)
(a) Plan view of shear stress ratio
-4 -3 -2 -1 0 1 2 3 4
X
-5
-4
-3
-2
-1
0
1
2
3
4
Y
F8-2 (Dr=55%)
F8-5 (Dr=53.4%)
F8-6 (Dr=52.5%)
(b) Plan view of shear strain
Fig. 7.16 Comparison of three figure-8 tests with the same CSRdir at 0.15
7.4 The development of shear strain in multi-directional simple shear tests 199
(a) Plan view of shear strain with excess pore pressure ratio
-1 -0.5 0 0.5 1 1.5 2 2.5
X
-1
-0.5
0
0.5
1
Y
stress ratio
strain
stress ratio increment
(b) Plan view of stress ratio vector, stress ratio increment vector and strain increment vector
Fig. 7.17 Plan-view shear strain analysis of the 58th cycle in test E2010-3
200 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
Figure 7.17a presents the plan view of shear strain of the 58th cycle of oval test E2010-3.
Excess pore pressure ratio is indicated by the colour of the data points. The strain of the 57th
cycle is also shown as a reference, without excess pore pressure ratio. This cycle is the one
where the strain path of the test starts to rotate considerably as shown in Figure 7.15. The
shear strain accumulates along an oval route in a similar alignment to the elliptical stress
path in the first half of the cycle, but dramatic deviation takes place subsequently. The most
distinct development of shear strain occurs when excess pore pressure ratio exceeds 0.65,
which is reflected in the increasing distance between neighbouring data points.
Figure 7.17b plots the same strain path with stress ratio vectors and stress ratio increment
vectors calculated from the plan view of the shear stress but plotted onto the strain path.
The length of the stress ratio vectors is equivalent to SSRtotal . Strain vector is not shown to
prevent the “zero” strain issue from misleading the interpretation of test results. Comparing
Figure 7.17b with 7.17a, there is a close connection between the length of strain increment
vectors and excess pore pressure ratio. The shortest strain increment vectors are seen in the
section where excess pore pressure is also the lowest, while the longest strain increment
vectors appear where excess pore pressure is the highest (the dark red part in Figure 7.17a).
Before excess pore pressure reaches 0.65, the variation of strain increment vectors follows
a roughly symmetrical pattern, maintaining a stable geometry of the strain path. When
excess pore pressure ratio exceeds 0.65, the difference between the smallest and largest strain
increment increases. Huge strain accumulates towards the direction of those long strain
increment vectors, while less strain develops in the direction of short ones. As a consequence,
the development of the strain path is diverted, appearing like a rotation.
Furthermore, there is a seemingly inverse relationship between the length of strain
increment vectors and stress ratio vectors. Where strain increment vectors are long, stress
ratio vectors are always shorter. However, such a relationship is in fact a biased perception of
that between excess pore pressure and strain increment. The reason why such a relationship
seems to exist is that the length of stress ratio vectors can reflect the general results of loading
and unloading, while the latter, as discussed in Chapter 6, affects the variation of excess pore
pressure within a shearing cycle. This can be demonstrated by the fact that the points of
7.4 The development of shear strain in multi-directional simple shear tests 201
equal SSRtotal do not necessarily have the same excess pore pressure ratio. In Figure 7.17b,
for instance, the initial point and the end point on the path have almost the same rightward
stress ratio vector with SSRtotal at 0.2, but the strain increment vector of the end point is
much longer than the initial one because the excess pore pressure ratio at the end of this cycle
is larger than that at the beginning.
The relationship between the increment of total shear strain versus excess pore pressure
is presented in Figure 7.18 for all tests, with the data of test E2010-3 coloured red. Despite
considerable scatter, it appears that an excess pore pressure threshold exists in all tests,
before which the strain increment is low and stable but after which strain increment starts
to rise. For test E2010-3, this threshold is around 0.65, consistent with the preceding
observations. The examination of all the tests indicates that the threshold ranges from 0.55 to
0.7 across different tests. This finding suggests that the softening of undrained sands reaches
a significant degree when excess pore pressure exceeds a certain threshold. Measures should
be taken in geotechnical engineering practice to prevent this threshold from being reached to
avoid detrimental deformation.
Close inspection of Figure 7.17b unveils more implications. The relationship between
strain increment vectors and stress ratio increment vectors is not clear. The insignificant
variation of the length of stress ratio increment vectors suggests that the amplitude of
stress ratio increment does not affect directly that of the strain increment. Discernible non-
coaxiality also exists between stress increment and strain increment, with strain increment
vectors rotating gradually towards the direction of shear stress ratio increment counterparts.
Clearly, the interaction between shear strain and stress involves not only amplitude but also
orientation in multi-direction loading conditions. The effects of directionality should be
taken into consideration if the objective of predicting multi-directional soil deformation is to
be achieved.
The pattern of shear strain development is further complicated if static shearing is
introduced. Figure 7.19 compares three circular tests with different static shearing. The shear
strain of post-liquefaction cycles is also included in Figure 7.19b in dotted form. Test C15-2
has no static shearing and thus has a spiral strain path. Test C15-X15-1, on the contrary, has
202 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
Fig. 7.18 The increment of total shear strain versus excess pore pressure ratio in all tests (test
E2010-3 is coloured red)
the largest static shearing that eliminates stress reversal in the X direction. Shear strain in
this test accumulates primarily along the X direction with only marginal strain developing in
the Y direction. By contrast, test C15-X7.5-1 has moderate static shearing, whose strain path
has off-centred spiral geometry that combines the features of the other two tests.
There is a clear tendency of shear strain to accumulate towards the direction of static
shearing. A similar observation has been reported for both sands and clay (Biscontin
et al., 2004; Kammerer, 2002; Rutherford, 2012). This implies that downhill deformation
(landslide) can take place in sloping ground no matter what direction the cyclic shearing is in.
The seismic safety of the structures and infrastructures constructed on sloping ground cannot
be taken for granted even if the inclination is small, especially when serviceability is crucial.
Another observation is related to the case with static shearing larger than the amplitude of
cyclic shearing resulting in no stress reversal in the downhill direction, such as test C15-X15-
1. The deformation potential in the strike direction is limited under such circumstance. After
7.4 The development of shear strain in multi-directional simple shear tests 203
liquefaction happens, shear strain increases rapidly in both dip (X) and strike (Y) direction in
test C15-2 and C15-X7.5-1. In test C15-X15-1, by contrast, shear strain rises only in the dip
direction, while in the strike direction is restrained though stress reversal occurs. Kammerer
(2002) suggested that this phenomenon is a result of limited stress rotation. The orientations
of stress vectors range roughly from−90◦ to 90◦ relative to positive X axis in test C15-X15-1,
while ranging from −180◦ to 180◦ in the other two tests. Full rotation of stress vector does
not occur when stress reversal is inhibited in a direction, resulting in limited development of
strain in test C15-X15-1. Evidently, the specific geometry of a loading path exerts influence
over the deformation of soil, which should not be overlooked by academic and practical
seismic research.
The findings regarding the development of shear strain in multi-directional loading
scenarios are summarised:
(1) Shear strain in multi-directional loading conditions develops in an analogous manner to
shear stress. The general geometry of the strain path reflects the features of the stress
path. Strain paths are diverted, depending on the specific loading path, when excess pore
pressure exceeds a certain threshold.
(2) A threshold in terms of excess pore pressure ratio is found to separate the period of
limited strain accumulation from the period of flow-type strain development. The excess
pore pressure ratio threshold ranges from 0.55 to 0.7 for the tests in this study. Within a
loading cycle, the parts with excess pore pressure higher than this threshold witness more
pronounced shear strain increments, while the other parts have less significant strain
increments. The variation of excess pore pressure within a shearing cycle can result in
the deviation of a strain path from the expected course.
(3) There is no clear correlation between the amplitude of strain increment and that of stress
increment. The direction of the stress increment vector is related to the stress increment
vector. The complex non-coaxiality between stress increment and strain increment
needs to be taken into consideration in predicting the development of shear strain in
multi-directional loading conditions.
204 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
-0.2 -0.1 0.1 0.2 0.3 0.4
SSRX
-0.25
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
SSRY C15-2 (Dr=57%)
C15-X7.5-1 (Dr=55.4%)
C15-X15-1 (Dr=56.4%)
(a) Plan view of shear stress ratio
-15 -10 -5 5 10 15 20 25 30
X
-25
-20
-15
-10
-5
5
10
15 Y C15-2 (Dr=57)
C15-X7.5-1 (Dr=55.4)
C15-X15-1 (Dr=56.4)
(b) Plan view of shear strain (post-liquefaction cycles are included)
Fig. 7.19 Comparison of three circular tests with different initial static shearing
7.5 The degradation of Shear modulus in multi-directional simple shear tests 205
(4) The existence of static shearing can further complicate the development of shear strain.
Downhill accumulation of shear strain happens no matter what direction cyclic shearing
is applied in, which can put the seismic safety and the serviceability of the structures on
sloping ground in peril. Over-large static shearing can also eliminate stress reversal and
constrain the rotation of shear stress and thus restrain the development of shear strain in
strike direction.
7.5 The degradation of Shear modulus in multi-directional
simple shear tests
Shear modulus degradation is one of most important characteristics of soil under shearing.
In uni-directional simple shear tests, secant shear modulus is commonly selected to produce
shear modulus degradation curves, which is defined conveniently through stress-strain hys-
teresis loops as the ratio of double-amplitude cyclic shear stress over double-amplitude shear
strain. In multi-directional shearing condition, this definition of shear modulus becomes
inadequate.
A unique stress-strain hysteresis loop cannot be obtained. If the direction of shear stress
and strain is taken into account, the relationship between stress and strain becomes a four-
dimension problem, and there is yet an effective method to derive such a relationship. If
the directionality is by-passed through the use of parameters such as total shear stress and
total shear strain, the meaning of stress-strain curves becomes doubtful. For instance, the
total shear stress-strain curve of circular test C20-3 as presented in Figure 5.11 is actually a
straight line. Should a secant total shear modulus be determined from it, the modulus will
always be zero, which does not give much information about the actual relationship between
shear stress and strain, if any.
Although a shear modulus can still be defined by the real-time shear stress and shear
strain, the significance of this modulus is ambiguous because of the non-coaxiality between
shear stress and shear strain, as well as between stress increment and strain increment, not to
mention the mathematical difficulty in coping with zero strain.
206 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
Furthermore, the definition of zero strain is another issue that can cause misleading
estimates of shear modulus. In the tests in this study, cyclic loading always starts from a
point on the loading paths rather than zero-shear state to avoid an abrupt increase in shearing,
and zero-strain is assigned correspondingly at the point where cyclic loading starts. If the
condition of zero-shear is selected as the point to begin the measurement of shear strain, all
the calculation related total shear strain changes, including total-strain-based shear modulus.
Under this circumstance, a multi-directional secant shear modulus G is proposed here for
multi-directional shearing condition as follows:
G =
τdir,DA
γdir,DA
(7.7)
where τdir,DA is the maximum distance between any two points on the plan view of shear stress
path of a loading cycle, as defined by equation 7.3, while γdir,DA represents the maximum
distance on strain path, following equation 7.1. It should be noted that τdir,DA and γdir,DA does
not necessarily occur simultaneously like τDA and γDA does in uni-directional tests, because
of the non-coaxiality.
Figure 7.20 presents the degradation of multi-directional secant shear modulus G with
multi-directional double-amplitude shear strain γdir,DA in all the tests in this study. A similar
plot for uni-directional tests was shown in Figure 4.22. Gmax is the small-strain shear modulus
proposed by Hardin and Drnevich (1972) based on uni-directional tests, and the cyan dotted
curves are their prediction curves (equation 4.6). What is impressive is that G/Gmax− γdir,DA
of multi-directional tests fits satisfactorily with the prediction of Hardin and Drnevich, even
though the latter was derived from hyperbolic stress-strain curve assumption which is tenable
only in uni-directional loading condition. Whether this finding holds true for small strain
cannot be examined with the tests in this study, but for the shear strain range over 1%, the
degradation of multi-directional secant shear modulus can be predicted adequately by Hardin
and Drnevich’s method.
In addition to shear strain, the degradation of shear modulus with excess pore pressure
is also worthy of examination because they are both related to soil softening. Figure 7.21
7.5 The degradation of Shear modulus in multi-directional simple shear tests 207
Fig. 7.20 The degradation of multi-directional secant shear modulus with multi-directional
double-amplitude shear strain
presents the degradation of multi-directional secant shear modulus with excess pore pressure
ratio. The shear modulus is normalised by the 2nd-cycle shear modulus with the same reason
as discussed in section 4.4. The bold black curve represents equation 4.7, which was fitted
from uni-directional test results in Figure 4.23.
Compared with Figure 4.23, the degradation curves of multi-directional tests are more
scattered at large excess pore pressures, especially in the cases where the maximum excess
208 THE EFFECTS OF MULTI-DIRECTIONAL SHEARING
0 0.2 0.4 0.6 0.8 1 1.2
r
u
0
0.2
0.4
0.6
0.8
1
1.2
G
/G
2
Uni-directional linear loading paths
Bi-directional linear loading paths
Circular loading paths
Oval loading paths
Figure-8 loading paths
Fig. 7.21 The degradation of shear modulus with excess pore pressure ratio
pore pressure ratio is far less than unity. However, the two-stage overall degradation tendency
is unchanged. When excess pore pressure ratio is lower than 0.3-0.4, the softening of sands is
not significant and shear modulus degrades with a slow but almost constant rate. After excess
pore pressure exceeds the threshold, the decline of shear modulus accelerates. Shear modulus
decreases generally to less than 40% of G2 when liquefaction takes place. Furthermore,
the degradation of normalised shear modulus with excess pore pressure is not affected
significantly by the category of loading paths. This suggests that such a degradation is a type
of potential characteristic of saturated sands in undrained simple shear tests, regardless of
loading paths.
In conclusion, multi-directional secant shear modulus following equation 7.7 is recom-
mended as an alternative to the uni-directional counterpart. The relationship between G/Gmax
and γdir,DA in multi-directional tests is in accordance with the prediction proposed by Hardin
7.5 The degradation of Shear modulus in multi-directional simple shear tests 209
and Drnevich (1972) for uni-directional tests. The degradation of multi-directional secant
shear modulus with either double-amplitude shear strain or excess pore pressure ratio can be
regarded as a characteristic of saturated sands which is not altered by loading paths.

Chapter 8
CONCLUSIONS AND FUTURE WORK
8.1 Conclusions
Four decades after it was investigated experimentally for the first time, the influence of
multi-directional shearing on soil behaviour, especially the undrained behaviour of saturated
sands, remains one of the least understood areas in geotechnics. This study presents a new test
database of Hostun sand using the modified multi-directional simple shear testing apparatus
and testing techniques. Remarkable differences between undrained soil behaviour under
uni-directional and multi-directional shearing were observed. The effectiveness, as well
as the necessity, of interpreting the stress paths of soil in a three-dimensional τ−σ ′v space
were illustrated. A conical failure surface is found to be the best description of the failure
envelope. Phase transformation was shown to exhibit bimodal features, with a conical phase
transformation surface observed in the tests with circular, oval and figure-8 loading paths but
not in bi-directional linear tests. Liquefaction criteria, liquefaction resistance assessment,
development of excess pore pressure and shear strain, as well as the degradation of shear
modulus, were investigated in multi-directional shearing conditions, with a discussion of
the definitions of shear stress, shear strain and shear modulus in multi-directional loading
scenarios.
The findings of this study are summarised as follows:
212 CONCLUSIONS AND FUTURE WORK
(1) The membrane problem in undrained simple shear tests can be solved by applying
physical constraints to the full height of sample assembly. The modified multi-directional
simple shear testing apparatus and testing techniques produce reasonable test data.
(2) Interpretation of multi-directional tests exclusively in 2D τ−σ ′v plane is not adequate
because the spatial features of the stress path and excess pore pressure development are
not fully captured. Multi-directional shearing should be analysed in a 3D τ−σ ′v space
where the soil response can be fully examined.
(3) A conical failure surface is derived from the 2D failure lines. The failure cone is a more
useful concept to describe the failure envelope of sand. Failure lines in 2D τ−σ ′v plane
are the cross section of the failure cone on a plane through its axis.
(4) The phenomenon that the stress paths in bi-directional linear tests cannot reach failure
lines in the 2D τ−σ ′v plane can be explained by the failure cone. The stress paths of
bi-directional linear tests are located on a plane that is parallel to the failure cone axis.
The conic section between the plane and the cone is a hyperbola. The stress paths of
bi-directional linear tests viewed in 2D τ−σ ′v plane cannot exceed the hyperbolic failure
envelope.
(5) The widely-observed inverse correlation between the limiting excess pore pressure ratio
and shear stress amplitude is also a result of the failure cone. The smaller the minimum
total shear stress ratio, the closer a stress path can approach the vertex of failure cone,
thus the higher the maximum excess pore pressure ratio. By contrast, the higher the
maximum total shear stress ratio, the farther away from the cone vertex a stress path
can reach after failure, hence the lower the minimum excess pore pressure ratio in
post-liquefaction cycles.
(6) Phase transformation in multi-directional shearing conditions exhibits a bimodal feature.
A conical phase transformation surface is found in the tests with circular, oval and
figure-8 loading paths, as evidenced by the changing location of phase transformation
8.1 Conclusions 213
state points, whereas in bi-directional linear tests, the phase transformation states on the
plane containing stress paths do not fall on a hyperbola.
(7) Liquefaction criteria based on either excess pore pressure ratio or shear strain need to be
modified for multi-directional tests. It was found in this study that ru,max = 1− SSRtotal,mintanφFL
and γdir,DA = 6% can be used as an effective liquefaction criterion in multi-directional
shearing conditions.
(8) Based on the liquefaction resistance assessment method proposed by Seed and Harder
(1990), a coefficient of multi-directional loading defined by Kµ was used in this study
for the convenience of analysis. The lowest Kµ obtained was 0.57, corresponding to
a reduction of 43% of the required cyclic shear stress ratio to trigger liquefaction. Kµ
is smaller than unity in most cases, indicating that multi-directional shearing tends to
reduce the liquefaction resistance of the medium to dense sand tested in this study.
(9) It is recognised that the cyclic shear stress ratio CSR equals the maximum total shear
stress ratio SSRtotal in the uni-directional tests on which Seed’s liquefaction assessment
method was based. There is a possibility that N− SSRtotal curves can be used as the
alternative to the N −CSR curves. The assessment of liquefaction resistance with
N − SSRtotal relationships indicated that contradictory conclusions may be obtained
regarding the effects of initial static shearing.
(10) The relationship between the excess pore pressure and normalised cycle number is not
altered significantly by the type of loading path, as is the relationship between the number
of cycles to trigger liquefaction and the excess pore pressure ratio generated in the first
loading cycle.
(11) The development of shear strain within a loading cycle is highly associated with the
variation of the excess pore pressure ratio in the same cycle. An excess pore pressure
ratio threshold was found at approximately 0.55-0.7, beyond which the strain increment
increases dramatically.
214 CONCLUSIONS AND FUTURE WORK
(12) When initial static shearing is applied, shear strain accumulates towards the direction of
static shearing regardless of the direction of the cyclic shearing. If the initial static shear
is large enough to eliminate stress reversal in the downhill direction, the shear strain
in the strike direction will be restrained, which can be attributed to the reduced stress
rotation reported by Kammerer (2002).
8.2 Future work
The insights gained in this study provide novel perspectives to understand undrained soil
behaviour under multi-directional loading and can be used to modify existing models to
better predict the response of sands of interest. Nonetheless, further research regarding
multi-directional shearing should be conducted to develop a more advanced understanding of
this topic. The recommendations for future work are summarised as follows:
(1) More in-depth investigation on the formation of shear band under multi-directional
loading is needed to develop insights into the mechanisms of phase transformation.
(2) Simple shear testing techniques could be further improved for more accurate measure-
ments of the critical state void ratio, thereby integrating the critical state theory to
consistently investigate the effects of the state parameter.
(3) The liquefaction assessment method based on N −CSR relationships should be re-
evaluated systematically to provide better safety guidelines for engineering design.
(4) Methods to characterise loading paths need to be developed so that the effects of an
arbitrary loading path can be identified.
(5) Tests with irregular multi-directional loading paths could be conducted to provide a more
consistent determination of the effects of realistic loading paths.
(6) The effects of the degree of saturation on multi-directional soil behaviour should be
examined to provide a more comprehensive understanding of unsaturated soil.
8.2 Future work 215
(7) The effects of the fines content should be explored in multi-directional loading scenarios.
The multi-directional deformation of soil matrix may have interactions with different
soil structure caused by the existence of the fines content.
(8) The anisotropy of soil in terms of stiffness and fabrics may affect the soil behaviour
under multi-directional loading and is worthy of investigation.
(9) Deformation-related models, especially those based on the degradation of shear modulus,
should be re-examined to account for the directionality of shear stress and strain.
(10) Multi-directional discrete element modelling could be developed to investigate the effects
of multi-directional shearing from a micro mechanical perspective.
(11) Dynamic soil-structure and soil-foundation-structure models need to be modified to take
the effects of multi-directional stress and strain into consideration. Multi-directional
dynamic deformation of soils, foundation and structures may result in unexpected torque
in high-rise structures with peculiar layout and facets, thus resulting in more severe
damage.
(12) Multi-directional physical modelling tests, particularly centrifuge tests, should be con-
ducted to explore the influence of loading paths from a more practical point of view.

References
Amini, F. and Qi, G. (2000). Liquefaction testing of stratified silty sands. Journal of
Geotechnical and Geoenvironmental Engineering, 126(3):208–217.
Ampadu, S. K. and Tatsuoka, F. (1993). A hollow cylinder torsional simple shear apparatus
capable of a wide range of shear strain measurement. Geotechnical Testing Journal,
16(1):3–17.
ASTM D6467-13 (2013). Standard test method for torsional ring shear test to determine
drained residual shear strength of cohesive soils. Standard, ASTM.
Been, K., Jefferies, M., and Hachey, J. (1991). Critical state of sands. Geotechnique,
41(3):365–381.
Been, K. and Jefferies, M. G. (1985). A state parameter for sands. Géotechnique, 35(2):99–
112.
Bernhardt, M. L. (2013). Examination of granular material behavior in a laminar-type direct
simple shear device using laboratory validated discrete element method simulations. PhD
thesis, Texas A&M University.
Biscontin, G., Pestana, J., and Nadim, F. (2004). Seismic triggering of submarine slides in
soft cohesive soil deposits. Marine Geology, 203(3-4):341–354.
Bjerrum, L. and Landva, A. (1966). Direct simple-shear tests on a norwegian quick clay.
Geotechnique, 16(1):1–20.
Black, D. K. and Lee, K. L. (1973). Saturating laboratory samples by back pressure. Journal
of the soil mechanics and foundations division, 99(1):75–93.
Boulanger, R. W. (1990). Liquefaction behavior of saturated cohesionless soils subjected
to uni-directional and bi-directional static and cyclic simple shear stresses, volume 2.
University of California, Berkeley.
Boulanger, R. W., Chan, C. K., Seed, H. B., Seed, R. B., and Sousa, J. B. (1993). A low-
compliance bi-directional cyclic simple shear apparatus. Geotechnical Testing Journal,
16(1):36–45.
Boulanger, R. W. and Seed, R. B. (1995). Liquefaction of sand under bidirectional monotonic
and cyclic loading. Journal of Geotechnical Engineering, 121(12):870–878.
218 References
Broms, B. and Casbarian, A. (1965). Effects of rotation of the principal stress axes and of
the intermediate principal stress on the shear strength. In Proceedings of the 6th ICSMFE,
volume 1, pages 179–183.
Carraro, J. A. H. (2004). Mechanical behavior of silty and clayey sands. PhD thesis, Purdue
University.
Casagrande, A. (1936). Characteristics of cohesionless soils affecting the stability of slopes
and earth fills. Journal of Boston Society of Civil Engineers, 23(1):13–32.
Casagrande, A. (1976). Liquefaction and cyclic deformation of sands-a critical review.
Harvard Soil Mechanics Series No. 88, Cambridge: Harvard University.
Casagrande, A. and Rendon, F. (1978). Gyratory shear apparatus; design, testing procedures,
and test results on undrained sand. Technical report, Harvard University Soil Mechanics
Lab, Cambridge.
Castro, G. (1969). Liquefaction of sands. PhD thesis, Harvard University.
Castro, G. (1975). Liquefaction and cyclic mobility of saturated sands. Journal of the
Geotechnical Engineering Division, 101(6):551–569.
Chang, C. S. and Whitman, R. V. (1988). Drained permanent deformation of sand due to
cyclic loading. Journal of geotechnical engineering, 114(10):1164–1180.
Clough, G. W., Iwabuchi, J., Rad, N. S., and Kuppusamy, T. (1989). Influence of cementation
on liquefaction of sands. Journal of Geotechnical Engineering, 115(8):1102–1117.
Coelho, P. (2007). In situ densification as a liquefaction resistance measure for bridge
foundations. PhD thesis, University of Cambridge.
DeGroot, D. J. (1992). The multidirectional direct simple shear apparatus with application
to design of offshore arctic structures. PhD thesis, Massachusetts Institute of Technology.
DeGroot, D. J., Germaine, J. T., and Ladd, C. C. (1993). The multidirectional direct simple
shear apparatus. Geotechnical Testing Journal, 16(3):283–295.
DeGroot, D. J., Germaine, J. T., and Ladd, C. C. (1994). Effect of nonuniform stresses on
measured dss stress-strain behavior. Journal of geotechnical engineering, 120(5):892–912.
Degroot, D. J., Ladd, C. C., and Germaine, J. T. (1996). Undrained multidirectional direct
simple shear behavior of cohesive soil. Journal of geotechnical engineering, 122(2):91–98.
Doanh, T., Ibraim, E., and Matiotti, R. (1997). Undrained instability of very loose hostun
sand in triaxial compression and extension. part 1: experimental observations. Mechanics
of Cohesive-frictional Materials, 2(1):47–70.
Duku, P. M., Stewart, J. P., Whang, D. H., and Venugopal, R. (2007). Digitally controlled
simple shear apparatus for dynamic soil testing. Geotechnical Testing Journal, 30(5):368–
377.
References 219
Duku, P. M., Stewart, J. P., Whang, D. H., and Yee, E. (2008). Volumetric strains of clean
sands subject to cyclic loads. Journal of geotechnical and geoenvironmental engineering,
134(8):1073–1085.
Escribano, D. E., Nash, D. F. T., and Diambra, A. (2018). Local and global volumetric strain
comparison in sand specimens subjected to drained cyclic and monotonic triaxial compres-
sion loading. Geotechnical Testing Journal. https://doi.org/10.1520/GTJ20170054.
Ezaoui, A. and Benedetto, H. D. (2009). Experimental measurements of the global anisotropic
elastic behaviour of dry hostun sand during triaxial tests, and effect of sample preparation.
Géotechnique, 59(7):621–635.
Franke, E., Kiekbusch, M., and Schuppener, B. (1979). A new direct simple shear device.
Geotechnical Testing Journal, 2(4):190–199.
Fukutake, K. and Matsuoka, H. (1989). A unified law for dilatancy under multi-directional
simple shearing. Doboku Gakkai Ronbunshu, 1989(412):143–151.
Georgiannou, V. and Konstadinou, M. (2014). Effects of density on cyclic behaviour of
anisotropically consolidated ottawa sand under undrained torsional loading. Géotechnique,
64(4):287.
Goodman, R. E. (1989). Introduction to rock mechanics. New York: Wiley.
Harder, J., Leslie, F., and Boulanger, R. (1997). Application of Kσ and Kα correction
factors. In Technical Report NCEER, volume 97, pages 167–179. US National Center for
Earthquake Engineering Research (NCEER).
Hardin, B. O. and Drnevich, V. P. (1972). Shear modulus and damping in soils: design
equations and curves. Journal of Soil Mechanics & Foundations Div, 98(sm7).
Høeg, K., Dyvik, R., and Sandbækken, G. (2000). Strength of undisturbed versus recon-
stituted silt and silty sand specimens. Journal of geotechnical and geoenvironmental
engineering, 126(7):606–617.
Iai, S., Matsunaga, Y., and Kameoka, T. (1992). Analysis of undrained cyclic behavior of
sand under anisotropic consolidation. Soils and Foundations, 32(2):16–20.
Ishihara, K. (1985). Stability of natural deposits during earthquakes. In Proceedings of 11th
ICSMFE, volume 1, pages 321–376.
Ishihara, K. (1993). Liquefaction and flow failure during earthquakes. Geotechnique,
43(3):351–451.
Ishihara, K. and Li, S.-I. (1972). Liquefaction of saturated sand in triaxial torsion shear test.
Soils and Foundations, 12(2):19–39.
Ishihara, K. and Nagase, H. (1988). Multi-directional irregular loading tests on sand. Soil
Dynamics and Earthquake Engineering, 7(4):201–212.
Ishihara, K., Tatsuoka, F., and Yasuda, S. (1975). Undrained deformation and liquefaction of
sand under cyclic stresses. Soils and foundations, 15(1):29–44.
220 References
Ishihara, K. and Yamazaki, F. (1980). Cyclic simple shear tests on saturated sand in multi-
directional loading. Soils and Foundations, 20(1):45–59.
Ishihara, K. and Yasuda, S. (1975). Sand liquefaction in hollow cylinder torsion under
irregular excitation. Soils and Foundations, 15(1):45–59.
Jefferies, M. (1993). Nor-sand: a simle critical state model for sand. Géotechnique, 43(1):91–
103.
Jefferies, M. and Been, K. (2015). Soil liquefaction: a critical state approach. London: CRC
press.
Jin, D., Luan, M.-t., and Yang, Q. (2008). Liquefaction and cyclic failure of sand under
bidirectional cyclic loading. Electronic Journal of Geotechnical Engineering, 13.
John, K. W. (1968). Graphical stability analysis of slopes in jointed rock. Journal of Soil
Mechanics & Foundations Div, 94(2):497–526.
Kammerer, A., Wu, J., Riemer, M., Pestana, J., and Seed, R. (2004a). A new multi-
directional direct simple shear testing database. In Proceedings of 13th World Conference
on Earthquake Engineering, number 2083.
Kammerer, A., Wu, J., Riemer, M., Pestana, J., and Seed, R. (2004b). Shear strain develop-
ment in liquefiable soil under bi-directional loading conditions. In Proceedings of 13th
World Conference on Earthquake Engineering, number 2081.
Kammerer, A. M. (2002). Undrained response of Monterey 0/30 sand under multidirectional
cyclic simple shear loading conditions. PhD thesis, University of California, Berkeley.
Kuerbis, R. and Vaid, Y. (1988). Sand sample preparation-the slurry deposition method. Soils
and Foundations, 28(4):107–118.
Ladd, R. (1978). Preparing test specimens using undercompaction. Geotechnical Testing
Journal, 1(1):16–23.
Lade, P. (1973). Cubical triaxial tests on cohesionless soils. Journal of the Soil Mechanics
and Foundation Division, 99(10):793–812.
Lashkari, A., Karimi, A., Fakharian, K., and Kaviani-Hamedani, F. (2017). Prediction of
undrained behavior of isotropically and anisotropically consolidated firoozkuh sand: insta-
bility and flow liquefaction. International Journal of Geomechanics, 17(10):04017083.
Lee, K. L. and Fitton, J. (1968). Factors affecting the cyclic loading strength of soil. Vibration
Effects of Earthquakes on Soils and Foundations, 450:71–95.
Li, X. S. and Dafalias, Y. F. (2011). Anisotropic critical state theory: role of fabric. Journal
of Engineering Mechanics, 138(3):263–275.
Li, Y., Yang, Y., Yu, H.-S., and Roberts, G. (2016). Monotonic direct simple shear
tests on sand under multidirectional loading. International Journal of Geomechanics,
17(1):04016038.
References 221
Matsuda, H., Hendrawan, A. P., Ishikura, R., and Kawahara, S. (2011). Effective stress
change and post-earthquake settlement properties of granular materials subjected to multi-
directional cyclic simple shear. Soils and foundations, 51(5):873–884.
Matsuda, H., Shinozaki, H., Okada, N., Takamiya, K., and Shinyama, K. (2004). Effects of
multi-directional cyclic shear on the post-earthquake settlement of ground. In Proceedings
of 13th World Conference on Earthquake Engineering, number 2890.
Matsuoka, H. and Nakai, T. (1974). Stress-deformation and strength characteristics of soil
under three different principal stresses. In Proceedings of the Japan Society of Civil
Engineers, volume 232, pages 59–70. Japan Society of Civil Engineers.
Meneses, J., Ishihara, K., and Towhata, I. (1998). Effects of superimposing cyclic shear
stress on the undrained behavior of saturated sand under monotonic loading. Soils and
Foundations, 38(4):115–127.
Mirbaha, K. (2017). Bi-directional cyclic behaviour and liquefaction analysis of a silica-
carbonate sand. Master’s thesis, The University of Western Ontario.
Mitrani, H. (2006). Liquefaction remediation techniques for existing buildings. PhD thesis,
University of Cambridge.
Miura, S. and Toki, S. (1982). A sample preparation method and its effect on static and
cyclic deformation-strength properties of sand. Soils and foundations, 22(1):61–77.
Mulilis, J., Townsend, F., and Horz, R. (1978). Triaxial testing techniques and sand liquefac-
tion. In Dynamic geotechnical testing. ASTM International.
Murthy, T., Loukidis, D., Carraro, J., Prezzi, M., and Salgado, R. (2007). Undrained
monotonic response of clean and silty sands. Géotechnique, 57(3):273–288.
Nagase, H. and Ishihara, K. (1987). Effects of load irregularity on the cyclic behaviour of
sand. Soil Dynamics and Earthquake Engineering, 6(4):239–249.
Oda, M. (1972). The mechanism of fabric changes during compressional deformation of
sand. Soils and foundations, 12(2):1–18.
Oda, M., Kawamoto, K., Suzuki, K., Fujimori, H., and Sato, M. (2001). Microstructural
interpretation on reliquefaction of saturated granular soils under cyclic loading. Journal of
Geotechnical and Geoenvironmental Engineering, 127(5):416–423.
Oda, M., Koishikawa, I., and Higuchi, T. (1978). Experimental study of anisotropic shear
strength of sand by plane strain test. Soils and foundations, 18(1):25–38.
Peacock, W. H. (1968). Sand liquefaction under cyclic loading simple shear conditions.
Journal of Soil Mechanics & Foundations Div, 94(3):689–708.
Priest, S. D. (1985). Hemispherical projection methods in rock mechanics. Allen & Unwin.
Pyke, R., Seed, H., and Chan, C. (1975). Settlement of sands under multidirectional
shaking. Journal of Geotechnical and Geoenvironmental Engineering, 101(ASCE# 11251
Proceeding).
222 References
Reddy, K. R. and Saxena, S. K. (1992). Liquefaction resistance of cemented sands under
multidirectional cyclic loading. Canadian Geotechnical Journal, 29(6):989–993.
Renouf, M., Acary, V., and Dumont, G. (2005). 3D frictional contact and impact multibody
dynamics. A comparison of algorithms suitable for real-time applications. In Mutlibody
Dynamics 2005, ECCOMAS Thematic Conference.
Roscoe, K. (1953). An apparatus for the application of simple shear to soil samples. In
Proceedings of 3rd ICSMFE, volume 1, pages 186–191.
Roscoe, K. H., Schofield, A., and Wroth, C. (1958). On the yielding of soils. Geotechnique,
8(1):22–53.
Rothenburg, L. and Bathurst, R. (1989). Analytical study of induced anisotropy in idealized
granular materials. Geotechnique, 39(4):601–614.
Rudolph, C., Grabe, J., and Albrecht, I. (2014). Simple shear tests with a varying shearing
direction during cyclic shearing. Géotechnique Letters, 4(2):102–107.
Rutherford, C. and Biscontin, G. (2013). Development of a multidirectional simple shear
testing device. Geotechnical Testing Journal, 36(6):858–866.
Rutherford, C. J. (2012). Development of a multi-directional direct simple shear testing
device for characterization of the cyclic shear response of marine clays. PhD thesis, Texas
A&M University.
Sako, N. and Adachi, T. (2004). Effect of two-directional input motion on charateristics
of sands liquefaction based on pseudo-dynamic tests. In Proceedings of 13th World
Conference on Earthquake Engineering, number 10.
Salgado, R., Bandini, P., and Karim, A. (2000). Shear strength and stiffness of silty sand.
Journal of Geotechnical and Geoenvironmental Engineering, 126(5):451–462.
Sauer, J. and Schömer, E. (1998). A constraint-based approach to rigid body dynamics
for virtual reality applications. In Proceedings of the ACM symposium on Virtual reality
software and technology, pages 153–162. ACM.
Schofield, A. N. (2005). Disturbed soil properties and geotechnical design. London: Thomas
Telford.
Schofield, A. N. and Wroth, P. (1968). Critical state soil mechanics. London: McGraw-Hill.
Seed, H., Arango, I., and Chan, C. (1975). Evaluation of soil liquefaction potential for level
ground during earthquakes. a summary report. Technical report, Shannon and Wilson.
Seed, H. B. (1979). Soil liquefaction and cyclic mobility evalution for level ground during
earthquakes. Journal of Geotechnical and Geoenvironmental Engineering, 105(2):201–
255.
Seed, H. B., Idriss, I., and Arango, I. (1983). Evaluation of liquefaction potential using field
performance data. Journal of Geotechnical Engineering, 109(3):458–482.
References 223
Seed, H. B. and Lee, K. L. (1966). Liquefaction of saturated sands during cyclic loading.
Journal of Soil Mechanics & Foundations Div, 92(6):105–134.
Seed, H. B., Martin, G. R., and Pyke, R. M. (1978). Effect of multidirectional shaking on
pore pressure development in sands. Journal of the Geotechnical Engineering Division,
104(1):27–44.
Seed, R. B. and Harder, L. (1990). SPT-based analysis of cyclic pore pressure and undrained
residual soil strength. In Proc., H. Boldon Seed Memorial Symp., volume 2, pages 351–376.
Sivathayalan, S. and Ha, D. (2011). Effect of static shear stress on the cyclic resistance of
sands in simple shear loading. Canadian Geotechnical Journal, 48(10):1471–1484.
Sladen, J., D’hollander, R., and Krahn, J. (1985). The liquefaction of sands, a collapse
surface approach. Canadian Geotechnical Journal, 22(4):564–578.
Stringer, M. and Madabhushi, S. (2009). Novel computer-controlled saturation of dynamic
centrifuge models using high viscosity fluids. Geotechnical Testing Journal, 32(6):559–
564.
Takahashi, H., Kitazume, M., Ishibashi, S., and Yamawaki, S. (2006). Evaluating the
saturation of model ground by p-wave velocity and modelling of models for a liquefaction
study. International Journal of Physical Modelling in Geotechnics, 6(1):13–25.
Talobre, J. (1957). La Mccanique ties Roches. Paris: Dunod.
Tatsuoka, F., Muramatsu, M., and Sasaki, T. (1982). Cyclic undrained stress-strain behavior
of dense sands by torsional simple shear test. Soils and Foundations, 22(2):55–70.
Tokimatsu, K. and Yoshimi, Y. (1982). Liquefaction of sand due to multidirectional cyclic
shear. Soils and Foundations, 22(3):126–130.
Ueng, T.-S., Wang, M.-H., Chen, M.-H., Chen, C.-H., and Peng, L.-H. (2005). A large
biaxial shear box for shaking table test on saturated sand. Geotechnical Testing Journal,
29(1):1–8.
Vaid, Y. and Chern, J. (1985). Cyclic and monotonic undrained response of saturated sands.
In Advances in the art of testing soils under cyclic conditions, pages 120–147. ASCE.
Vaid, Y. and Negussey, D. (1988). Preparation of reconstituted sand specimens. Advanced
Triaxial Testing of Soil and Rock, pages 405–417.
Vaid, Y. and Sivathayalan, S. (2000). Fundamental factors affecting liquefaction susceptibility
of sands. Canadian Geotechnical Journal, 37(3):592–606.
Vaid, Y., Stedman, J., and Sivathayalan, S. (2001). Confining stress and static shear effects in
cyclic liquefaction. Canadian Geotechnical Journal, 38(3):580–591.
Vaid, Y. and Thomas, J. (1995). Liquefaction and postliquefaction behavior of sand. Journal
of Geotechnical Engineering, 121(2):163–173.
Verdugo, R. and Ishihara, K. (1996). The steady state of sandy soils. Soils and foundations,
36(2):81–91.
224 References
Vucetic, M. and Lacasse, S. (1982). Specimen size effect in simple shear test. Journal of the
Geotechnical Engineering Division, 108(12):1567–1585.
Wriggers, P. and Krstulovic´-Opara, L. (2004). The moving friction cone approach for
three-dimensional contact simulations. International Journal of Computational Methods,
1(01):105–119.
Wu, J. (2002). Liquefaction triggering and post-liquefaction deformation of Monterey
0/30 sand under uni-directional cyclic simple shear loading. PhD thesis, University of
California, Berkeley.
Yamamuro, J. A. and Covert, K. M. (2001). Monotonic and cyclic liquefaction of very loose
sands with high silt content. Journal of geotechnical and geoenvironmental engineering,
127(4):314–324.
Yamamuro, J. A. and Lade, P. V. (1998). Steady-state concepts and static liquefaction of silty
sands. Journal of geotechnical and geoenvironmental engineering, 124(9):868–877.
Yamazaki, F., Towhata, I., and Ishihara, K. (1985). Numerical model for liquefaction problem
under multi-directional shearing on horizontal plane. In Proceedings of 5th International
Conference on Numerical Methods in Geomechanics, pages 1–5.
Yee, E., Duku, P. M., and Stewart, J. P. (2013). Cyclic volumetric strain behavior of sands
with fines of low plasticity. Journal of Geotechnical and Geoenvironmental Engineering,
140(4):04013042.