Slope Stability and Reliability Analysis (Mechanical Engineering Theory and Applications) 1536129356, 9781536129359

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MECHANICAL ENGINEERING THEORY AND APPLICATIONS

SLOPE STABILITY AND RELIABILITY ANALYSIS

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MECHANICAL ENGINEERING THEORY AND APPLICATIONS

SLOPE STABILITY AND RELIABILITY ANALYSIS

YUNG MING CHENG

Copyright © 2018 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected]. NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN: H%RRN

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

vii

Chapter 1

Introduction

1

Chapter 2

Slope Stability Analysis Methods

Chapter 3

Advanced Topics in Slope Stability Analysis

119

Chapter 4

Three-Dimensional Slope Stability Analysis

189

Chapter 5

Reliability Slope Stability Analysis

285

13

References

381

Index

405

PREFACE Slope stability is always a very important topic in many developed and highly congested cities, particularly for many cities in China where slope failures have killed many people with a significant loss of property. The author has also participated in different types of slope stability research and consultancy works in different countries, and has published two books entitled Soil Slope Stability Analysis and Stabilization – New Methods and Insights and Frontier in Civil Engineering, Vol.1, Stability Analysis of Geotechnical Structures, which are well-favored by many students, engineers and researchers. The author also frequently receives emails about the details of the more innovative slope stability analysis methods, stabilization and monitoring systems, as well as the procedures in the numerical implementation of some of the stability analysis methods. In view of the various improvements in the theory of slope stability analysis over the years, the author would like to write a new book on slope stability analysis and slope reliability analysis, and aims this new book at students, engineers as well as researchers. In this book, different methods of slope stability analysis will be discussed in a broad sense. Following that, the limit equilibrium and finite element methods will be discussed in more detail, as these two methods are the methods commonly used for practical works. Detailed procedures for limit equilibrium analysis will be provided to aid the students in learning, while the program SLOPE2000 will be introduced for the solution of more complicated problems. Some interesting engineering cases will be illustrated in this book. The author will also try to introduce the use of a distinct element slope stability method, which is a technique still far from practical applications, but it does offer some insights that are not possible with the other methods. Following that, the author will introduce the importance of a reliability slope stability analysis, which is an important issue for cities with complicated ground conditions and a high water table. Due to the intensive computation required for reliability analysis, the author has proposed many improvements to various reliability assessment methods in order to maintain a balance between accuracy and time of computation. The central core of SLOPE 2000 and SLOPE 3D for two-dimensional and threedimensional slope stability analysis as introduced in this book are developed mainly by the author, while there are many research personnel who have helped in various works associated with the research works. The author would like to thank Yip C.J., Wei W.B., Li N., Li L. Li D.Z. and Liu L.L. for their help in preparing parts of this book and the preparation of some of the figures as well.

Chapter 1

INTRODUCTION 1.1. INTRODUCTION The total area of Hong Kong is approximately 1,103 km2, accommodating a population of 7.4 million with a hilly terrain in many places. To suit for the population and the economic development, many slopes are formed for land development to cope with the rapid development of Hong Kong. Natural hillsides have been transformed into residential and commercial areas for infrastructural development. The stability of man-made and natural slopes is of major concern to the Government and the public. In Hong Kong, there are about 60000 registered slopes (40000 for government slopes and 20000 for private slopes), and each year about 500-1000 slopes can be stabilized (GEO 1976, 1984, 1994, 1995, 1996a, 1996b, 1996c, 1998, 2001, 2003). Slope stabilization is hence a major construction activity in Hong Kong. Hong Kong’s steeply hilly terrain, heavy rain and dense development make it prone to risk from landslides. Hong Kong has a history of tragic landslides with serious loss of life and property damage. In the 50 years after 1947, more than 470 people died, mostly as a result of failures associated with man-made cut slopes, fill slopes and retaining walls. Even though the risk to the community has been greatly reduced by concerted Government action since 1977, on average about 300 incidents affecting man-made slopes, walls and natural hillsides are reported to the Government every year. Most of these incidents are minor; many are just washouts and erosion on the surfaces of slopes and hillsides, but a significant proportion are larger failures which can threaten life and property, block roads and disrupt the lives of the community. On 18 June 1972, the 40m high road embankment failure at Sau Mau Ping Estate in Kowloon (GEO 1976) killed 71 people (see Figure 1). This accident was followed by the collapse of the hillside above a steep temporary excavation on Po Shan Road in the MidLevels area of Hong Kong Island, which triggered a landslide that destroyed a 12-storey residential building and killed 67 people a few hours later after the Sau Mau Ping collapse. Four years later, another severe rainstorm hit Hong Kong and brought down three fill slopes in Sau Mau Ping Estate again which were constructed without proper compaction. The resulting landslides killed 18 people. Some more famous slope failure cases in Asia are given in Figure 1.1 to 1.5, and these cases clearly illustrate the importance of proper slope stability analysis and stabilization.

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Yung Ming Cheng

Figure 1.1. Fill slope failures in Hong Kong, 1972.

Figure 1.2. Fill slope failure in Hong Kong, 1972.

Introduction

Figure 1.3. Natural slope failure in Hong Kong 1995.

Figure 1.4. Natural slope failure in Philippine, 2006.

3

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Yung Ming Cheng

Figure 1.5. Fill slope failure in China, 2015.

Table 1.1. Annual Rainfall in Hong Kong Year 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Annual Rainfall (mm) 3343 2565 2129 2752 3092 2490 1942 1739 3215 2628 1707 3066 2182 2372 1487 1925 2847 2638 1875 3027

Introduction

5

The average annual rainfall in Hong Kong is about 2500mm which falls mostly in the summer season between May and September, with an average of 300 slope failures each year. Some of the annual rainfall measurements are shown in Table 1.1 for illustration. Recently, the number of slope failure has reduced, and there are only 226 failure cases in 2016 and 161 failure cases in 2015. The reduction in the number of slope failures can be attributed to the slope stabilization works that are carried out in Hong Kong. Landslide risk to the community has been substantially reduced since the Government established the Geotechnical Engineering Office (GEO) on 1977 and the Land Preventive Measure Programme. In order to enhance slope safety, the Government embarked some long term programmes for slope stabilization works since the late 1970s. The programmes have been accelerated and expanded in recent years to a Landslip Preventive Measures (LPM) Programme with an annual expenditure of about HK$870 million to upgrade substandard Government slopes and undertake safety-screening of old private slopes. A total expenditure of about HK$600 million each year by the maintenance departments to properly maintain all Government slopes, and other measures have been taken to achieve landslide risk reduction by the Hong Kong Government (GEO 2002, 2004a, 2004b, 2007, 2008, 2011a, 2011b). Besides Hong Kong, many major cities in China are also facing similar situations, and slope stabilization becomes a very important activity with different types of materials and construction techniques being adopted for slope stabilization. Hong Kong’s steep hilly terrain and limited access to many steep slopes have created great difficulties in soil nail installation. Besides Hong Kong, many cities in China and other Asian countries are also suffering from slope failure problems, and many people are killed each year due to slope failures and debris flow. It is hence not surprising that there are many active research works in slope stability analysis, stabilization and protective measures, innovative construction materials and installation methods in Hong Kong and China. The author has carried out many research works in these areas, and has also been involved in many large scale slope stability problems in different countries. Besides that, based on the author’s research works, powerful two-dimensional and three-dimensional computer programs for slope stability analysis have also been developed. This book can be viewed as a collection of the various research works and teaching materials by the author, and this book should be useful to both students, engineers and researchers. This book aims to critically review the various slope stability analysis methods and slope reliability engineering. In particular, the book will consider the fundamental assumptions of both limit equilibrium, finite element method and other methods in assessing the stability of a slope with guidance on their limitations. Hong Kong is now renowned for its urban slope engineering and landslide risk management that enable its safe and sustainable development. The slope engineering and landslide management practice in Hong Kong has advanced considerably with time. Broadly speaking, it may be classified into three notable stages (Wong and Ho 2006): 1) Empirical slope design before 1977 - Slope design and construction were based on rule of thumb, such as 55 ° steep for soil cut slopes and 35 ° steep for fill embankments. There was little geotechnical input, except for critical facilities, such as dams. About 39,000 sizable man-made slopes were formed in this period. The vast majority of these slopes do not meet the current slope design standards, and are

6

Yung Ming Cheng particularly vulnerable to landslide at times of heavy rain. These un-engineered manmade slopes bring a long-term landslide problem to Hong Kong. 2) Geotechnical slope design and landslide prevention from 1977 to mid-1990s: In the aftermath of several disastrous man-made slope failures, the Geotechnical Engineering Office (GEO, which is currently part of the Civil Engineering and Development Department of the Government of the HKSAR) was set up in 1977 as the central body to regulate geotechnical engineering and slope safety in Hong Kong. Man-made slopes formed after 1977 in Hong Kong are subject to geotechnical design and checking, to ensure that they meet the required design standards. A total of about 18,000 sizeable man-made slopes have been formed since then. The GEO also operates a Landslip Preventive Measures (LPM) Programme, to systematically assess the stability of the pre-1977 man-made slopes according to their ranked order of priority and upgrade substandard Government slopes to the required design standards (GCO 1984). The conventional deterministic approach of slope stability analysis was adopted in slope design. Landslide prevention was primarily aimed at, and based on, achieving the required design factor of safety, although risk management concept was implicit in the strategy adopted. Risk consideration, if carried out, was made in a qualitative manner. 3) Enhanced landslide risk management since mid-1990s: The GEO has pioneered the development and introduction of an explicit risk-based approach and strategy, in addition to the deterministic approach, for slope assessment and landslide management. The risk-based methodology embraces a holistic consideration of the likelihood of landslide and its adverse consequences. It can be applied in a qualitative or quantitative framework. The quantitative applications, in particular, have been instrumental in formulating the overall slope safety strategy for Hong Kong, as well as managing the landslide risk at individual vulnerable sites. This approach aligns slope engineering and landslide mitigation with other fields that practice the state-ofthe-art risk management in a more explicit manner.

The slope engineering practice in Hong Kong can also reflect the development and advancement of both the theory and field practice in slope stability problems. Actually, all the three stages are still practiced in Hong Kong and many countries at present. For rapid stabilization of slopes (particularly for quick remedial of failed slopes), the use of prescriptive design approach is still adopted by many engineers in Hong Kong. On the other hand, the use of deterministic approach to carry out slope stability analysis and design is the most commonly adopted approach in Hong Kong. It is however find that up to 5% of the slopes which have been stabilized according to the current design practice will still fail eventually. In this respect, the risk-based methodology and the reliability analysis becomes essential for some critical slopes.

1.2. SOME CONTROL MEASURES IN HONG KONG A landslip warning will be issued by the Hong Kong Observatory in conjunction with Geotechnical Engineering Office when there is a high risk of many landslips as a result of

Introduction

7

persistent heavy rainfall. The warning is aimed at predicting the occurrence of numerous landslips, and isolated landslips which cannot be predicted will occur from time to time in response to less severe rainfall when the Warning is not in force. On the issuance of such a warning, a Landslip Special Announcement will be sent to the local radio and television stations for broadcast to the public, and the announcement will be updated at regular intervals until the likelihood of landslips has diminished. The Landslip Warning supplements routine weather forecasts by drawing attention to risk from landslips due to heavy rain. It is intended to prompt the public to take precautionary measures to reduce their exposure to risk posed by landslips, and to assist engineers, contractors and others who are likely to suffer losses from landslips. The warning also alerts the relevant government departments and organisations to take appropriate actions, such as opening of temporary shelters, search and rescue operations, closure of individual schools and relief work. It is issued irrespective of whether other severe weather warnings, e.g., tropical cyclone signals or rainstorm warning signals, are in force. Like all forecasts, a Landslip Warning represents an assessment of the weather based on the latest information available at the time. There will unavoidably be false alarms as well as occasions when heavy rain which may cause landslips develops suddenly and affects parts of Hong Kong before a warning can be issued. Although heavy rainstorms are not uncommon at any time of the year in Hong Kong, most of them happen during the summer months. In fact, close to 80 per cent of the annual rainfall occurs between May and September. The highest ever hourly rainfall recorded at the Hong Kong Observatory in Tsimshatsui is 145.5 millimetres which occurred during the rainstorm on 7 June 2008, but higher hourly rainfalls have been recorded in some other parts of Hong Kong. Every year heavy or prolonged rain causes landslips in Hong Kong. Property owners, engineers, architects, contractors and others concerned should take all necessary precautions against damage, and the public should take precautions against risk of personal injury. On average, the Landslip Warning is issued about three times every year, and on average two or three hundred landslips occur each year. Most of these are small in scale, but many are large enough to cause injury to people, damage to property and blockage of roads. Hong Kong has a bad history of landslips. It has steep hilly terrain, intense seasonal rainfall and very dense hillslope development. In particular, many thousands of substandard man-made slopes were constructed in the past, largely during the years of rapid development which followed World War II, and many of these slopes are prone to failure at times of heavy rainfall. In the past 50 years, a total of more than 470 people have been killed by landslips. On two days alone, during severe rainstorms in July 1966 and June 1972, 86 and 148 lives were lost respectively due to landslips. Since the introduction of effective geotechnical control in Hong Kong in 1977, and with concerted government effort over the past 20 years to reduce the risk from landslips to the community, casualties and disruption from landslips have been greatly reduced. However, the risk can never be entirely eliminated, and some tragedies continue to occur. The collapse of an old masonry wall in an estate in Kennedy Town (Kung Lung House) after torrential rain July 1994 killed five people and seriously injured three, and caused the evacuation of more than 2,500 people. Heavy rain following the passage of Severe Tropical Storm Helen in August 1995 brought widespread landslips with three deaths. 1997 was the wettest year on record to that time, resulting in 548 landslips being reported, with the loss of two lives.

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Yung Ming Cheng

In order to issue timely warnings to the public, the Hong Kong Observatory keeps a continuous watch of the weather in and around Hong Kong. Readings from a network of more than 100 automatic rain-gauges covering the whole of the Special Administrative Region are telemetered to the Observatory Headquarters to provide real-time information essential for assessing the likelihood of landslips. In addition to the conventional meteorological observations, the Observatory's weather radar system provides a good means to continuously monitor the movement and development of rain-bearing clouds. Meteorological satellite imagery at high resolution are received at frequent intervals, providing a bird's eye view of cloud patterns over Asia and the western Pacific. The Civil Engineering and Development Department (CEDD) has developed a computerized Slope Information System (SIS) containing data on all of those slopes. The SIS is a geographic information system which integrates different types of slope information linked to geographic and textual databases. It provides valuable information to geotechnical engineers, property owners, management companies and the general public. To support the SIS, the Survey Division of CEDD is commissioned to maintain and update a database of the registered slopes by the Computerized Slope Registration & Location Plan (CSRLP) System. The system was firstly introduced in 1996 and has been recently revamped in 2006. Besides supporting the SIS, CSRLP System also supports the Squatter Control Warning System involving analysis of slope, land status and dwellings which are susceptible to landslide hazard. The computerized Slope Information System (SIS) developed by the Slope Safety Division contains textual and spatial information of man-made slope features ready for online analysis of the spatial relation between slopes and surrounding topography. With a view to support the SIS, the Survey Division is commissioned to maintain the computerized Slope Registration and Location Plan (CSRLP) System. It is a digital slope mapping system which contains slope feature boundaries, registration numbers and basic map details. Survey Division also develops Squatter Slope Information System involving analysis of slopes, land status and dwellings which are susceptible to landslide hazard. To prepare for computerization in July 1994, information is collected and retrieved for identifiable manmade features in Hong Kong. The registrable features are classified into eight categories: F – Fill slope or platform with a surface sloping at >15° and a minimum fill thickness of 1m and without any associated retaining wall(s). C – Cut slope without any associated retaining wall(s). R – All retaining walls supporting a slope or platform with a surface sloping at 0,

For c=0,

For =0,

2 D D  f 0  1  0.5   1.4     l    l 2 D D  f 0  1  0.3   1.4     l    l 2 D D  f 0  1  0.6   1.4     l    l

(2.69)

For the correction factor shown above and Figure 2.3, l is the length joining the left and right exit points while D is the maximum thickness of the failure zone with reference to this line. Since the correction factors by Janbu (1973) are based on limited case studies and are purely empirical in nature, the uses of these factors to complicated nonhomogeneous slopes are questioned by some engineers. Since the intereslice shear force can sometimes generate a high factor of safety for some complicated cases which may occur in dam and hydropower projects, the use of the Janbu simplified method (1957) is preferred over other methods in these kinds of projects in China. The Lowe and Karafiath method (1960) and the Corps of Engineers method are based on the interslice force functions type 5 and type 6. These two methods satisfy only force equilibrium but not moment equilibrium. In general, the Lowe and Karafiath method (1960) will give results close to those from the “rigorous” methods even though the moment equilibrium is not satisfied. For the Corps of Engineers method, it may lead to a high factor of safety in some cases, and some engineers actually adopt a lower interslice force angle to account for this problem (Duncan and Wright 2005), and this practice is also adopted by some engineers in China. Similarly, the load transfer method in China satisfies only the force equilibrium, and the factor of safety from this method appears to be slightly lower than that from the other methods in general.

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Yung Ming Cheng

Figure 2.3. Definitions of D and l for the correction factor in Janbu simplified method.

2.5. SIMPLE ILLUSTRATION ON SLOPE STABILITY ANALYSIS For illustration of the numerical process, a simple slope as shown in Figure 2.4 is studied. The slope is given by coordinates (4,0), (5,0), (10,5), and (12,5) while the water table is given by (4,0), (5,0), (10,4) and (12,4). The soil parameters are: unit weight = 19 kN/m3, c’=5 kPa and ’=36. To define a circular failure surface, the coordinates of the centre of rotation and the radius should be defined. Alternatively, a better method is to define the x-ordinates of the left and right exit ends and the radius of the circular arc. The latter approach is better as the left and right exit ends can usually be estimated easily from engineering judgment. In the present example, the x-ordinates of the left and right exit ends are defined as 5.0m and 12.0m while the radius is defined as 12m. The soil mass is divided into 10 slices for analysis and the details are given below: Slice 1 2 3 4 5 6 7 8 9 10

weight (kN) 2.50 7.29 11.65 15.54 18.93 21.76 23.99 25.51 32.64 11.77

base angle () 16.09 19.22 22.41 25.69 29.05 32.52 36.14 39.94 45.28 52.61

base length (m) 0.650 0.662 0.676 0.694 0.715 0.741 0.774 0.815 1.421 1.647

base pore pressure (kPa) 1.57 4.52 7.09 9.26 10.99 12.23 12.94 13.04 7.98 0.36

Slope Stability Analysis Methods

39

Figure 2.4. Numerical examples for a simple slope.

Table 2.3. Factors of safety for the failure surface shown in Figure 2.4 F

Bishop

Janbu Janbu Swedish Load Sarma Morgenstern simplified rigorous factor Price 1.023 1.037 1.024 0.991 1.027 1.026 1.028 Note: The correction factor is applied to Janbu simplified method. The results for Morgenstern-Price method using f(x)=1.0 and f(x)=sin(x) are the same. Tolerance in iteration analysis is 0.0005.

The results of analyses for the problem in Figure 2.4 are given in Table 2.3. For the Swedish method or the Ordinary method of slices where only the moment equilibrium is considered while the interslice shear force is neglected, the factor of safety from the global moment equilibrium takes the simple form as:

Fm 

cl  W cos   ul  tan  W sin 

(2.70)

A factor of safety 0.991 is obtained directly from the Swedish method for this example without any iteration. Based on an initial factor of safety 1.0, the successive factors of safety during the Bishop iteration analysis are 1.0150, 1.0201, 1.0219, 1.0225, 1.0226. For the Janbu simplified method (1957), the successive factors of safety during the iteration analysis using the Janbu simplified method are 0.9980, 0.9974, 0.9971. Based on a correction factor of 1.0402, the final factor of safety from Janbu simplified analysis is 1.0372. If the double QR

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Yung Ming Cheng

method is used for the Janbu simplified method (1957), a value of 0.9971 is obtained directly from the first positive solution of the Hessenberg matrix without using any iteration analysis. For the Janbu rigorous method (1973), the successive factors of safety based on iteration analysis are 0.9980, 0.9974, 0.9971, 1.0102, 1.0148, 1.0164, 1.0170, 1.0213, 1.0228, 1.0233, and 1.0235. For the Morgenstern-Price method (1965), a factor of safety 1.0282 and the internal forces are obtained directly from the double QR method without any iteration analysis. The variation of Ff and Fm with respect to  using the iteration analysis for this example is shown in Figure 2.5. It should be noted that Ff is usually more sensitive to  than Fm in general, and the two lines may not meet for some cases which can be considered as no solution to the problem. There are cases where the lines are very close but actually do not intersect. If a tolerance large enough is defined, then the two lines can be considered as having an intersection and the solution converge. This type of “false” convergence is experienced by many engineers in Hong Kong. These two lines may be affected by the choice of the moment point, and convergence can sometimes be achieved by adjusting the choice of the moment point. The results as shown in Figure 2.5 assume the interslice shear forces to be zero in the first solution step, and this solution procedure appears to be adopted in many commercial programs. Cheng et al. (2008a) have however found that the results shown in Figure 2.5 may not be the true result for some special case, and this will be further discussed in later section.

Figure 2.5. Variation of Ff and Fm with respect to  for the example in Figure 2.4.

From Table 2.3, it is clear that the Swedish method is a very conservative method as first suggested by Whitman and Bailey (1967). Besides, Janbu simplified method (1957) will also give a smaller factor of safety if the correction factor is not used. After the application of the correction factor, Cheng find that the results from Janbu simplified method (1957) are usually close to those “rigorous” methods. In general, the factors of safety from different methods of analysis are usually close to each other, as pointed out by Morgenstern (1992).

Slope Stability Analysis Methods

41

2.6. MISCELLANEOUS CONSIDERATIONS ON SLOPE STABILITY ANALYSIS 2.6.1. Acceptability of the Failure Surfaces and Results of Analysis Based on an arbitrary interslice force function, the internal forces which satisfy both the force and moment equilibrium may not be kinematically acceptable, but this issue is seldom considered by the engineers. The acceptability conditions of the internal forces include: 1) Since the Mohr-Coulomb relation is not used along the vertical interfaces between different slices, it is possible though not uncommon that the interslice shear forces and normal forces may violate the Mohr-Coulomb relation. 2) Except for the Janbu rigorous method and the extremum method for which the resultant of the interslice normal force must be acceptable, the line of thrust from other “rigorous” methods which are based on overall moment equilibrium may lie outside the failure mass which is not possible. In fact, all the other methods of analysis cannot guarantee that the thrust line location must be acceptable, as the thrust line location is not used in most of the rigorous formulation. The extremum approach by Cheng et al. (2010) can overcome this problem at the expense of intensive computation. 3) The interslice normal forces should not be in tension. For the interslice normal forces near to the crest of the slope where the base inclination angles are usually high, if c’ is high, it is highly likely that the interslice normal forces will be in tension in order to maintain the equilibrium. This situation can be eliminated by the use of tension crack. Alternatively, the factor of safety with tensile interslice normal forces for the last few slices may be accepted (which is the practice as adopted in most of the commercial programs), as the factor of safety is usually not sensitive to these tensile forces. On the other hand, tensile interslice normal forces near the slope toe are usually associated with special shape failure surfaces with kinks, steep upward slope at the slope toe or unreasonably high/low factor of safety. The factors of safety associated with these special failure surfaces need special care in the assessment and should be rejected if the internal forces are unacceptable. 4) The base normal forces may be negative near the toe and crest of the slope. For negative base normal forces near to the crest of the slope, the situation is similar to the tensile interslice normal forces and may be tolerable (as adopted in most commercial programs). For negative base normal forces near to the toe of the slope which is physically unacceptable, it is usually associated with deep seated failure with a high upward base inclination. Since very steep exit angle is not likely to occur, it is possible to limit the exit angle during automatic location of the critical failure surface. The factor of safety associated with negative base normal forces near to the toe of slope should also be rejected in the calculation. If the above criteria are strictly enforced to all the slices of a failure surfaces, many slip surfaces will fail to converge. One of the reasons is the effect of the last slice when the base angle is large. Based on the force equilibrium, tensile interslice normal force will be created

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Yung Ming Cheng

easily if c’ is high. This result can propagate so that the results for the last few slices will be in conflict with the criteria above. If the last few slices are not strictly enforced, the factor of safety will be acceptable when compared with other methods of analysis. A suggested procedure is that if the number of slices is 20, only the first 15 slices are checked against the criteria above.

2.6.2. Tension Crack As the condition of limiting equilibrium develops with the factor of safety close to 1, a tension crack shown in Figure 2.3 may form near the top of the slope through which no shear strength can be developed. If the tension crack is filled with water, a horizontal hydrostatic force Pw will generate additional driving moment and driving force which will reduce the factor of safety. The depth of a tension crack zc can be estimated as:

zc 

2c K a



(2.71)

where Ka is the Rankine active pressure coefficient. The presence of tension crack will tend to reduce the factor of safety of a slope, but the precise location of a tension crack is difficult to be estimated for a general problem. It is suggested that if tension crack is required to be considered, it should be specified at different locations and the critical results can then be determined. Sometimes, the critical failure surface with and without tension crack can differ appreciably, and the location of the tension crack needs to be assessed carefully. In SLOPE 2000 by Cheng or some other commercial programs, the location of the tension crack can be varied automatically during the search for the critical failure surface.

2.6.3. Earthquake Earthquake loadings are commonly modelled as vertical and horizontal loads applied at the centroid of the sliding mass, and the values are given by the earthquake acceleration factors kv/kh (vertical and horizontal) multiply with the weight of the soil mass. This quasistatic simulation of earthquake loading is simple in implementation but should be sufficient for most design purposes, unless the strength of soil is reduced by more than 15% due to the earthquake action. Beyond that, a more rigorous dynamic analysis may be necessary which will be more complicated, and more detailed information about the earthquake acceleration as well as the soil constitutive behaviour are required. Usually, a single earthquake coefficient may be sufficient for the design, but a more refined earth dam earthquake code is specified in DL5073-2000 in China. The design earthquake coefficients will vary according to the height under consideration which will be different for different slices. Though this approach appears to be more reasonable, most of the design codes and existing commercial programs do not adopt this approach. The program SLOPE 2000 by Cheng can however accept this special earthquake code.

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2.6.4. Water and Seepage Increase in pore water pressure is one of the main factors for slope failure. Pore water pressure can be defined in several ways. The classical pore pressure ratio ru is defined as u/h, and an average pore pressure for the whole failure mass is usually specified for the analysis. Several different types of stability design charts are also designed using an average pore pressure definition. The use of a constant averaged pore pressure coefficient is obviously a highly simplified approximation. With the advancement in computer hardware and software, the uses of these stability design charts are now mainly limited to the preliminary designs only. Pore pressure coefficient is also defined as a percentage of the vertical surcharge applied on the ground surface in some countries. This definition of the pore pressure coefficient is however not commonly used. If pore pressure is controlled by the ground water table, u is commonly taken as whw, where hw is the height of the water table above the base of the slice. This is the most commonly used method to define the pore pressure, which assumes that there is no seepage and the pore pressure is hydrostatic. Alternatively, a seepage analysis can be conducted and the pore pressure can be determined from the flow-net or the finite element analysis. This approach is more reasonable but is less commonly adopted in practice due to the extra effort to perform a seepage analysis. More importantly, it is not easy to construct a realistic and accurate hydrogeological model to perform the seepage analysis. Pore pressure can also be generated from the presence of perched water table. In a multilayered soil system, a perched water table may exist together with the presence of a watertable if there are great differences in the permeability of the soil. This situation is rather common for the slopes in Hong Kong. For example, slopes at mid-levels in Hong Kong Island are commonly composed of fill at the top which is underlain by colluvium and completely decomposed granite. Since the permeability of completely decomposed granite is 1 to 2 orders less than that for colluvium and fill, a perched water table can easily established within the colluvium/fill zone during heavy rainfall while the permanent water table may be within the completely decomposed granite zone. Consider Figure 2.6, a perched water table may be present in soil layer 1 with respect to the interface between soils 1 and 2 due to the permeability of soil 2 being 10 times less than that of soil 1. For slice base between A and B, it is subjected to the perched water table effect and pore pressure should be included in the calculation. For slice base between B and C, no water pressure is required in the calculation while the water pressure at slice base between C and D is calculated using the ground water table only. For the problem shown in Figure 2.7, if EFG which is below ground surface is defined as the ground water table, the pore water pressure will be determined by EFG directly. If the ground water table is above the ground surface and undrained analysis is adopted, ground surface CDB is impermeable and the water pressure arising from AB will become external load on surface CDB. For drained analysis, the water table given by AB should be used, but vertical and horizontal pressure correspond to the hydrostatic pressure should be applied on surfaces CD and DB. Thus a trapezoidal horizontal and vertical pressure will be applied to surfaces CD and DB while the water table AB will be used to determine the pore pressure.

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Figure 2.6. Perched water table in a slope.

Figure 2.7. Modelling of ponded water.

The use of hydrostatic water pressure is usually on the safe side and is commonly adopted by the engineers. Some programs (SLOPE2000) accept the definition of the seepage field from a seepage analysis. The actual water pressure from the seepage analysis can be used to define the water pressure, which is then superimposed by the forces induced from the hydraulic gradient. The average horizontal/vertical component of the hydraulic gradient multiplied by the unit weight of water will be applied at the centroid of the wetted zone of each slice for the analysis. For the treatment of the interslice forces, usually the total stresses instead of the effective stresses are used. This approach, though slightly less rigorous in the formulation, can greatly simplify the analysis and is adopted in virtually all the commercial programs. Greenwood

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(1987) and Morrison and Greenwood (1989) have reported that this error is particularly significant where the slices have high base angles with a high water table. King (1989) and Morrison and Greenwood (1989) have also proposed revisions to the classical effective stress limit equilibrium method. Duncan and Wright (2005) have also reported that some “simplified” methods can be sensitive to the assumption of the total or effective interslice normal forces in the analysis.

2.6.5. Saturated Density of Soil The unit weights of soil above and below water table are not the same and may differ by 1-2 kN/m3. For computer programs which cannot accept the input of saturated density, this can be modelled by the use of 2 different types of soil for a soil which is partly submerged. Alternatively, some engineers assume the two unit weights to be equal in views of the small differences between them.

2.6.6. Moment Point For simplified methods which satisfy only the force or moment equilibrium, the Janbu (1957) and the Bishop methods (1955) are the most popular methods adopted by the engineers. There is a perception among some engineers that the factor of safety from the moment equilibrium is more stable and is more important than the force equilibrium in stability formulation (Abramson et al. 2002). However, true moment equilibrium depends on the satisfaction of force equilibrium. Without force equilibrium, there is actually no moment equilibrium. Force equilibrium is, however, totally independent of the moment equilibrium. For methods which satisfy only the moment equilibrium, the factor of safety actually depends on the choice of the moment point. For circular failure surface, it is natural to choose the centre of the circle as the moment point, and it is also well known that the Bishop method can yield very good result even when the force equilibrium is not satisfied. Fredlund et al. (1992) have discussed the importance of the moment point on the factor of safety for the Bishop method, and the Bishop method cannot be applied to general slip surface because the unbalanced horizontal force will create different moment contribution to different moment point. Baker (1980) has pointed out that for the “rigorous” methods, the factor of safety is independent of the choice of the moment point. Cheng et al. (2008a) have however found that the mathematical procedures to evaluate the factor of safety may be affected by the choice of the moment point. Actually, some commercial programs allow the users to choose the moment point for analysis. The double QR method by Cheng (2003) is not affected by the choice of the moment point in the analysis and is a very stable solution algorithm.

2.6.7. Use of Soil Nail/Reinforcement Soil nailing is a slope stabilization method by introducing a series of thin elements called nails to resist tension, bending and shear forces in the slope. The reinforcing elements are usually made of round cross-section steel bars. Nails are installed sub-horizontally into the

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soil mass in a pre-bore hole, which is fully grouted. Occasionally, the initial portions of some nails are not grouted but this practice is not commonly adopted. Nails can also be driven into the slope, but this method of installation is uncommon in practice. The fundamental principle of soil nail is the development of tensile force in the soil mass and renders the soil mass stable. Although only tensile force is considered in the analysis and design, soil nail function by a combination of tensile force, shear force and bending action which is difficult to be analyzed. The use of finite element by Cheng et al. (2007b) has demonstrated that the bending and shear contribution to the factor of safety is generally not significant, and the current practice in soil nail design should be good enough for most cases. Nails are usually constructed at an angle of inclination from 10 to 20. For ordinary steel bar soil nail, a thickness of 2mm is assumed as the corrosion zone so that the design bar diameter is totally 4mm less than the actual diameter of the bar according to Hong Kong practice. Nail is usually protected by galvanization, paint, epoxy and cement grout. For critical location, protection by expensive sleeving similar to that in rock anchor may be adopted. Alternatively, GFRP, CFRP may be used for soil nail which are currently under consideration by the author. The practical imitations of soil nails include: 1) Lateral and vertical movement may be induced from excavation and the passive action of soil nail is not as effective as the active action of anchor. 2) Difficulty in installation under some and groundwater condition. 3) Suitability of soil nail in loose fill is doubted by some engineers – stress transfer between nail and soil is difficult to be established. 4) Collapse of drill hole before the nail is installed can happen easily in some ground conditions. 5) For very long nail hole, it is not easy to maintain the alignment of the drill hole. There are several practices in the design of soil nails. One of the precautions in the adoption of soil nails is that the factor of safety of a slope without soil nail must be greater than 1.0 if soil nail is going to be used. This is due to the fact that soil nail is a passive element, and the strength of the soil nails cannot be mobilized until soil tends to deform. The effective nail load is usually taken as the minimum of: a) the bond strength between cement grout and soil; b) the tensile strength of the nail, which is limited to 55% of the yield stress in Hong Kong, and 2mm sacrificial thickness of bar surface is allowed for corrosion protection; c) the bond stress between the grout and the nail. In general, only factors (a) and (b) are controlling in the actual design. The bond strength between cement grout and soil are usually based on one of the following criteria: a) The effective overburden stress between grout and soil control the unit bond stress on soil nail, and it is estimated from the formula (c’D+2Dv’tan’) for Hong Kong practice, while Davis method (Shen et al. 1981) allows an inclusion of angle of inclination, and D is the diameter of the grout hole. A safety factor of 2.0 is commonly applied to this bond strength in Hong Kong. During the calculation of the

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bond stress, only the portion behind the failure surface (effective bond length) is taken into the calculation. b) Some laboratory tests suggest that the effective bond stress between nail and soil is relatively independent of the vertical overburden stress. This is based on the stressredistribution after the nail hole is drilled and the surface of the drill hole should be stress free. The effective bond load will then be controlled by the dilation angle of the soil. Some of the laboratory tests in Hong Kong have shown that the effective overburden stress is not important for the bond strength. On the other hand, some field tests in Hong Kong have shown that the nail bond strengths depend on the depth of embedment of the soil nails. It appears that the bond strength between cement grout and soil may be governed by the type of soil, method of installation and other factors, and the bond strength may be dependent on the overburden height and time in some cases, but this is not a universal behaviour. c) If the bond load is independent of the depth of embedment, the effective nail load will then be determined in a proportional approach shown in Figure 2.8. For a soil nail of length L, bonded length Lb and total bond load Tsw., Le for each soil nail and Tmob for each soil nail are determined from the formula below: For slip 1: Tmob = Tsw

(2.72)

In this case the slip passes in front of bonded length and the full magnitude is mobilized to stabilize the slip For slip 2: Tmob = Tsw x (Le/Lb)

(2.73)

In this case the slip intersects the bonded length and only a proportion of the full magnitude provided by the nail length behind the slip is mobilised to stabilise the slip. The effective nail load is usually applied as a point load on the failure surface in the analysis. Some engineers however model the soil nail load as a point load at the nail head or as a distributed load applied on the ground surface. In general, there is no major difference in the factors of safety from these minor variations in treating the soil nail forces.

Figure 2.8. Definition of effective nail length in the bond load determination.

The effectiveness of soil nail can be illustrated by adding 2 rows of 5m length soil nails inclined at an angle of 15 to the problem shown in Figure 2.4 which is shown in Figure 2.9. The x-ordinates of the nail heads are 7.0 and 9.0. The total bond load is 40 kN for each nail

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which is taken to be independent of the depth of embedment, while the effective nail loads are obtained as 27.1 kN and 24.9 kN considered by a simple proportion as given on Figure 2.8. The results of analysis as shown in Table 2.4 have illustrated that: (1) Swedish method is a conservative method in most cases; (2) Janbu rigorous method (1973) is more difficult to converge as compared with other methods. It is also noticed that when external load is present, there are greater differences between the results from different methods of analysis.

Figure 2.9. Two rows of soil nail are added to the problem in Figure 2.4.

Table 2.4. Factor of safety for the failure surface shown in Figure 2.4 (correction factor is applied in the Janbu simplified method) F

Bishop 1.807

Janbu simplified 1.882

Janbu rigorous fail

Swedish 1.489

Load factor 1.841

Sarma 1.851

Morgenstern Price 1.810

2.6.8. Failure to Converge Fail to converge during the solution of the factor of safety is sometimes found for ‘rigorous’ methods which satisfy both force and moment equilibrium. If this situation is found, the initial trial factor of safety can be varied and convergence is sometimes achieved (but no guarantee). Alternatively, the double QR method by Cheng (2003) can be used, as this is the ultimate method in the solution of the factor of safety. If no physically acceptable answer can be determined from the double QR method, then there is no physically acceptable result for the specific method of analysis with the given assumptions (f(x) or thrust line). Under such condition, the simplified methods can be used to estimate the factor of safety or the extremum principle by Cheng et al. (2010) may be adopted to determine the factor of safety. The convergence problem of the “rigorous” method will be studied in more details in later sections, and there are more case studies which are provided in the user guide of SLOPE 2000. The seriousness of convergence problem can be illustrated by the use of double QR method in SLOPE 2000 as compared with the iteration analysis in two famous commercial programs. As shown in Tables 2.5 and 2.6, 30 prescribed non-circular failure surfaces with soil nails are considered using the Spencer analysis (f(x)=1). SLOPE 2000 by the author can

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give satisfactory answers for all the 30 cases (no convergence requirement in double QR analysis), but the two commercial programs fail to give answers in many cases. From the author’s experience, if the failure surfaces are relatively non-smooth with heavy external loads, many commercial programs fail to converge in the analysis easily. Such failure to converge is however purely false alarm, as SLOPE 2000 using double QR method is able to give satisfactory results in all the 30 test cases as well as many other problems. If double QR method cannot give an answer to a problem, then by nature of the problem, there is no answer to the problem with the given interslice force function f(x). The engineers can however vary f(x) and get an answer in most cases, but such tuning will not be automatic by nature, as the form of f(x) for which an answer exist cannot be auto-evaluated by nature. The results as shown have clearly illustrated the limitations of the classical limit equilibrium formulation. Table 2.5. Comparisons of the results from program A with SLOPE 2000 for 30 non-circular failure surfaces with soil nail, using Spencer analysis Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Program A Fail Fail Fail Fail Fail Fail Fail Fail 1.586 Fail Fail Fail Fail Fail Fail 1.7584 Fail Fail Fail Fail Fail Fail Fail Fail Fail 1.266 1.471 1.232 1.493 1.357

Slope 2000 1.2401 1.439 1.1308 1.3295 1.494 1.4826 1.2726 1.2446 1.5722 1.1457 1.8187 2.396 1.9984 1.5873 1.4557 1.7584 1.3991 1.5732 1.495 1.4074 1.3438 1.5323 1.7703 1.9606 1.2796 1.264 1.473 1.2367 1.4761 1.3221

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Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Program B Fail 1.407 1.1 1.382 Fail Fail 1.303 Fail 1.575 1.103 1.753 Fail Fail Fail Fail 1.542 1.381 1.527 Fail Fail 1.362 1.572 1.771 1.829 Fail 1.275 1.483 1.238 1.483 1.373

Slope 2000 1.2401 1.439 1.1308 1.3295 1.494 1.4826 1.2726 1.2446 1.5722 1.1457 1.8187 2.396 1.9984 1.5873 1.4557 1.7584 1.3991 1.5732 1.495 1.4074 1.3438 1.5323 1.7703 1.9606 1.2796 1.264 1.473 1.2367 1.4761 1.3221

For the test cases as given Tables 2.4 and 2.5 as well as other test cases not shown in these two tables, interested readers can obtain the input files for the two commercial programs and for SLOPE 2000 for study from the author at [email protected]. These test cases can illustrate the difficulties in ensuring convergence as well as numerical problems that may come up from the computation in many commercial programs using iteration analysis.

2.6.9. Location of the Critical Failure Surface The minimum factor of safety as well as the location of the critical failure surface are required for the proper design of a slope. For a homogeneous slope with a simple geometry and no external load, the log-spiral failure surface will be a good solution for the critical

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failure surface. In general, the critical failure surface for a sandy soil with a small c’ value and high ’ will be close to the ground surface while the critical failure surface will be a deep seated one for a soil with a high c’ value and small ’. With the presence of external vertical load or soil nail, the critical failure surface will generally drive the critical failure surface deeper into the soil mass. For a simple slope with a heavy vertical surcharge on top of the slope (typical abutment problem), the critical failure surface will be approximately a twowedge failure from the non-circular search. This failure mode is also specified by the German code for abutment design. For simple slope without any external load or soil nail, the critical failure surface will usually pass through the toe. Based on the above characteristics of the critical failure surface, the engineers can manually locate the critical failure surface with ease for simple problem. The use of the factor of safety from the critical circular or log-spiral failure surface (Frohlich 1953, Chen 1975) which will be slightly higher than that from the non-circular failure surface is also adequate for simple problems. For complicated problems, the above guidelines may not be applicable, and it will be tedious to carry out the manual trial and error in locating the critical failure surface. Automatic search for the critical circular failure surface is available in nearly all of the commercial slope stability programs. A few commercial programs also offer the automatic search for non-circular critical failure surface with some limitations. Since the automatic determination of the effective nail load (controlled by the overburden stress) appears to be not available in most of the commercial programs, engineers often have to perform the search for the critical failure surface by manual trial and error, and the effective nail load is separately determined for each trial failure surface. To save time, only limited failure surfaces will be considered in the routine design. The author has found that reliance only on the manual trial and error in locating the critical failure surface may not be adequate, and the adoption of the modern optimization methods to overcome this problem will be discussed later.

2.6.10. 3D Analysis All failure mechanism is 3D in nature but 2D analysis is performed at present. The difficulties associated with true 3D analysis are: (1) sliding direction, (2) satisfaction of 3D force and moment equilibrium, (3) relating the factor of safety to the previous two factors, (4) great amount of computational geometrical calculations are required. At present, there are still many practical limitations in the adoption of 3D analysis, and there are only a few 3D slope stability programs which are suitable for ordinary use. Simplified 3D analysis for symmetric slope is available in SLOPE 2000 by Cheng, and true 3D analysis for general slope has been developed which is named as SLOPE3D. 3D slope stability analysis will be discussed in details in later section.

2.7. LIMIT ANALYSIS METHOD - DLO The limit analysis adopts the concept of an idealized stress-strain relation, i.e., the soil is assumed as a rigid perfectly plastic material with an associated flow rule. Without carrying out the step-by-step elasto-plastic analysis, the limit analysis can provide solutions to many

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problems. Limit analysis is based on the bound theorems of classical plasticity theory (Drucker et al., 1951; Drucker and Prager, 1952). The general procedure of limit analysis is to assume a kinematically admissible failure mechanism for an upper bound solution or a statically admissible stress field for a lower bound solution, and the objective function will be optimized with respect to the control variables. Early efforts of limit analysis were mainly made on using the direct algebraic method or analytical method to obtain the solutions for slope stability problems with simple geometry and soil profile (Chen, 1975). Since closed form solutions for most practical problems are not available, later attention has been shifted to employing the slice techniques in traditional limit equilibrium to the upper bound limit analysis (Michalowski, 1995; Donald and Chen, 1997). Limit analysis are based on two theorems: (a) the lower bound theorem, which states that any statically admissible stress field will provide a lower bound estimate of the true collapse load; and (b) the upper bound theorem, which states that when the power dissipated by any kinematically admissible velocity field is equated with the power dissipated by the external loads, then the external loads are upper bounds on the true collapse load (Drucker and Prager, 1952). A statically admissible stress field is one that satisfies the equilibrium equations, stress boundary conditions, and yield criterion. A kinematically admissible velocity field is one that satisfies strain and velocity compatibility equations, velocity boundary conditions, and the flow rule. When combined, the two theorems provide a rigorous bound on the true collapse load. Application of the lower bound theorem usually proceeds as stated next: (a) First, a statically admissible stress field is constructed. Often it will be a discontinuous field in the sense that we have a patchwork of regions of constant stress that together cover the whole soil mass. There will be one or more particular value of stress that is not fully specified by the conditions of equilibrium. (b) These unknown stresses are then adjusted so that the load on the soil is maximized but the soil remains unyielded. The resulting load becomes the lower bound estimate for the actual collapse load. Stress fields used in lower bound approaches are often constructed without a clear relation to the real stress fields. Thus, the lower bound solutions for practical geotechnical problems are often difficult to find. Collapse mechanisms used in the upper bound calculations, however, have a distinct physical interpretation associated with actual failure patterns and thus have been extensively used in practice.

2.7.1. Lower Bound Approach The lower bound method is a stress field approach for finding a bound solution to the actual limit load, i.e., the load is calculated from a statically admissible stress field which will be a lower bound and on the safe side. This concept is similar to the slip line theory, and the similarities between the two techniques lie in: (1) both methods address the collapse load directly without considering the deformation and (2) the information of the yield criterion is introduced in the formulation of the problem in the first place of the formulation. In this sense, it is reasonable to consider the slip line theory as the early development of the lower bound theorem. Kotter (1903) was believed to be the first to derive the slip-line equations for the plane deformation. The first analytical closed form solution was obtained by Prandtl (1920) who

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developed the solution with a singular point with a pencil of straight slip-lines passing through it. These results were later applied by Reissner (1924) and Melan (1938) to the bearing capacity of footings on a weightless soil. Considering the self-weight of the soil would render the solution so complicated that a numerical procedure is required. Numerical solutions to the slip line equations were obtained by finite difference method by Sokolovskii (1960; 1965), who studied a number of interesting geotechnical problems such as the bearing capacity of footings and slopes as well as pressure of a fill on retaining walls. De Jong (1957) adopted a different approach and developed a graphical procedure for the solutions of the slip line equations. Other forms of approximate solutions include the application of perturbation methods (Spencer 1961) and series expansion methods. More recently, numerical results considering seismic effects, axisymmetry etc. were given by Cheng (2003b; 2005a; 2007d) using slip-line method for bearing capacity and lateral earth pressure problems. The application of the conventional analytical limit analysis was usually limited to simple problems. Numerical methods therefore have been employed to compute the lower and upper bound solutions for the more complex problems. The first lower bound formulation based on the finite element method was proposed by Lysmer (1970) for plane strain problems. The approach uses the concept of finite element discretization and linear programming. The soil mass is subdivided into simple three-node triangular elements where the nodal normal and shear stresses are taken as the unknown variables. The stresses are assumed to vary linearly within an element, while stress discontinuities are permitted to occur at the interface between adjacent triangles. The statically admissible stress field is defined by the constraints of the equilibrium equations, stress boundary conditions, and linearized yield criterion. Each nonlinear yield criterion is approximated by a set of linear constraints on the stresses that lie inside the parent yield surface, thus ensuring that the solution is a strict lower bound. This lead to an expression for the collapse load which is maximized subjected to a set of linear constraints on the stresses. The lower bound load can be solved by optimization, using the techniques of linear programming. Other investigations have worked on similar algorithms (Anderheggen and Knopfel, 1972; Bottero et al., 1980). The major disadvantage of these formulations is the linearization of the yield criterion which generates a large system of linear equations and requires excessive computational times, especially if the traditional simplex or revised simplex algorithms are used (Sloan, 1988a, 1988b). Therefore, the scope of the early investigations is mainly limited to small-scale problems. Efficient analyses for solving numerical lower bounds by the finite element method and linear programming method have been developed recently (Bottero et al., 1980; Sloan, 1988a and 1988b). The key concept of these analyses is the introduction of an active set algorithm (Sloan, 1988b) to solve the linear programming problem where the constraint matrix is sparse. Sloan (1988b) has shown that the active set algorithm is ideally suited to the numerical lower bound formulation and can solve a large-scale linear programming problem efficiently. A second problem associated with the numerical lower bound solutions occurrs when dealing with statically admissible conditions for an infinite-half space. Assdi and Sloan (1990) solved this problem by adopting the concept of infinite elements, and hence obtained rigorous lower bound solutions for general problems. Lyamin and Sloan (1997) proposed a new lower bound formulation which used linear stress finite elements, incorporating nonlinear yield conditions, and exploiting the underlying convexity of the corresponding optimization problem. They showed that the lower bound

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solution could be obtained efficiently by solving the system of nonlinear equations that define the Kuhn-Tucker optimality conditions directly. Recently, Zhang (1999) presented a lower bound limit analysis in conjunction with another numerical method - the rigid finite element method (RFEM) to assess the stability of slopes. This formulation satisfies both static and kinematical admissibility of a discretized soil mass without requiring any assumption. The nonlinear programming method is employed to search for the critical slip surface.

2.7.2. Upper Bound Approach Generating a kinematically admissible velocity field is comparatively easier than generating a statically admissible stress field. Therefore, in the literature, there are a number of variants of the applications of upper bound approaches for geotechnical problems. Each variant differs from the others primarily in the approximation of the velocity field. Implementation of the upper bound theorem is generally carried out as follows: (a) First, a kinematically admissible velocity field is constructed. No separations or overlaps should occur anywhere in the soil mass. (b) Second, two rates are then calculated: the rate of internal energy dissipation along the slip surface and discontinuities that separate the various velocity regions, and the rate of work done by all the external forces, including gravity forces, surface tractions and pore water pressures. (c) Third, the above two rates are set to be equal. The resulting equation, called energy-work balance equation, is solved for the applied load on the soil mass. This load would be equal to or greater than the true collapse load. The first application of upper bound limit analysis to slope stability problem was by Drucker and Prager (1952) in finding the critical height of a slope. A failure plane was assumed, and analyses were performed for isotropic and homogeneous slopes with various angles. In the case of a vertical slope, it was found that the critical height obtained by the upper bound theorem was identical with that obtained by the limit equilibrium method. Similar studies have been done by Chen and Giger (1971) and Chen (1975). However, their attention was mainly limited to a rigid body sliding along a circular or log-spiral slip surface passing both through the toe and below the toe in cohesive materials. The stability of slopes was evaluated by stability factor, which could be minimized using an analytical technique. A comprehensive study of the conventional upper bound limit analysis in soil mechanics is presented in the work by Chen (1975), in which simplified velocity fields consisting of rigid blocks sliding along assumed failure surface are considered. The collapse load is obtained by equating the external work done to the internal power dissipation that is assumed to occur along the slip surface. Results obtained with the simplified upper bound are found with high accuracy for simple stability problems of homogeneous soil condition if the slip surfaces are carefully chosen. Later, Chen and Liu (1990) extended the technique to take into account of seismic loading and anisotropy soil properties. Karel (1977a, 1977b) presented an energy method for soil stability analysis. The failure mechanisms used in the method included: (a) A rigid zone with a planar or a log-spiral transition layer, (b) A soft zone confined by plane or log-spiral surfaces, and (c) A composed failure mechanism consisting of rigid and soft zones. The internal dissipation of energy occurred along the transition layer for rigid zone, within zone and along transition layer for

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the soft zone. However, no numerical technique was proposed to determine the least upper bound of the factor of safety. Izbicki (1981) presented an upper bound approach to slope stability analysis. A translational failure mechanism, which was confined by a circular slip surface in the form of rigid blocks similar to the traditional slice method was used. The factor of safety was determined by an energy balance equation and the equilibrium conditions of the field of force associated with the assumed kinematically admissible failure mechanism. However, no numerical technique was provided to search for the least upper bound of factor of safety in the approach. Michalowski (1995) presented an upper bound (kinematical) approach of limit analysis in which the factor of safety for slopes is derived associated with a failure mechanism in the form of rigid blocks analogous to the vertical slices used in traditional limit equilibrium methods. A convenient way to include pore water pressure has also been presented and implemented in the analysis of both translational and rotational slope collapse. The strength of the soil between blocks was assumed explicitly that it was taken as zero or its maximum value set by the Mohr-Coulomb yield criterion. Inspired by the slicing techniques in the limit equilibrium method, Donald and Chen (1997) introduced the so called “multi-block” technique in which the sliding soil mass (traditionally treated as a whole rigid block in Chen’s work (Chen 1975; Chen and Liu 1990) is divided into a number of smaller blocks. Velocity discontinuities are permitted between the linear interfaces between those smaller blocks. The collapse load is sought similarly from work balance equation. Following this line of research, Wang (2001) extended the multiblock approach to three-dimensional analysis and applied it to the collapse load of material obeying non-associated flow rule. Michalowski (2001) proposed a three-dimensional failure mechanism using blocks with conical surfaces in the determination of the bearing capacity of square and circular footing. These methods suffer from the inherent limitation as in the conventional upper bound analysis in that the failure mechanism needs to be predefined. To overcome this difficulty, Chen (2004) has employed the finite element discretisation for the generation of the blocks in which each element is viewed as a rigid block. Velocity discontinuities are allowed between the blocks and energy is dissipated only along the velocity discontinuities between two adjoining elements. This general version of the multiblock technique is named as the “rigid-finite-element-based upper bound analysis.” The resulting problem in their formulation is then solved by SQP method. (Chen 2004). Using the finite element discretisation for the generation of velocity field is more robust and general; however, the accuracy of the solution obtained is sensitive to the mesh used for the analysis.

2.7.3. Discontinuity Layout Optimization Discontinuity layout optimization (DLO) is a novel limit analysis based slope stability analysis method which is proposed by Smith and Gilbert (2007). DLO can identify a critical layout of lines of discontinuity, which form a failure, by using the rigorous mathematical optimization techniques. These lines of discontinuity are virtually the slip lines in twodimensional geotechnical stability problems. DLO can be used to identify critical translational sliding block failure mechanisms which are familiar to the method of slices solution for slope stability problem. This traditional method, however, works only with failure soil mass in a

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few sliding blocks. DLO can overcome this limitation by the lines of discontinuity which define the boundaries between the rigid moving blocks of material being divided into a large number of sliding blocks. The failure mechanisms are actually the upper bound theory corresponding to a load factor or strength factor. The load factor or strength factor obtained from the trial process for any arbitrarily assumed mechanism is increased if necessary in order to find the exact solution which brings a slope to collapse. Based on this theory, DLO procedure, in essence, can be applied to a wide range of geotechnical stability problems by trial and error process. In the DLO procedure, the discontinuities are assumed as variables that considers the relative displacement along the discontinuities. By using the relative displacement, compatibility can be checked at each node by applying the simple linear equation involving these variables. Finally, an objective function can be defined in accordance with the energy dissipation along all discontinuities when the failure occurs. A linear optimization problem can then be determined by a linear function of the slip displacement variables. DLO procedure expresses the limit analysis problem entirely in terms of lines of discontinuity instead of elements as in the classical continuum problem (Smith and Gilbert 2007). Using DLO, a large number of potential discontinuities are set up at different orientations; while the continuum based element formulations, discontinuities are typically restricted to lie only at the edges of elements. With the use of modern optimization algorithms, an optimized solution can be achieved easily. After the initial success by Smith and Gilbert (2007), there are different works in DLO by Clarke et al. (2013), Smith and Gilbert (2013), Bauer and Lackner (2015), Al-Defae and Knappett (2015), Leshchinsky (2015), Vahedifard et al. (2014), Leshchinsky and Ambauen (2015). The original DLO formulation suffers from the limitation that only the translation mechanism can be considered. In view of such limitation, Gilbert et al. (2010) and later Smithy and Gilbert (2013) have extended the DLO formulation to cover the rotational formulation. Since DLO is actually a numerical form of limit analysis, the basic limitation of limit analysis is similar to that for DLO. As discussed above, the compatibility of the displacement of the nodes is important to achieve the potential critical mechanisms. There are three cases to demonstrate the equilibrium and compatibility conditions at a node in Figure 2.10. Figure 2.10a consider an unloaded node in a truss: 5

(2.74)

5

(2.75)

i qi  0  i 1 i qi  0  i 1

where i = cosi and i = sini. Compatibility at node in slip is considered for translation only in Figure 2.10b: 5

i si  0  i 1

(2.76)

Slope Stability Analysis Methods 5

i si  0  i 1

57 (2.77)

where si is the shear displacement for i discontinuity. Compatibility at node in slip considers both the translation and dilation in Figure 2.10c: 5

i si   n  i 1

i i

5

i si   n  i 1

i i

0

(2.78) 0

(2.79)

where n is normal displacement for i discontinuity.

Figure 2.10. Equilibrium and compatibility conditions at a node: (a) Equilibrium at (unloaded node in a truss; (b) compatibility at node in slip mechanism (translation only); (c) compatibility at node in slip mechanism (translation and dilation) (from Smith and Gilbert, 2007).

2.7.4. Some Studies on Discontinuity Layout Optimization The details of the DLO are covered by Smith and Gilbert, and this method is currently available in the program LimitState GEO. Interested readers should consult this commercial program about the operations and the underlying principle. It should be noted that there is only one commercial program for DLO up to the present, and 3D DLO is still not available for practical purposes, due to various technical problems and limitations. A typical output from DLO is given in Figure 2.11 for reference. It should be noted that both the factor of safety and critical slip surface from DLO match very well with the results from LEM or SRM for this case, but the author has also found that there are many cases for which noticeable differences between the results from DLO and LEM/SRM are found, particularly when some special soil parameters are defined.

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Yung Ming Cheng

Figure 2.11. An output from DLO analysis.

With reference to the standard slope stability problem by Cheng et al. (2007) as shown in Figure 2.12, 23 cases with different soil parameters are studied. It is noticed that the results from DLO and LEM match very well except for several cases (cases 6, 11, 16, 21, 22, 23). On the other hand, the results from SRM1 (zero dilation angle) and SRM2 (dilation angle=friction angle) match even better with that from the results using LEM. Actually, the author has carried out many case studies with the DLO and SRM and has found that the DLO at present has more problem cases than SRM, particularly for some special combinations of soil parameters. For normal cases, the differences between DLO and other methods are actually small.

Figure 2.12. Geometry of a standard problem by Cheng et al. (2007b).

59

Slope Stability Analysis Methods Table 2.6. Comparisons between the results from LEM, SRM and DLO

1

Cohesion c' (kPa) 2

Friction angle Φ' (degree) 5

FOS (LEM) 0.25

2

2

15

3

2

4

0.26

FOS by SRM (SRM1) 0.25

FOS by SRM (SRM2) 0.25

0.50

0.51

0.52

0.50

25

0.74

0.76

0.77

0.75

2

35

1.01

1.04

1.07

1.03

5

2

45

1.35

1.40

1.42

1.36

6

5

5

0.41

0.44

0.41

0.41

7

5

15

0.70

0.73

0.72

0.71

8

5

25

0.98

1.00

1.02

0.99

9

5

35

1.28

1.31

1.33

1.30

10

5

45

1.65

1.70

1.71

1.69

11

10

5

0.65

0.71

0.66

0.65

12

10

15

0.98

1.04

1.00

0.995

13

10

25

1.30

1.35

1.34

1.33

14

10

35

1.63

1.70

1.69

1.65

15

10

45

2.04

2.09

2.09

2.07

16

20

5

1.06

1.22

1.12

1.12

17

20

15

1.48

1.61

1.51

1.50

18

20

25

1.85

1.96

1.87

1.87

19

20

35

2.24

2.34

2.26

2.27

20

20

45

2.69

2.79

2.72

2.74

21

5

0

0.20

0.24

0.22

0.22

22

10

0

0.40

0.47

0.44

0.44

23

20

0

0.80

0.95

0.88

0.88

Case

FOS (DLO)

Yu et al. (1998) have given a very detailed comparison between the use of limit analysis and LEM, and it is found that the results from the two methods are similar and comparable in most cases for relatively simple problems. Recently, DLO has been adopted for slope stability analysis by Leshchinsky and Ambauen (2015), and it is found that the results by DLO and LEM are comparable in general. Leshchinsky and Ambauen (2015) have however found some cases for which there are noticeable differences between the DLO and LEM, and they have concluded that DLO requires less assumption on the location of collapse, and therefore may be more preferable than LEM, especially for complex, yet realistic geotechnical problems. After reviewing the examples by Leshchinsky and Ambauen (2015), the author tends to disagree with the results and comments by Leshchinsky and Ambauen (2015). There are some limitations in the works by Leshchinsky and Ambauen (2015) which include: 1) use of classical LEM method which are greatly affected by convergence problem (Cheng et al.

60

Yung Ming Cheng

2008, Cheng et al. 2010); 2) critical failure surface has not been determined (Figure 12 from Leshchinsky and Ambauen (2015) has only considered 151 surfaces); 3) interslice force function can be critical in complex problems. As discussed by Cheng (2003), Cheng et al. (2010) and Cheng et al. (2013a), these three problems can lead to relatively poor solution in some cases by the classical LEM, and the extremum principle by Cheng et al. (2010), Cheng et al. (2011b) and Cheng et al. (2013a) have overcome these problems and can provide solutions similar to some classical plasticity problems which are not possible with the classical LEM. A fair comparison and commentary on these methods must be based on reliable and robust analyses that identify the differences between DLO and LEM. With reference to Figure 2.13 which is Figure 5a by Leshchinsky and Ambauen (2015), the soil parameters are unit weight=19 kN/m3, c’=28 kPa and ’=20. The critical result by DLO pass below the toe of the slope at the right hand side of Figure 2.13 by Leshchinsky and Ambauen (2015). On the other hand, the critical result by the author using the heuristic optimization method and Spencer method developed by Cheng et al. (2007a) and Cheng and Lau (2014) pass through the toe of the slope. The critical result by Baker (1980) also pass through the toe of the slope while the critical result by Krahn and Fredlund (1997) (not shown for clarity) is similar to that by DLO but extends further to the right of the toe. The result by Krahn and Fredlund (1997) is not determined by the use of advanced optimization algorithm, and the adequacy of the result has not been confirmed. The author has tried several updated commercial programs and have obtained results similar to that by the author as shown in Figure 2.13. Since the friction angle of the soil is 20 which is not a small value, the critical result by limit analysis should pass through the toe of slope as demonstrated by Chen (1975) using limit analysis. In views of the above discussion, the author will suggest that the result by DLO is not the critical solution for such a simple case which is surprising to the author. Leshchinsky-Ambauen (2015) Spencer Method

Figure 2.13. Comparisons of DLO and LEM for Figure 5a by Leshchinsky and Ambauen (2015), FOS=2.03 by DLO and 1.94 by Spencer method.

With reference to Figure 2.14 where there is a 0.5m thickness of soft material for soil layer 2, the soil parameters are unit weight=19 kN/m3, c’=28 kPa and ’=20 for layer 1 and unit weight=19 kN/m3, c’=0 kPa and ’=10 for soil layer 2. The results by DLO and the author are very similar except that the critical result by the author lies at the bottom of the soft layer while the results by Leshchinsky and Ambauen (2015) lie at the top of the soft layer. The critical results by Baker (1980) and Krahn and Fredlund (1997) are also at the bottom of the soft band but are mistaken to be at the top of the soft band by Leshchinsky and Ambauen (2015). The author has reduced the thickness of the soft layer to 1mm, and the factor of safety as well as the critical result will then be equal to that by Leshchinsky and Ambauen (2015),

Slope Stability Analysis Methods

61

Baker (1980), and Krahn and Fredlund (1997). When the author increases the thickness of the soft layer to 1.5m, the critical result will still lie at the bottom of the soft layer with a factor of safety 1.14. Since the shear strength parameters at soil layer 2 are low, the weight of the soil tends to push the soft material to the right so that the critical slip surface should lie within the soft band, and the results by the author are more reasonable as compared with other results. As discussed by Cheng (2007) and Cheng et al. (2012), the presence of a soft band is mathematically equivalent to a Dirac function, for which many optimization algorithms fail to work. The domain transformation technique by Cheng (2007) and the coupled optimization algorithm by Cheng et al. (2012) have effectively overcome this problem without any special precaution required by the engineers in the analysis. In Figure 2.15 which is same as that for Figure 2.14 with a pore pressure ratio 0.25 (Figure 5d by Leshchinsky and Ambauen, (2015), the critical result by Leshchinsky and Ambauen (2015) lies at the top of the soft band while the critical results by the author, Baker (1980), and Krahn and Fredlund (1997) (mistaken to be at the top of the soft band by Leshchinsky and Ambauen (2015) lie at the bottom of the soft band, and the inability to locate the critical result for a soft band by DLO is clearly illustrated. In Figure 2.16 which is same as that for Figure 2.14 with a prescribed water table (Figure 5f by Leshchinsky and Ambauen, (2015), the critical result again lie at the bottom of the soft band which are different the critical result by Leshchinsky and Ambauen (2015) (again Baker 1980 and Krahn and Fredlund (1997) also get critical result at bottom of soft band but are mistaken to be top of soft band by Leshchinsky and Ambauen 2015). In Figure 2.17 (Figure 12 by Leshchinsky and Ambauen 2015), there are great differences between the critical result by the author and Leshchinsky and Ambauen (2015). The soil parameters are unit weight=20 kN/m3, c’=0 kPa and ’=30 for soil layer 1, unit weight=19 kN/m3, c’=0 kPa and ’=45 for soil layer 2 and unit weight=19 kN/m3, c’=10 kPa and ’=0 for soil layer 3. On the left hand side of the critical slip surface by Leshchinsky and Ambauen (2015), there is a very sudden change in the slope of the critical failure surface which seems unlikely to happen. At the right hand side of the critical slip surface by Leshchinsky and Ambauen (2015), the critical slip surface is nearly vertical, which is also highly unlikely, as the friction angle of soil layer 1 and 2 are 30 and 40 respectively with zero cohesive strength. When this same slip surface by Leshchinsky and Ambauen (2015) is considered with the M-P method using f(x)=sin(x), the author actually gets a factor of safety of 1.05, which is significantly greater than the result of 0.95 by Leshchinsky and Ambauen (2015). This problem is then reanalyzed by the author using LEM to locate the critical slip surface. For this problem, the use of f(x)=1 is poor in convergence, and the author gets a slightly different critical slip surface and a factor of safety of 0.97 by using f(x)=1.0. As mentioned by Cheng et al. (2008, 2010), f(x) can be critical in some cases which will affect the optimized solution. In this respect, the author has also adopted the extremum principle (Cheng et al. 2010, 20110, 2013) and have obtained a critical solution 0.915 which is close to that by using f(x)=sin(x). Since the extremum principle, which is practically equivalent to the lower bound approach, satisfies all the force and moment equilibrium with acceptable internal forces and is free from convergence problem, this result can be considered as a very good estimation of the critical result for the present problem. The author views that the critical result by Leshchinsky and Ambauen (2015) is possibly a local minimum instead of being the global minimum.

62

Yung Ming Cheng

Leshchinsky-Ambauen (2015) Spencer Method SLOPE 2000 - Version 2.5 TITLE: DESCRIPTION: DATE: Soil Profile Phreatic WT Perch WT

a

Bedrock Level Soil Nail

24 External Load

22

Failure Surface

20

Search Range Left boundary

18 16

Search Range Right boundary

14 12 10

Line of Thrust (Janbu Rigorous)

A B Bedrock

8

2D Morgenstern-Price

6 4

min.FOS =

1.1381

2

lambda =

0.1470

0 0

5

10

15

20

25

30

35

40

45

50

b Figure 2.14. Comparisons of DLO and LEM for Figure 5b by Leshchinsky and Ambauen (2015), FOS=1.28 by DLO and 1.21 by Spencer method. The thickness of the soft band has been increased to 1.25m for the result in Figure 2b with FOS=1.14.

Leshchinsky-Ambauen (2015) Spencer Method

Figure 2.15. Comparisons of DLO and LEM for Figure 5d by Leshchinsky and Ambauen (2015), FOS=1.01 by DLO and 0.99 by Spencer method.

Slope Stability Analysis Methods

63

Leshchinsky-Ambauen (2015) Spencer Method

Figure 2.16. Comparisons of DLO and LEM for Figure 5f by Leshchinsky and Ambauen (2015), FOS=1.14 by DLO and 1.07 by Spencer method.

Limit Analysis (2015) M-P

Figure 2.17. Comparisons of DLO and LEM for Figure 12 by Leshchinsky and Ambauen (2015), FOS=0.95 by DLO and 1.0 by Spencer method by Leshchinsky and Ambauen (2015) using 151 trial surface, 0.92 by M-P method using f(x)=sin(x) and 0.97 by f(x)=1.0 by the author using about 20000 trials with simulated annealing optimization method.

The author has adopted an accuracy of 0.001 in all the global optimization search in the present study, and the global minima of each example has been tested with different optimization algorithms for confirmation. Based on the above case studies, it can be concluded that some of the past reported results in literature which are not optimized with the modern optimization algorithms may not be reliable enough for comparisons. In particular, for the presence of a soft band which is a difficult problem, the present study and the works by Cheng (2007b), Cheng et al. (2012) have demonstrated that great care must be taken in order to obtain a good result. Furthermore, as a relatively new computational method, DLO has been demonstrated to be affected by the soft band or local minima problem. Overall, the author views that the problems presented in this section are not fundamental deficiencies of DLO. Instead, they highlight the limitations of the numerical technique in implementing the DLO up to the present moment. With refined and improved numerical technique coupled with DLO, the author expects that better results will be produced by DLO in the future. On the other hand, it is dangerous to compare the advantages and limitations of different stability methods based on old results or computer programs which may have various limitations. Some of the comments in previous literature are possibly distorted by the limitations of the computational technique in computer programs instead of being the actual comparisons of different stability analysis methods.

64

Yung Ming Cheng Table 2.7. FOS for different nodal number and tolerance of analysis (c’=0, ’=25for the soft band) for the problem in Figure 2.18

Case 1 2 3 4

Case 1 2 3 4

Case 1 2 3

28m domain size, solution tolerance 0.01, different nodal density FOS difference with nodal No. FOS by DLO FOS by LEM LEM(DLO %) 250 1.356 0.927 -46.28 500 1.069 0.927 -15.32 1000 1.082 0.927 -16.72 2000 1.055 0.927 -13.81 28m domain size, nodal density 500, different solution tolerance FOS difference with solution tolerance FOS by DLO FOS by LEM LEM(DLO %) 0.01 1.069 0.927 -15.32 0.001 1.069 0.927 -15.32 0.004 1.069 0.927 -15.32 0.005 1.069 0.927 -15.32 solution tolerance 0.01, nodal density 500, different domain size FOS difference with Domain Size (m) FOS by DLO FOS by LEM LEM(DLO %) 28 1.069 0.927 -15.32 20 1.093 0.927 -17.91 12 1.025 0.927 -10.57 28,15

20,15

Soil1 28,10 8,8

28,9.5

8,7.5 Soil2

0,5 y

5,5 5,4.5

8,7.1 Soil3

0,0 x

28,0

Figure 2.18. A slope problem with a soft band as discussed by Cheng et al. (2007b).

For the problem with a soft band at soil layer 2 as discussed by Cheng et al. (2007b), surprising results are again obtained by DLO. The unit weight of the soils are 19 kN/m3, and c’=20 kPa and ’=35 for soil layer 1, c’=0 kPa and ’=25 for soil layer 2 and c’=10 kPa and ’=35 for soil layer 3. As discussed by Cheng et al. (2007a), it appears that some SRM programs are affected by the size of the solution domain. The factor of safety for LEM is obtained as 0.927 by the Spencer method by Cheng et al. (2007b), and this value lie within

Slope Stability Analysis Methods

65

the SRM1 and SRM2 results by Plaxis and the new version of Phase (8.0). On the other hand, the factor of safety appears to be highly dependent on the nodal number adopted in the analysis. Even if 2000 nodal number is adopted, the factor of safety from DLO still appears to be unsatisfactory which is given in Table 2.7. The results by DLO are higher than those by LEM or SRM under all cases in Table 2.7, and the differences are not minor. Surprisingly, the critical failure surface from DLO as shown in Figure 2.19 is similar to that by LEM or SRM (Cheng et al. 2007b). From Table 2.7, it can be concluded that the most influential factor in a proper DLO analysis is the nodal number.

Figure 2.19. Critical failure surface from DLO for the problem in Figure 2.18.

If the third layer of soil instead of the second layer of soil is a soft material, the factor of safety has been established to be 1.29 from Spencer method, 1.27 from Extremum principle and 1.33 for SRM2 for all the SRM programs as discussed by Cheng et al. (2007b), and f(x) is relatively important for the present case (in general f(x) is not negligible if the friction angle is low). On the other hand, the result by DLO will approach the above factor of safety when the nodal number is large enough (Table 2.8). However, while the critical failure surfaces from LEM and SRM agree quite well as shown in Figure 2.20a and 2.20b, the critical failure surface from DLO extends further to the right in Figure 2.20c. To further examine these results, the author has found another local minimum 1.29 with the Spencer method for the failure surface as shown in Figure 2.21, which is very similar to that one by DLO as shown in Figure 2.20. The author view that a local minimum has been obtained from the DLO analysis. It should also be notes that the failure surfaces in Figure 2.20a and Figure 2.21 bear virtually the same factors of safety, and the differences between the two values are small so that it can be viewed that there are two global minimum for this problem. LEM can analyzed such problem easily while it takes more effort for SRM to detect the result in Figure 2.21. Actually, without the previous knowledge about the existence of another global minimum, engineers will miss the result in Figure 2.20 easily. For DLO, the failure surface as given by Figure 2.20a or 2.20b cannot be obtained even increasing nodal number as given in Table 2.8. In this respect, there are some inherent limitation in the present development of DLO.

66

Yung Ming Cheng Table 2.8. DLO, SRM and LEM analysis for the problem in Figure 2.18, when the third layer of soil has low shear strength

Case 1 2 3

DLO analysis of soft soil layer 3 with different nodal density (28 domain, solution tolerance 0.01) SLOPE 2000 - Version 2.5 FOS FOS FOS FOS difference FOS difference with TITLE: DESCRIPTION: by DLO by SRM2 by LEM with LEM(DLO%) LEM(SRM2%) DATE: 1.405 1.33 1.27 -8.91 -3.10 Soil Profile 1.358 1.33 1.27 -5.27 -3.10 Phreatic WT 1.35 1.33 1.27 -4.65 -3.10

nodal No. 250 500 1000

Perch WT soil1

14

Bedrock Level Soil Nail External Load

12

Failure Surface 10

soil2 soil3

Search Range Left boundary

8

Search Range Right boundary

6

Line of Thrust (Janbu Rigorous)

4 2D Morgenstern-Price min.FOS =

2

1.2633

lambda = -0.1000 0 0

5

10

15

a

B Figure 2.20 (Continued).

20

25

67

Slope Stability Analysis Methods

SLOPE 2000 - Version 2.5 TITLE: DESCRIPTION: DATE:

c

Figure 2.20. Critical failure surface for the problem in Figure 2.18 when the third layer of soil is soft material (a) result from Extremum principle; (b) result from SRM2; (c) result from DLO.

Soil Profile Phreatic WT Perch WT

soil1

14

Bedrock Level Soil Nail External Load

12

Failure Surface 10

soil2 soil3

Search Range Left boundary

8

Search Range Right boundary

6

Line of Thrust (Janbu Rigorous)

4 2D Morgenstern-Price 2

min.FOS =

1.8623

lambda =

0.4882

0 0

5

10

15

20

25

Figure 2.21. Another LEM global minimum for the problem in Figure 2.18, when the third layer of soil is soft material.

2.8. FINITE ELEMENT LIMIT ANALYSIS (FELA) In this section, the Finite-Element-based Limit Analysis (FELA) in which the stress and velocity field are discretised with finite dimensional element spaces as proposed by the author is discussed. Stability problems are formulated as the solution of convex optimizations in the present study. The Mohr-Coulomb (MC) yield criterion for plane strain analysis is

68

Yung Ming Cheng

transformed to second order conic constraints such that the resulting large scale optimization problem could be solved by the standard Second Order Cone Programming (SOCP) solvers with great efficiency. For MC materials under the full three-dimensional condition, hyperbolic smoothing in the meridional plane and round-off of corners in the octahedral plane can be applied to render the yield function to be everywhere differentiable. The NLP optimization stemming from the formulation is then solved efficiently by the primal-dual interior point algorithms. This technique will ensure that the developed method can achieve high accuracy while maintaining a fast solution suitable for complicated practical problems. Slip bands for most of the geotechnical problems are highly localized and the quality of the FELA solution is considerably affected by the configuration of the mesh. A variety of strategies for error estimation and mesh adaptation are investigated by the author. The residual error estimator and recovery-based error estimators that have been proposed for the FEM are tailored for the FELA. Comparisions on the performances of these error estimators are made to the yield criterion slackness and the deformation based method in this study. With an iterative local refinement technique, a coarse mesh without prior knowledge on the approximate solution can be used as the initial discretisation input. The optimal mesh distribution and the failure mechanism will be obtained as part of the solution, which enables solutions with satisfactory degree of accuracy to be obtained from a poor initial mesh and the need for the prior knowledge in generating high quality mesh is then removed, which is of great practical value for complicated problems. The method as developed in the present section can usually solve a problem within five to fifteen minutes (for several thousand elements), which is considered to be acceptable so that the present work can be useful to both theoretical study as well as practical application.

2.8.1. Literature Review of FELA Numerical lower bound analysis of the FELA in soil mechanics appears to be first carried out by Lysmer (1970). In Lysmer’s formulation, discretisation with three-node triangles is used and normal and shear stresses are taken as the optimal variables. To furnish the problem to be solvable by the linear programming method (LP), the Mohr-Coulomb yield criterion is linearised with an internal polyhedral approximation. This early formulation of the FELA was later improved by Anderheggen and Knopfel (1972), Pastor and Turgeman (1982) and Bottero et al. (1980). Important modifications include the use of Cartesian stresses to replace the normal and shear stresses as the optimizing variables, the introduction of extension elements to account for the semi-infinite domain and others. Despite these improvements, applications of the numerical lower bound analysis remained limited due to the difficulty in the solution of the large-scale LP optimization. A significant development of the FELA as LP is due to Sloan (1988a,b), who introduced active set algorithm for the solution of the resulting LP in the plane strain analysis. It was demonstrated that the active set algorithm was ideally suited for the optimization problems generated from the finite element formulation in the lower bound analysis. To consider the axial limit analysis, Pastor and Turgeman (1982) proposed a lower bound formulation in conjunction with the linear finite element for the axisymmetric problems of the material of von Mises or Coulomb yield criterion. Recently, similar problem is re-studied by Khatri and Kumar (2009) who extended the FELA with linear programming based on the

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assumption that the magnitude of the hoop stress remains close to the least compressive stress 3. In this formulation, the Mohr Coulomb yield criterion as well as the formulation is linearised for the plane strain formulation by Sloan (1988a,b). They proposed a formulation that only extra 3 constraints are required instead of 3 times the number of the sides of the polygon in formulation of Pastor and Turgeman (1982). With this formulation, the bearing capacity of circular footing on purely cohesive soils (Khatri and Kumar 2009) and for c-phi soils (Kumar and Khatri 2011) have been studied respectively. LP formulation of the FELA is simple in the concept and generally efficient in plane strain and axisymmetric analysis. However, extension of these formulations to threedimensional analysis is almost impractical as the linearization of the nonlinear yield criteria in the full three-dimensional analysis leads to a huge number of constraints. In order to treat the yield criteria as their natural form, Lyamin (1999) developed a lower bound formulation of the FELA based on the NLP for the MC material for both 2D and 3D analyses. The resulting optimization problems are then solved with a quasi-Newton algorithm that was originally proposed by Zouain et al. (1993). It is shown by comparison studies (Lyamin 1999) that the NLP formulation with a quasi-Newton algorithm is much more efficient and stable than its LP analogue as in Sloan (1988a,b). When applying the NLP in the FELA, yield criteria are required to be smooth such that gradients with respect to the Cartesian stresses are able to be computed everywhere. The widely accepted yield criteria for geomaterials such as the MC yield criterion and HB yield criterion are notorious for the singularities at the apex and corners, and thus smoothing is required in the application of the optimization algorithm. Lyamin (1999) applied a hyperbolic approximation in the meridional plane, and a round-off technique in the deviatoric plane for three dimensional analysis was first proposed by Sloan and Booker (1986). Threedimensional problems in the FELA with the MC material have been solved satisfactorily. A similar approach was applied to smooth the HB yield criterion in the limit analysis of rock masses by Merifield et al. (2006) for the plane strain analysis. It has recently been realized that a class of yield criteria in geomaterials, including the MC yield criterion, could be cast into conic constraints and consequently the FELA could be formulated in the form of conic programming. This concepts had been studied for the von Mises material with lots of studies (Andersen et al. 2000; Ciria 2002). The implication of this manipulation of yield criteria is that the singularities presents in the MC yield criterion will pose no difficulties in the algorithm specialized for the conic programming. Makrodimopoulos and Martin (2006) proposed a conic form of lower bound formulation of the FELA for plane strain analysis for the MC material. The resulting optimization problems are efficiently solved by a primal-dual interior point algorithm with the optimization package MOSEK-Aps (2010), which has been proven to be more robust and efficient than the NLP counterpart. In addition to the numerical benefits of the conic formulation, the yield criteria in a conic form will be treated as its natural form rather than a smoothed approximation in the NLP formulation. Since these works, the solution of plane strain problems of the MC material is much more efficient than they were used to be. Later, Krabbenhoft et al. (2008) and Martin and Makrodimopoulos (2008) independently proposed the Positive Semidefinite Programming (SDP) formulation of three dimensional limit analysis of the MC material, which is inspired by the fact that the MC yield criterion can naturally be expressed in terms of the linear combination of the principal stresses (eigenvalues of the symmetric stress tensor). Moderate 3D problems with several thousand of

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tetrahedral elements are solved with encouraging efficiency by algorithms specialized for SDP such as Sedumi (Strum 1999). Currently, no large-scale model analysis has been reported and there are also some convergence problems encountered in the analysis. With the increase in the model size, the optimization problems in 3D would grow appreciably. In spite of the shortcomings of the SDP formulation, it is still attractive in three-dimensional limit analysis as it is a relatively new field of optimization and has numerous applications in disciplines such as engineering, mathematics, computer science, operational research, etc. More powerful code and algorithm will definitely arise in the future to overcome the mathematical difficulties of the current SDP. Recently, Kammoun et al. (2010) presented a decomposition technique for the lower bound in the FELA. The problem is partitioned into finite element sub problems, and with an auxiliary interface problem, very large-scale problem with millions of variables and constraints are solved using the interior point algorithm. The best lower bound solution Nc=3.7752 is obtained with this technique for the vertical cut of purely cohesive material, and this value is commonly taken as the benchmark for this type of problem. In the numerical formulation of the lower bound theorem, discretisation using the finite element space is not the only choice. Recently, Chen et al. (2008) presented a lower bound formulation based on the element-free Galerkin (FEG) method. The collapse load is computed iteratively by solving a sequence of sub-problems generated by a reduced-basis technique, and the Complex method is adapted to solve the resulting nonlinear programming problem. Similarly, Le et al. (Le et al. 2010a; Le et al. 2010b) presented a similar lower bound formulation based on element-free Galerkin method for the plates and slabs of von Mises and Nielsen material. The stress/moment field is constructed by using a moving leastsquare approximation. The main limitation to this approach is the application of the method to nonhomogeneous problems which are commonly found for many geotechnical problems. Upper bound formulation of the FELA is pioneered by Anderheggen and Knopfel (1972) and Bottero et al. (1980). In their formulation, three-node triangular elements are utilized to discretize the velocity field and kinematically admissible discontinuities are allowed between two adjacent elements to compensate for the low order of velocity interpolation. To render the resulting optimization problem to be linear programming problem that could be solved by linear programming algorithm, the yield criterion is linearized as an external polygon to ensure the solution to be an upper bound. Shortcomings in these early formulations are: (1) the revised simplex algorithm which is used is generally slow in computation; and (2) the direction of the shearing are required to be specified for each discontinuities a priori, making the large number of arbitrary discontinuities impossible. To tackle the first problem, Sloan (1989) improved the solution technique by solving the dual problem of the upper bound formulation with an active set algorithm that is originally developed in his lower bound formulation (Sloan 1988a,b). To address the second shortcoming of the pioneering formulation, Sloan and Kleeman (1995) introduced a method to automatically determine the direction of shearing by describing the velocity jump with an additional set of four unknowns. The amended formulation due to Sloan and Kleeman (1995) has been proven to be efficient and robust and has been studied in details by Kim et al. (2002) and Bandini (2003) with applications in geotechnical engineering. Noting the large number of the linear constraints due to the linearization of the yield criterion in the upper bound analysis, Lyamin and Sloan (2002a, 2002b) proposed general nonlinear formulation in which simplex element is used for the discretisation of the velocity

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field and each element is associated with constant stresses. A two-stage quasi-Newton algorithm is used to solve the KKT conditions of the optimization problem arising from the discretised upper bound problem. In their formulation, examples in both two-dimensional (2D) and three-dimensional (3D) conditions are presented to demonstrate the superiority of the NLP formulation over the LP analogue. Li and Yu (2005) proposed an NLP formulation for the MC and DP yield criteria using purely the kinematic variables. The MC and DP yield criterion are rewritten in a quadratic form with which the stresses variables are eliminated from the expression of the rate of the power dissipation by applying the normality condition. This formulation is conceptually simpler than the NLP formulation proposed by Lyamin and Sloan (2002b), however, one difficulty of this formulation is that the objective function obtained is not everywhere smooth and is nondifferentiable in the rigid area such that an iterative algorithm based on distinguishing rigid/plastic area was adopted in finding the solution. The algorithm that was originally proposed by Burland et al. (1977) and Huh and Yang (1991) was adopted in finding the solution. In finite element formulation of the upper bound analysis, higher order element in addition to the use of constant strain elements has been studied. Linear strain elements with straight edges have been be tailored for the upper bound discretisation formulation by Yu et al. (1994) and Pastor et al. (2002) for plane strain and axisymmetric analysis respectively. More recently, following similar idea, Makrodimopoulos and Martin (2007) adopted the discretisation using simplex elements for the upper bound discretisation and formulated the upper bound analysis as standard SCOP for the MC material under plane strain condition and Drucker-Prager material under full three-dimensional condition. The optimization problem resulting from this formulation is then solved with primal-dual interior point algorithm specialized for the SOCP. The significance of the conic formulation of the FELA with the MC material is that the expression of the power dissipation can be derived easily and the singularities of the MC yield criterion will not pose any difficulty at the solution stage. Later, Makrodimopoulos and Martin (2007, 2008) extended their formulation to include velocity discontinuities in the discretisation of the velocity field, and the velocity jump is constrained to vary linearly by introducing additional constraints. It has been shown that the introduction of the velocity discontinuities will dramatically increase the size of the resulting problem. However, if the discontinuities are arrange with the knowledge of the slip bands, very impressive results could be obtained. Formulating the upper bound analysis in velocities require the expression of the rate of power dissipation such that the stress field is not explicitly included in the analysis. However, to express the power dissipation in terms of the velocity variables will become complicated for general yield criterion. A new formulation of the upper bound theorem in terms of the stress variables was proposed by Krabbenhoft et al. (2005), and similar ideas can be found in the works by Ciria (2002). The upper bound theorem is formulated directly from the dual form, i.e., the stresses are used as the optimization variables and the upper bound solution is obtained by maximising the load multiplier instead of minimising the load as in the conventional upper bound analysis. By the duality theory in convex programming, it could be guaranteed that the formulation yields the rigorous upper bound. More recently, da Silva and Antao (2008) proposed a parallel mixed finite element upper bound, where the velocity and strain field were independently defined and compatibility condition is imposed by augmented Lagrange method. The upper bound FELA has been

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applied to masonry works by Cavicchi and Gambarotta (2006) and Milani (2008; 2011), Milani et al. (2007; 2010) using the LP formulation. In recent years, attempts have also been made to seek alternatives of discretisation other than finite element space. For example, Smith and Gilbert (2007) proposed a discontinuity layout optimization procedure to determine the discontinuity directly with interesting and promising applications. Le et al. (2010a) and Le et al. (2010b) presented a meshfree approach to formulate the upper bound theorem where the yield at the interior of the solution may be violated in some cases. In the present research, discussion will be restricted to finite element based limit analysis. The mathematical procedures for the FELA is complicated, and interested readers can refer to the Ph.D. thesis by Li (2014) for details of the treatment and the mathematical procedures.

2.8.2. Mesh Adaptation in Limit Analysis Since the 1970s, error estimators have been developed to assess the discretisation error of the finite element solutions. Posterior error estimators in the literature of the FEM can generally be classified into two groups, namely, residual-based error estimators and the recovery-based error estimators (post processing estimate). The idea of the residual error is to compute the residual of the equilibrium within an element and jumps at element boundaries to obtain an estimate of the error in the energy norm. Errors are estimated in a sense of how much the discretisation of the continuous field has failed to satisfy the equilibrium condition and the boundary condition for a particular element. Fundamental works regarding the residual error estimate are presented in the works by Babuska and Miller (1978), Babuska and Rheinboldt (1979), Kelly et al. (1983), Babuska and Miller (1987). Recovery-based error estimate defines the error by evaluating the smoothed derivatives with the original ones inspired by the smoothing procedure to recover more accurate nodal derivatives in the finite element analysis (Zienkiewicz and Zhu 1992a). A good review on the developments of the error estimate in the FEM is presented in the works by Gratsch and Bathe (2005), and more mathematical details on the error estimate are discussed in the works by Brenner and Scott (2008). Mesh adaptation techniques in the FELA are primarily the extension of the existing procedures in the FEM. Borges et al. (2001) extended a recovery-based error estimator (Borges et al. 1999) to local directional interpolation error in mixed limit analysis formulation which is based on the recovering scheme to compute the second derivatives of the finite element solution. The scalar field of the Lagrangian multiplier is used as the control variables. The global re-meshing technique by Lyamin et al. (2005) and Sloan et al. (2008) tailored the error estimator (Borges et al. 1999) and applied it to the lower bound formulation of the limit analysis. The advancing front mesh generator with the adaptive mesh refinement has also been studied, and it is shown that when the scheme is coupled with the automatic fan zone general, results with high accuracy can be produced. It is also demonstrated by Lyamin et al. (2005) that the anisotropy of the mesh adaptation does not seem to provide better lower bounds for the unstructured mesh consisting of 3-noded triangles. Both of these methods can be regarded as an extension of the recovery-based error estimate scheme, which implicitly assumes that the smoothness of the solution is equivalent to the accuracy in solution.

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Christiansen and Edmund (2001) proposed a mesh adaptation procedure for the von Mises material, which is based on the localized mesh refinement on the unstructured triangular mesh. The mesh update is steered based on the yield slack and equivalent deformation. A computationally economy procedure is tested on plane strain problems with very accurate solutions obtained. The idea of guiding the degree of freedom concentration in the mesh adaptation is simple but adequate in the context of the limit analysis as the FELA generally assumes the rigid plasticity condition, which implies that most of the regions are actually rigid bodies. An essentially different error estimator was proposed by Ciria et al. (2008). In his work, the total gap between the lower and upper bounds is decomposed into the sum of elemental contributions and elements are marked for refinement based on their contributions to the global error. This procedure has been proven to be successful in applying to 2D analysis. As an extension of the approach by Ciria (Ciria 2002; Ciria et al. 2008), Munoz et al. (2009) further considered the error contribution from internal edges.

2.8.3. Error Estimate The error of a discrete solution due to the discretisation is defined by the difference between the numerical approximate and the exact solution. When the exact solution of a system is not known, which is often the case; the accuracy of the numerical approximate could be assessed by performing a sequence of analyses with meshes of increasing intensity. The exact solution is then predicted by the Richardson’s extrapolation and thus the error is calculated. However, this process is overly computationally involved. Various error estimators that are exclusively determined based on the current solution and the problem data have been proposed in the literature of the FEM. Comprehensive reviews regarding error estimate for the FEM are provided in works by Gratsch and Bathe (2005) and Zienkiewicz and Taylor (2005). Error estimators in the FEM can be classified into three groups (Gratsch and Bathe 2005): 1) Explicit error estimator that directly makes use of the finite element interpolation and the data of the problem; 2) Implicit error estimator where auxiliary local boundary problem is required to be solved; 3) Recovery-based error estimators that make use of the difference between the smoothed gradient and unsmoothed gradient. Since the finite element spaces are used for the discretisation for stress and velocity field in the FELA, it is natural to ask whether similar error estimators can be tailored to steer the mesh adaptation in the FELA. Firstly, it is worthwhile to review the error estimation in the displacement-based FEM. The error of the solution due to interpolation is defined as the difference between the exact solution and the numerical approximation. Explicit error estimators are concerned with the direct evaluation of the residual. For the displacement-based FEM, the displacement field is continuous while the stress and strain fields calculated by differentiation are discontinuous across the inter-element boundaries. Discontinuities of stresses and strains are the direct

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results of the introduction of the finite element discretisation and could be viewed as an indicator of error. In the recovery-based error estimate, post-processing of the gradient of the solution is first performed and then the estimate of the error is obtained by comparing the post-processed gradients with the original one. For some particular problems, the recovery can be better justified by the fact that there exist certain points within an element that have higher accuracy of derivatives, the so-called superconvergence point. A well-known patch recovery error estimator has been proposed by Zienkiewicz and Zhu (1992a, 1992b). The deformation pattern of a solid body in elastoplastic FEM differs from that in the limit analysis. In limit analysis, the domain at the ultimate limit state consists of the plastic deformed region and the plastic rigid region (Christiansen 1996), while the deformation is more continuous in the elastoplastic finite element analysis. Nevertheless, there have been a number of attempts in applying the error estimate concepts developed in FEM in limit analysis (Borges et al. 2001; Lyamin et al. 2005). In limit analysis, the choice of the control variables in the calculation of the error is not obvious. The scalar field of the Lagrangian multipliers have been adopted by Borges et al. (2001) and Lyamin et al. (2005) based on the patch recovery technique. In this research, we will extend the work and also study the performance of the residual-based error estimators evaluating the jumps of the multiplier field. For the FELA formulated as NLP, the Lagrangian multipliers associated with the yield criteria serve as a suitable option for the control variable, as it indicates which inequalities are active and which point is undergoing plastic flow. Adopting the Lagrangian multipliers as the control variable was adopted by Borges et al. (2001) in the mixed limit analysis based on the recovery Hessian matrix technique. Following similar approach, Lyamin et al. (2005) tailored the error estimator for the lower bound analysis and studied various recovery schemes of the recovery of the Lagrangian multiplier field, recovery of the Lagrangian field, recovery of the gradient of Lagrangian field and Hessian of Lagrangian. From their results, it appears that different recovery techniques do not provide major noticeable differences. In views of the previous findings, the gradient recovery scheme is adopted by the author, as it resembles the procedures in recovery process in the displacement-based FEM. In the author’s formulation, multipliers are discontinuous due to the introduction of the stress discontinuities. For the SOCP formulation of the FELA, the dual variable field is very much like the stress field and belongs to a dual cone. In this case, the dual variable could be regarded as the equivalent strains (Christiansen and Edmund 2001). The recovery based error estimator could be obtained by the error of the sum of n components in the dual cone. Similarly, the residual based error estimator for the FELA could be obtained by replacing the continuous displacement field with the discontinuous dual variable field. The author has applies the residual error estimator to the dual variable field. In the upper bound FELA, if the continuous velocity field is adopted in the formulation, the recovery-based or the residual-based error estimate could be utilized directly following the procedure in the FEM, as the velocity field is exactly the same as that in the FEM. When the velocity field is discontinuous, the gradients of the velocity field are calculated with the same procedure as that in the multiplier field scheme. Due to the absence of the superconvergence in the FELA, it seems difficult to assess the local error from either a lower bound or upper bound analysis alone. However, it can be argued that the error of the solution is contributed exclusively from the plastically deformed region. This point is easier to understand from the viewpoint of an upper bound analysis for

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which the objective function consists of the terms due to internal power dissipation and external work done by body forces. The contribution to power dissipation comes exclusively from the plastic region and the refinement of the rigid zone does not give any improvement to the bound solution. This argument holds for lower bound as well because the lower bound problem can be transformed into an equivalent kinematic form. This is different from the FEM in which the deformation exhibits in a continuous form. The distinctions between the deformation fields calculated with the FELA and FEM have been discussed in length by (Christiansen 1996). Therefore, refinement of the plastic zone is both a practical and reasonable approach to be adopted. An error estimator, which is conceptually different from the residual or recovery based approaches in the FEM, was proposed by Ciria (2002). The error associated with a given triangulation is defined by the gap between the upper bound and lower bound solution calculated from the same mesh. The mesh is updated based on the elemental contribution to the error defined by (Ciria 2002; Ciria et al. 2008). Ciria (2002) and Ciria et al. (2008) applied the error estimator of to a number of problems of von Mises material under plane stress and plane strain conditions and promising results were obtained. However, it is clear that requires that both the stress field from the lower bound analysis and velocity field from the upper bound analysis to be calculated, which implies that the lower- and upper bound analysis need to be performed together. This poses a great challenge to the computer capacity for large models and will thus considerably limit the model size in practical applications. In this research, we will seek computationally cheaper mesh update scheme under which it is possible to compute separately the lower and upper bounds for large scale models. It could be argued and will be demonstrated in later sections that the error in the FELA comes exclusively from the slip bands because of the assumption of the rigid plasticity of the material. It is therefore reasonable to distribute the grids on the slip bands. To this end, evaluating the yield function provides the most straightforward approach to identify the yield zone of a domain. After the solution of a lower bound problem, the slackness of the yield functions at each vertex in the mesh can be back calculated. Other than evaluating the yield function, the dual variables corresponding to the yield function offers another feasible indicator of the yield zone. Dual variables associated with the yield function can be regarded as the equivalent strains. As rigid perfectly plasticity is assumed in limit analysis, strain components would become zero in rigid portion of the domain, and only the portion undergoing plastic flow has non-zero strains.

2.8.4. Localized Mesh Refinement with Unstructured Mesh Methods for mesh adaptation include remeshing, reorienting or splitting elements. Remeshing generates new meshes based on the computed solution, which is comparatively computationally demanding because the mesh generation subroutine is called in each refinement and vast amount of data are to be transferred from one mesh to another. The merits of this technique are that the decedent meshes are not restricted by the initial mesh. This technique was adopted in the works by Borges et al. (2001) for the mixed limit analysis and Lyamin et al. (2005) for the lower bound limit. In elements splitting method, elements marked according to the prescribed rules are subdivided into a number of children. This technique is simple in concept but the potential

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problem inherited is that the subdivision of elements will generate hanging nodes which requires more sophisticated data structure and complicated refinement algorithm to perform the refinement. Various element splitting methods can be designed for local refinement. The embedded refinement in Figure 2.22 is the simplest to implement since no hanging nodes will be generated during the refinement. However, the regularity of the mesh is not maintained, and more significantly, it may experiences “locking” because the discontinuities will be severely restricted by the initial mesh. The regular refinement or red-green refinement and the bisection method (Rivara 1984; Sewell 1972) have been applied in the FEM and proven to be robust. The regular refinement divides the target element into 4 children (red procedure) and a green procedure is required to eliminate the hanging nodes on the edge. The bisection refinement divides the target element into two children and similarly a recursive algorithm is required to address the hanging nodes. For three-dimensional problem, the bisection divides a tetrahedron into two.

Figure 2.22. (a) regular refinement, (b) bisection, (c) directed section and (d) embedment refinement.

The basic idea of bisection-based local refinement is to divide the target element into two children and the added extra node is either eliminated with a recursive algorithm or by a closure procedure. Depending on the choices of the edge to split, two variants of element bisection algorithms have been developed in the literature. The newest vertex bisection method was first proposed by Sewell (1972) in which the target element is split into two smaller children by connecting one of the vertexes (called peak) and midpoint of the opposite edge (called base or refinement edge). The newly inserted node is assigned as the peak of the child element, i.e., the edge opposite to the newly-added node in each child will be split. After the bisection of an element, an extra node (hanging nodeP) will be generated on the edge of the adjoining element as shown in Figure 2.22. It is necessary to eliminate the hanging node by further bisecting the neighbouring element. It is expected that one step of closure may generate more hanging nodes and thus the closure may propagate. Mitchell (1988) proved that the propagation stops after a finite number of iterations. Four similar classes will be generated as shown in Figure 2.22, thus the regularity of the triangulation is guaranteed.

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Figure 2.23. Bisection of a triangle.

Figure 2.24. Four similar classes generated by the newest-vertex bisection.

The longest-edge bisection proposed and studied by Rivara’s group (Rivara 1984; Rivara 1989; Rivara and Iribarren 1996) always bisects one of the longest edges of the triangle to guarantee the regularity of meshes during the refinement. Indeed, it has been proved by Rosenberg and Strenger (1975) that the smallest angle in the descendants of the original element is bounded below by the half of the smallest angle in the initial triangulation. When the longest edge is always opposite to the newest vertex, the longest-edge is equivalent to the newest-vertex bisection. One of the examples is for uniform meshes: the mesh obtained by dividing rectangles into triangles using their diagonals. The peaks are always at the right angles and the longest edges are opposite to the peaks. A recursive algorithm based on the compatible division of an element for the bisection method has also been described in the work by Kossaczky (1994). An element is divided only when it is compatibly divisible. An element is compatibly divisible if the edge marked for division is the refinement edge of the neighbour opposite to the peak or on the boundary of the domain. The refinement process works only on the compatible element and obviously the hanging node is avoided automatically. Another method to perform local mesh refinement is the red-green-refinement (Bank et al. 1983) which is commonly applied to two-dimensional applications. During the refinement, a marked triangle is divided into 4 smaller children (called red refinement or regular refinement) as shown in Figure 2.25. The generated hanging nodes on the edges of the neighbouring triangles that are not flagged for the refinement are refined irregularly by bisection (green refinement). If more than two sides are split for a neighbouring element, it should also be marked for the red-refinement. By continuation of these two strategies, a hierarchical mesh is produced. The pleasant property of this method is that the newly generated children are geometrically similar to the original one and therefore the element quality of the new triangulation is the same as the original ones.

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Figure 2.25. Four similar classes generated by the newest-vertex bisection.

In the FEM, the choice of the element refinement scheme is primarily a matter of taste of the engineer and the convenience for the computer implementation. Bisection and red-green refinement strategies have both found applications in the FEM. For instance, the bisection algorithm has been implemented in the FEM code ALBERT (Schmidt and Siebert 2000) and the red-green refinement has been adopted by Bey (1995). To investigate the performance of these two categories of the subdivision in the limit analysis for the von Mises material, Christiansen and Edmund (2001) have compared the performances of the two types of refinement procedures in two dimensional cases. From their results, no significant differences are noted except that the bisection refinement appeared more likely to generate ill-conditioned optimization problems in comparison to the regular refinement for limit analysis. Consider a refinement of one single triangle after three times of refinements by bisection and regular division respectively. A hidden dependence of the linear constraints will be generated due to the elements in the shaded area (Makrodimopoulos and Martin 2006). This situation is unlikely to occur in the regular refinement scheme Figure 2.26b. However, it could be avoided locally by traversing the mesh from node to node rather than by edge during the assembly of the inter-element equilibrium equations as suggested by Makrodimopoulos and Martin (2006). The advantage in using the bisection method is that it allows the initial mesh, in particularly a very coarse mesh, to adapt in a moderate rate and distribute the error more optimally as it bisect the target element into two children instead of 4 in the regular refinement. For stability problems with large transitional zone, e.g., the bearing capacity problem with large frictional angle, a huge number of elements can be generated by a regular refinement algorithm in several refinement steps. Based on the current study, it is found that the bisection method can be an attractive approach if the limitation of this method as mentioned is carefully eliminated during adaptive refinement.

Figure 2.26. (a) mesh after three bisection of a triangle and (b) the mesh after three regular refinements.

It is well known that for problems with singularities, the fan elements would results in much better bound solutions. The local mesh refinement has been studied by Munoz et al.

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(2009), and two strategies have been proposed to solve the “locking” phenomenon due to stress singularities. However, the major problem of these two strategies is the regularity of the element not being maintained, and ill-conditioned optimization problem would be generated as the refinement proceeds. To improve the performance of the local mesh refinement in such cases, it is considered more convenient to apply a generic semi-circular zone with fan element at the stage of mesh preparation. This zone can generated easily with common mesh generation algorithms and regular refinement can be applied thereafter. Alternatively, using the fan elements generation algorithms (Lyamin and Sloan 2003) for the initial mesh is another approach for simple problems.

2.9. FINITE ELEMENT METHOD In the classical limit equilibrium and limit analysis methods, the progressive failure phenomenon cannot be estimated except for the method by Pan (1980) or the variable factor of safety approach by Cheng et al. (2011b). Some researchers propose to use the finite element method to overcome some of the basic limitations in the traditional methods of analysis. At present, there are two major applications of the finite element in slope stability analysis. The first approach is to perform an elastic (or elasto-plastic) stress analysis by applying the body force (weight) due to soil to the slope system. Once the stresses are determined, the local factors of safety can be determined easily from the stresses and the Mohr-Coulomb criterion. The global factor of safety can also be defined in a similar way by determining the ultimate shear force and the actual driving force along the failure surface. Pham and Fredlund (2003) adopted the dynamic programming method to perform this optimization search, and they suggested that this approach can overcome the limitations of the classical limit equilibrium method. The author however views that the elastic stress analysis is not a realistic picture of the slope at the ultimate limit state. In views of these limitations, the author does not think that this approach is really better than the classical approach. It is also interesting to note that both the factor of safety and the location of the critical failure surface from such analysis are usually close to that by the limit equilibrium method. To adopt the elasto-plastic finite element slope stability analysis, one precaution should be noted. If the deformation is too large so that the finite element mesh is greatly modified, the geometric non-linear effect may induce a major effect on the results. The second finite element slope stability approach is the strength reduction method (SRM). The shear strength reduction technique is proposed as early as 1975 by Zienkiewicz et al. (1975), and has since been used by Naylor (1982) and Matsui and San (1992) and many others. The application of SRM occurs mostly in the past decade due to the increasing speed of computers. In the SRM, the gravity load vector for a material with unit weight s is determined from eq.(2.80) as:

{ f }   s  [ N ]T dv

(2.80)

where {f} is the equivalent body force vector and [N] is the shape factor matrix. The constitutive model adopted in the non-linear element is usually the Mohr-Coulomb criterion,

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but other constitutive models are also possible, though seldom adopted in practice. The material parameters c’ and ’ are reduced according to cf=c’/F; f=tan-1{tan(’/F)}

(2.81)

The factor of safety F keeps on changing until the ultimate state of the system is attained, and the corresponding factor of safety will be the factor of safety of the slope. The termination criterion is usually based on one of the following: 1) the nonlinear equation solver cannot achieved convergence after a pre-set maximum number of iteration; 2) there is a sudden increase in the rate of change of displacement in the system; 3) a failure mechanism has developed. The location of the critical failure surface is usually determined from the contour of the maximum shear strain or the maximum shear strain rate. Among a number of commonly used linear and non-linear constitutive models, the elastic-perfectly plastic model with Mohr-Coulomb failure criterion is mostly adopted. It has been shown that the sudden transition of soil behavior from elastic state to plastic state in the elastic-perfectly plastic model helps identify the factor of safety when applying strengthreduction technique (Dawson et al. 1999). Others have used more complicated constitutive models in the strength reduction method. For example, Matsui and San (1992) used the Duncan and Chang’s hyperbolic model (Duncan and Chang 1970), and Zhang et al. (2011) used a revised Duncan and Chang’s model. However, identifying the failure becomes more difficult. In addition, the factor of safety by strength reduction is found insensitive to the constitutive model used, even though more complicated constitutive models have advantages in predicting the loading-displacement behavior. This is because the stress-dependent stiffness behavior and hardening effects are excluded when using the strength reduction technique (Plaxis manual 2014; Alkasawneh et al. 2008). Zhang et al. (2011)’s study also revealed that the factors of safety by using Duncan and Chang’s model and using the revised model are almost the same. One difficulty to the strength-reduction stability analysis lies in the definition of slope failure criteria. So far there is no uniform definition in the literature. Matsui and San (1992) proposed to use the contour of shear strain as failure indicator; for example, their study of an embankment slope showed that a well-defined failure zone develops from the toe to top surface when the contour of shear strain exceeds 15%. In contrast, Griffiths and Lane (1999) and Dawson et al. (1999) took non-convergence (out-of-balance force) as an indicator for failure in finite element method and finite difference method, respectively. Others considered slopes failed when there are contiguous plastic zones through the toe to the crest (e.g., Zheng et al. 2009). The main advantages of the SRM are as follows: (i) the critical failure surface is found automatically from the localized shear strain arising from the application of gravity loads and the reduction of shear strength; (ii) it requires no assumption on the interslice shear force distribution; (iii) it is applicable to many complex conditions and can give information such as stresses, movements, and pore pressures which are not possible with the LEM. Griffiths

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and Lane (1999) pointed out that the widespread use of the SRM should be seriously considered by geotechnical practitioners as a powerful alternative to the traditional limit equilibrium methods. One of the important critics on SRM is the relative poor performance of the finite element method in capturing the localized shear band formation. While the determination of the factor of safety is relatively easy and consistent, many engineers find that it is not easy to determine the critical failure surfaces in some cases as the yield zone spread over a wide domain instead of localizing within a soft band. Other limitations of the SRM include the choice of an appropriate constitutive model and parameters, boundary conditions and the definition of the failure condition/failure surface. Although SRM appears to have many advantages, it is not easy to identify the critical slip surface from the analysis, and the results can be sensitive to many factors so that the users may need to carry out many trial and errors to obtain a reasonable result. As shown in Figure 2.27, a change in the dilation angle has a very small effect on the factor of safety, but a very major change in the critical slip surface. In fact, it is not easy to identify a clear slip surface for the results in Figure 2.27. On the other hand, if the dilation angle is revised to 35, a clear slip surface can be identified. In this respect, the engineers may need to use LEM to assess the acceptability of the results from SRM! Furthermore, after trying many commercial SRM programs, the author find that different program may produce different shear strain contours, critical slip surfaces or even factors of safety. The differences between the results from different SRM programs are actually not small, and such phenomenon is commonly found. In this respect, the LEM appears to be more stable than the SRM for general cases.

(a) SRM1

(b) SRM2

Figure 2.27. SRM analysis of the problem as shown in Figure 2.12 (a) c’=2 kPa, ’=45, ’=0; (b) c’=2 kPa, ’=45, ’=45.

Figure 2.28. SRM analysis of the problem as shown in Figure 2.12, c’=2 kPa, ’=45, ’=35.

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2.10. DISTINCT ELEMENT METHOD It can be considered that there are two versions of distinct element method which can be used for slope stability analysis. The first approach is proposed by Chang (1992) where each slice is connected with the others through normal and tangential springs along the interfaces. The relative displacements between adjacent slices can be related to the interslice normal and shear forces and moment by the normal and tangential springs. In this formulation, elastoplastic behaviour can be incorporated into the springs, and the concept of residual strength and stress redistribution can also be implemented. The equilibrium of the whole system is then assembled, and this will result in a 3N simultaneously equations to be solved. Cheng has programmed this method (Chan 1999) and found that the factor of safety from this method is very close to that by the other methods. In fact, if the normal and shear stiffnesses are kept constant throughout the system, the factor of safety is practically independent of the values of the stiffnesses. This method appears to be not used for practical or research purpose, and the selection of the normal and shear stiffnesses for general nonhomogeneous condition is actually difficult as no guideline or theoretical background can be provided for this approach. Although Cheng has found that this approach will give results similar to the classical LEM methods for normal problems, the suitability of this method for highly irregular problem is however unknown. When a soil slope reaches its critical failure state, the critical failure surface is developed and FOS of the slope is 1.0. The continuum methods like FEM or LEM will fail to characterize the post-failure mechanisms in the subsequent failure process. For a rock slope comprises multiple joints sets which control the mechanism of failure, a discontinuum modeling approach is actually more appropriate. Discontinuum methods treat the problem domain as an assemblage of distinct, interacting bodies or blocks which are subjected to external loads and are expected to undergo significant motion with time. This methodology is collectively named as the discrete element method (DEM). DEM will be a more suitable tool for the study of progressive failure and the flow of the soil mass after initiation of slope failure, though this method is not efficient for the analysis of a stable slope. The development of DEM represents an important step in the modeling and understanding of the mechanical behavior of joint rock masses or soil slope with FOS0, f(x) moves towards the left-hand side of the figure with an increasing . These results are also obvious because the failure zone will become longer with an increasing . It is also interesting to note that f(x) is the same for factors Nc and Nq. These results are not surprising as the failure mechanisms for Nc and Nq are the same based on the classical plasticity solution. On the other hand, the principal stresses for N are the vertical and horizontal stresses only in the passive wedges, which can be observed from the results in Figures 3.33 and 3.34. Hence, f(x) is zero only for the right-hand side in Figure 3.35. 1.2

f(x)

1 Ø=10

0.8

Ø=20

0.6

Ø=30 Ø=40

0.4

Ø=0

x 0.2 0 0

0.2

0.4

0.6

0.8

Figure 3.33. f(x) against different dimensionless distance x in case of Nc.

1

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Yung Ming Cheng 1.2

f(x)

1 Ø=10

0.8

Ø=20

0.6

Ø=30 Ø=40

0.4

Ø=0

x 0.2 0 0

0.2

0.4

0.6

0.8

1

Figure 3.34. f(x) against different dimensionless distance x in case of Nq. 1.2

f(x)

1 Ø=10

0.8

Ø=20

0.6

Ø=30 Ø=40

0.4 x 0.2 0 0

0.2

0.4

0.6

0.8

1

Figure 3.35. f(x) against different dimensionless distance x in case of N.

The results of the thrust line for Nc (and Nq) at = 0, 10, 20, 30 and 40 are given in Figure 3.36. The horizontal axis is dimensionless distance x in the range of 0 to 1. In the passive wedge region on the right-hand side, the thrust line ratio (LOT) is always 0.5. In the very beginning of the active wedge, the thrust line ratios are very close to 0.5, but deviate slightly from 0.5 due to the minor error arising from the iteration analysis in the slip line analysis. Outside the foundation, LOT will go below 0.5 and then gradually rebound to 0.5. The fluctuation in LOT outside the foundation is mainly caused by the radial shear zone. It should be noted that while the ultimate bearing capacity factors from SLIP are relatively insensitive to the grid sizes used in the analysis, the thrust line is more sensitive to the size of the grid. This situation is particularly important for the case of N, and thus, a very fine grid is adopted for the case of N or else there will be a larger fluctuation in the location of the thrust line.

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LOT

0.6 0.5 Ø=0

0.4

Ø=10 Ø=20

0.3

Ø=30

0.2

x

Ø=40

0.1 0 0

0.2

0.4

0.6

0.8

1

Figure 3.36. The thrust line for different  angles for case Nc (x coordinate is x ratio) (same result for Nq).

LOT

0.6 0.5 0.4 Ø=10

0.3

Ø=20 Ø=30

0.2

x

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.37. The thrust line for different  angles for case N (x coordinate is x ratio).

The results of the thrust line ratio for Nq are the same as those for Nc and are also given in Figure 3.36. These results are in line with those for which f(x) is the same for the cases of Nq and Nc. For the case of N, the results are different from those of the previous two cases. Just beneath the foundation, the stresses are mainly controlled by the ground pressure so that the thrust line is slightly less than 0.5, which is obtained in Figure 3.7. At the passive zone, where there is no imposed pressure, the vertical pressure is totally controlled by the weight of the soil. Therefore, the thrust line ratio is 1/3, which implies a triangular pressure distribution; this is consistent with the recommendation by Janbu (1973). It should be noted that the linear distribution of the ground pressure, as determined from the slip line analysis in Figure 3.31, applies only when c is taken as zero, which is also the way N is defined. For simplicity, the coupling effect between the unit weight and c is not considered in the present study, but the present study is not limited to the case of c=0.

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The results for the thrust line are in line with the suggestion from Janbu (1973). Janbu (1973) suggested that LOT could be determined based on the earth pressure theory. For a general slope from the frictional material, the lateral earth pressure distribution is largely controlled by the unit weight of the soil and will be close to a triangular shape; hence, a generally referred value of 1/3 for LOT is suggested. In the present study, for both Nc and Nq, where the unit weight of the soil is zero, the horizontal and vertical pressure under half of the footing will be constant, and thus, LOT should be exactly 0.5. The later part of the slip surface represents a passive earth pressure state in which the earth pressure distribution is similarly constant, and again LOT = 0.5. For N, there is a triangular-shaped earth pressure distribution in the passive zone, and hence, LOT = 1/3. When f(x) or the thrust line is defined, the problem can be back-analyzed in the following ways. For the case of Nc, a uniform pressure corresponding to cNc is applied on ground surface without any surcharge outside the foundation. The unit weight of the soil is set to zero in the slope stability analysis. For the case of Nq, a surcharge of 1 unit is applied outside the foundation, and the uniform foundation pressure is given by unit Nq, while the unit weight of the soil and the cohesive strength are set to zero in the analysis. For the case of N, a triangular pressure (see Figure 3.62) with a maximum equal to BN is applied on the ground surface (average pressure is 0.5 BN), while the cohesive strength and the surcharge outside the foundation are set to zero. Based on the f(x) shown in Figures 3.33, 3.34 and 3.35, and the failure surfaces given by the slip line solutions, Cheng et al. (2013a) have back-computed the factors of safety to be nearly 1.0 (only 0.001 to 0.002 less than 1.0) for all the cases. These results are obvious as the solutions from the slip line equations are the ultimate solutions of the system. On the other hand, when the thrust line ratios shown in Figures 3.36 and 3.37 are used in Janbu’s rigorous method (1973), after some revisions in the Janbu’s rigorous method, all the problems can converge nicely with the factors of safety close to 1.0 for all cases (only 0.001 to 0.002 less than 1.0). It should be noted that for normal slopes, the convergence of Janbu’s rigorous method (1973) is actually not too bad, although it is not very good either. On the other hand, if a more rigorous consideration of the moment equilibrium as suggested is adopted, the factor of safety is 1.0, while the f(x) and internal forces obtained are virtually the same as those from the slip line analysis. That means, a correct thrust line will correspond to a correct f(x), and the choice of internal or external variables is not important under the ultimate condition. It is interesting to note that all the factors of safety for the three bearing capacity factors are very close to 1.0 (0.001 to 0.002 different from 1.0) using either f(x) or the thrust line from the ultimate condition. That means, as long as the ultimate condition is given consideration, there is no difference between the uses of f(x) or the thrust line in defining a problem. In this respect, Morgenstern-Price’s method (1965) and Janbu’s method (1973) are judged to be equivalent methods when specifying a problem under the ultimate condition. Other than the ultimate condition, the choice of f(x) or the thrust line will give different factors of safety (well known in limit equilibrium analyses) as the solutions are only typical lower bound solutions. Since iteration analyses are sensitive to the thrust line location, the use of f(x) for normal routine engineering analyses and designs is advantageous in that it is easier to achieve convergence for normal cases.

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3.10.2. Boundary Forces in Limit Equilibrium Method Baker and Garber (1978) proposed to use the base normal forces as the variables in the variational principle formulation of slope stability problems. For Nq where =30, the base normal stresses under an external surcharge of 1 kPa outside the foundation are determined by the slip line method which are shown in Figure 3.38.

Figure 3.38. Base normal stresses distribution along the slip surface for Nq when =30 and q=1 kPa.

Based on the stresses determined from the slip line analysis, the base normal stresses and hence forces for the slices can be determined correspondingly. Once P is known, based on ( Er  E L )  0 , the factor of safety can be computed by force equilibrium as

F

  cl  P tan   cos    P sin 

(3.12)

Based on the base normal stress in Figure 3.38 which has been tested against different grid size used for the slip line analysis, the factor of safety from eq.(3.12) is exactly equal to 1.0 which is as expected. Using P or the thrust line as the control variables are however less satisfactory as compared with the use of the interslice force function in the optimization analysis. When the thrust line is defined, the moment equilibrium of the last slice is not used in the analysis so that true moment equilibrium cannot be satisfied (the well known problem for Janbu’s rigorous method (1973). It should be pointed out that for the original Janbu’s moment equilibrium (1973), the moment equilibrium of the slice interface instead of the slice is considered so that there is no problem for last slice, but moment equilibrium for each slice is not strictly enforced. The use of P also suffers from this limitation. Once P is prescribed, F will be known from eq.(3.12). Based on force equilibrium in horizontal and vertical directions as well as the moment equilibrium, the interslice normal and shear forces as well as the thrust line will be defined for the first slice. These results can then be used to compute the internal forces between slice 2 and 3. The computation progresses until to the last slice for which both the force and moment equilibrium cannot be enforced automatically. To apply the base normal force as the control variables in the extremum evaluation, the base normal forces for N-1 slices (N= total number of slices) are taken as the variables while the base normal forces

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for the last slice will be determined from a trial and error process when the equilibrium of the last slice is satisfied. The same principle can also be applied to the thrust line, where thrust line for only N-2 interfaces are prescribed and the thrust line for the last interface is obtained by a trial and error process until moment equilibrium is achieved. The use of f(x) is simpler in that majority of the back-computed thrust lines are acceptable so that the solution for the prescribed f(x) can be adopted directly without trial and error process. If the thrust line is not acceptable, then the solution is simply rejected and another trial f(x) can be considered. Based on these studies, it can be concluded that the use of external variables along the failure surface or the internal variables as given by the interslice normal/shear forces will be the same in the limit equilibrium formulation, if the extremum of the system is considered. The choice of the interslice force function or basal stress distribution will not affect the factor of safety of the system, as these are now variables for which the extremum condition to be achieved. On the other hand, if the extremum condition is not fulfilled, the choice of the interslice force function or the basal stress distribution will affect the value of the factor of safety.

3.11. UNIFICATION OF BEARING CAPACITY, LATERAL EARTH PRESSURE AND SLOPE STABILITY PROBLEMS Lateral earth pressure, ultimate bearing capacity and slope stability problems are three important and classical problems which are well considered by the use of the limit equilibrium method, limit analysis method and method of characteristic in the past. It is interesting to note that these three topics are usually considered separately in most of the books or research studies, and different methods of analyses have been proposed for individual problem even though they are governed by the same requirements for the ultimate conditions. Since the governing equations and boundary conditions for these problems are actually the same, the author view that each problem can be viewed as the inverse of the other problems which will be demonstrated in this section. For bearing capacity and lateral earth pressure problems, the use of slip line method is more common, as the geometry under consideration is usually more regular in nature. For slope stability problem where the geometry is usually irregular with complicated soil reinforcements and external loads, the use of the slip line method is practically not possible. Engineers usually adopt the limit equilibrium method (an approximate slip line form) with different assumptions on the internal force distribution (f(x)) for the solution of the problems. Morgenstern (1992) commented that for most practical problems, the uses of different assumptions on the internal forces are not important. Cheng et al. (2010) and Cheng et al. (2011b) developed the extremum principle and have demonstrated the equivalence of the maximum extremum from the limit equilibrium method and the classical plasticity solution by a simple footing on clay. Cheng et al. (2010) treated f(x) as a variable to be determined, and complete equilibrium is enforced during the search for the maximum extremum of a system. Cheng et al. (2010) also pointed out that as long as a f(x) is prescribed, the limit equilibrium solution will be a lower bound to the ultimate limit state which is the lower bound theorem. Under the ultimate condition where the strength of a system is fully mobilized, f(x) is actually determinate by this requirement which is a boundary condition which has not been used in the past. Cheng et al. (2010) pointed out that every kinematically

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acceptable failure surface should have a factor of safety. Failure to converge in the classical limit equilibrium analysis is caused by the use of an inappropriate f(x) in the analysis. In this section, the author will first demonstrate the equivalence between the classical lateral earth pressure and bearing capacity problem by the slip line method. The equivalence between the lateral earth pressure problem and slope stability problem will then be illustrated by the use of extremum principle. Based on these results, it can be concluded that the three classical problems are equivalent in the basic principles, and each problem can be viewed as the inverse of the other problems. Bearing capacity for shallow foundation has been studied by many researchers using different methods. Except for the term N (weight term), Nc and Nq terms for the cohesive strength and surcharge are the same among different researchers (except Terzaghi 1943) by using different methods of analysis. Sokolovskii (1965), Booker and Zheng (2000) and Cheng and Au (2005) solved the bearing capacity problem under various conditions by the method of characteristics. Martin (2005) pointed out that the method of characteristics can be used to establish the exact plastic collapse load for any combination of the parameters c, , , B and q for the exact bearing capacity calculations, and a powerful program ABC has been produced for such analysis. For lateral earth pressure, Chen (1975), Kerisel and Absi (1990), Cheng (2003), Subra and Choudhury (2006), Shukla et al. (2009), Liu et al. (2009a, 2009b), Peng and Chen (2013), Liu (2014), Vo and Russell (2014, 2016) and many others have also obtained three active/passive earth pressure coefficients for plane strain problem based on the limit analysis and the method of characteristic. Cheng et al. (2007d) also obtained the three lateral earth pressure coefficients for axi-symmetric problem based on the method of characteristic. In general, the basic numerical method is adopted by all the authors with various modifications or tricks applied to individual case.

3.11.1. Basic Slip Line Theory Slip line method (or method of characteristic) considers the yield and equilibrium of a soil mass controlled by the Mohr-Coulomb’s criterion, and it is a typical lower bound method. A typical slip line system is shown in Figure 3.39, and the equations are governed by the  and  characteristic equations given by Sokolovskii (1965) in eq.(3.5) and (3.6). y

A S

P S C B x

Figure 3.39.  and  lines in slip line solution.

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If the self-weight of soil is neglected (i.e., γ=0), eqs.(3.5) and (.3.6) can be simplified as follows:

 sin 2

sin 2

p   2R 0 S S

p   2R 0 S  S 

(3.13) (3.14)

Integrates the above equations along the  and  lines gives:

ln p  2 cot 2  C  ln p  2 tan   C (along α line)

(3.15)

ln p  2 cot 2  C   ln p  2 tan   C  (along β line)

(3.16)

where p  p  c cot  , c is cohesive strength of soil and Cα and Cβ are constants. The sign convention used in this book is compressive stress being positive, while the angles  and others are taken as positive if measured in a counter-clockwise direction. Eqs.(3.15), (3.16) will be used to derive the expressions for Nq, Nc, Kaq and Kac in the following sections. Generally speaking, by solving equations (3.5) and (3.6), solutions to many geotechnical problems have been developed and used. It is interesting to note that while there are many works towards the application of method of characteristic in various geotechnical problems, there is none which is devoted to the unification of these geotechnical problems. Bearing capacity, slip line and slope stability problems are treated as individual topics in all the previous works. For a more general condition, the friction along the interface between the soil and the footing or the retaining wall is considered. If a pressure p’ act on one point M on the soil surface and the angle between p’ and the normal line to boundary is δ (i.e., friction angle, Figure 3.40, p’ can be described as:

p 

 n  c cot  2   t2

where σn and τt are normal and shear stress acting on the point M respectively.

(3.17)

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τ

ccotφ

τt

σn

τt

p’

τ

D

δ

D C’

p ’

p’’

A

p C ’ 2θ1 C τ φ δ φ δ E 2θE2θ 1 2θ t σ OA σB O σ B n

ccot φ

M (a) stresses at boundary

n

n

C’ τt σ

σ

σ

n

p=(σ1+σ 2)/21+σ2)/ ccot p=(σ 2 φ (b) Mohr circle for stress at boundary

Figure 3.40. The stress state of point M on a rough surface.

The stress state at the point M can be represented by the Mohr circle as shown in Figure 3.40b. AD is the Mohr-Coulomb line and the angle between AD and horizontal line AB is φ. AC represents stress p’, and the angle between AC and horizontal line is δ (CAB), and  is taken as positive in Figure 3.40a in a clockwise direction, which corresponds to an active pressure condition. The line AC intersects the Mohr circle at point C, and the abscissa and ordinate of the point C are σn and τt respectively. Line OB denotes the characteristic stress p, ABD = 2θ, ABC = 2θ1, and angle θ1 is inclination angle between the major principal stress and boundary. It can easily be seen from the geometrical relationship in the Figure 3.40b that:

 t  p sin 

(3.18)

 n   n  c cot   p cos 

(3.19)

Therefore, eq. (3.18) divided by eq. (3.19) gives,

t

 n  c cot 



sin   tan  cos 

(3.20)

For convenience,  in eq.(3.20) is a boundary condition which is defined in terms of f, n, c and  instead of purely f, n for simplicity. This boundary condition is usually the footing or the back of a retaining wall (surface AO in Figure 3.41) where n is constant ( is neglected here). The true friction can be transformed to the apparent friction  easily. According to the geometrical relationship in Figure 3.40b,

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Figure 3.41. Unified model of bearing capacity and lateral earth pressure problem.

 n   n  c cot   AB  BE  AB  BC cos 21  AB  BD cos 21

(3.21)

 t  CE  BC cos 21  BD sin 21

(3.22)

From triangle ABD, it is known that,

AB  p  c cot 

(3.23)

BD  AB sin    p  c cot  sin 

(3.24)

Substitute the above equations in eqs. (3.21) and (3.22),

 n   n  c cot    p  c cot     p  c cot  sin  cos 21

(3.25)

 t   p  c cot  sin  sin 21

(3.26)

So, eq. (3.26) divided by eq. (3.25) gives:

 p  c cot  sin  sin 21  n  c cot   p  c cot     p  c cot  sin  cos 21 t



Solve eq. (3.20) and (3.27) simultaneously,



sin  sin 21 (3.27) 1  sin  cos 21

Advanced Topics in Slope Stability Analysis

sin  sin 21 sin   cos  1  sin  cos 21

173

(3.28)

Rearrange eq. (3.28) give,

sin   sin 21    sin 

(3.29)

The general solution to eq. (3.29) is,

21    2m  

(3.30)

and 21    2m  1  

(3.31)

where   arcsin

sin  . Combine the above two solutions, sin 

21    2m  k  1  k 

or

1  m 

 2

1 k     1  k   2 4

(3.32)

(3.33)

where k=±1; m is an integer and generally m=0 or m=±1. According to the triangle ABC in Figure 3.40b,

p  c cot   p

  c cot   sin  sin   n sin   k  cos  sin   k 

(3.34)

Based on eqs. (3.33) and (3.34), the pressure and friction angle on the soil boundary can be converted into the characteristic pressure p and θ1. For the angle θ between the major principal stress σ1 and boundary, if the boundary is horizontal, θ=θ1. If the boundary inclines to the horizon at α, θ=θ1+α. It is noted here that the sign (±) of k is defined based on the direction of displacement of the boundary when the soil attains the limit equilibrium state. When the boundary of soil move in the direction of p’, k=-1; when the boundary of soil move in the opposite direction of p’, k=1. In other words, if the soil is in active failure state, k=-1; otherwise, the soil is in the passive failure state and k=1. For the case of a smooth soil surface (δ=0) with tangential stress (shear stress) τt= 0, the characteristic pressure on the boundary OA can be simplified as,

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p

 n  c cot   1  sin 

 cot  

 n  c cos  1  k sin 

Equation (3.35) is obtained because of the relation

(3.35)

sin 

1

 . Lim 1  sin   0 sin     

3.11.2. Bearing Capacity Factors and Lateral Earth Pressure Coefficients for Rough Interface The ultimate limit state of a bearing capacity and lateral earth pressure problem is shown in Figure 3.41. For the ultimate bearing capacity problem, q2 is the surcharge on the ground surface, and q1 is an unknown to be determined. By contrast, when active lateral earth pressure problem is considered, q1 is the surcharge acting behind the retaining wall, and q2 is the lateral earth pressure to be determined. Except for the known and unknown variables and the direction in solving the problem (from left to right or from right to left), the governing equations and conditions for the bearing capacity and lateral earth pressure problems are actually similar. Friction angle is considered in the present study, i.e., the friction angle at the interface OA is δ1; and the friction angle at the interface OD is δ2. The total slipping wedge can be divided into three limit equilibrium regions, i.e., active failure region OAB, transition region OBC and passive failure region OCD. It is also interesting to note that the present concept is somewhat similar to the concept of Ppc, Ppq, and Ppγ as adopted by Terzaghi (1943), but the pressure on the inclined surface in the present work is normal to the inclined surface (or with an angle of friction to the normal) while the concept of Ppc, Ppq, and Ppγ are always vertical pressures. Furthermore, the failure mechanism as assumed by Terzaghi (1943) is different from many other researchers or the plasticity solution as obtained in the present work. For a plane strain case, the ultimate bearing capacity of shallow foundation can be determined from the ‘superposition’ approach which has been shown to be an approximate but good assumption for normal problems by Michalowski (1997) and Cheng (2002). For a strip footing loaded vertically in the plane of symmetry, the ultimate bearing capacity pressure qu is given by the bearing capacity factors Nγ, Nc and Nq as:

1 qu  qu  quc  quq  BN r  cN c  qN q 2

(3.36)

Similarly, the total active earth pressure is considered a combination of the effects due to the weight of the soil (paγ), the cohesive strength of soil (pac) and the surcharge loading (paq). The lateral active earth pressure coefficients Kaγ, Kac, and Kaq and the total active earth pressure can be expressed as,

pa  pa  pac  paq   hKar  cKac  qKaq

(3.37)

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To derive Nq from the bearing capacity model, assume c=0 and γ=0 and a uniform surcharge q2 is applied along interface OD. The boundary conditions along the slope surface OA (active boundary) are:

p  p

quq sin 1 sin   sin   k  cos  1 sin 1   1 

  1   

1 1  1    2 2

(3.38)

(3.39)

The boundary conditions along the slope surface OD (passive boundary) are:

p

q2 sin  2 cos  2 sin  2   2 

  1     

1  2   2    2

(3.40)

(3.41)

According to equations (3.15) or (3.16), substitute the above boundary conditions into the equation (3.16) respectively and the following relation can be obtained,

 quq sin 1    1  ln   2      1  1   tan     cos  sin        2 2  1 1 1    q   sin  2 1  ln  2  2     2   2     tan   cos  sin        2  2 2 2  

(3.42)

From the above equation, Nq can be derived,

Nq 

cos  1 sin 1   1 sin  2 exp 2    1   1   2   2  tan   cos  2 sin  2   2 sin 1

(3.43)

Similar procedure can be used to derive Nc. Consider the case where q2=0 and γ=0, the boundary conditions at the base of the footing become:

p  c cot  

quc  c cot  sin 1 cos  1 sin 1   1 

(3.44)

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θ is unchanged and is same as eq.(3.36). The boundary conditions along the slope surface OD become:

p  c cot  

sin  2 c cot  cos  2 sin  2   2 

(3.45)

θ is unchanged and is same as eq.(3.41). Similarly to the previous process in deriving Nq, substitute the above boundary conditions into eq. (3.16) gives Nc as:

 cos  1 sin 1   1 sin  2  Nc   exp 2    1   1   2   2  tan    1 cot  (3.46)  cos  2 sin  2   2 sin 1  For the lateral earth pressure problem, Kaq is derived by assuming c=0 and γ=0. The boundary conditions on the surface OA are:

p  p

q1 sin 1 sin   sin   k  cos  1 sin 1   1 

(3.47)

θ is the same as eq.(3.39). The boundary conditions on the interface OD are:

p

sin  2 cos  2 sin  2   2  p aq

(3.48)

θ is the same as eq.(3.41). Substitute the above boundary conditions into equation (3.16), and Kaq can be derived as:

K aq 

cos  2 sin  2   2 sin 1 exp   2  1   1   2   2  tan   (3.49) cos  1 sin 1   1 sin  2

Considering another case where q1=0 and γ=0, the boundary conditions on the surface OA become,

p  c cot  

c cot  sin 1 cos  1 sin 1   1 

(3.50)

θ is the same as eq.(3.38). The boundary conditions along the slope surface OD become,

p  c cot  

 p ac  c cot  sin  2 cos  2 sin  2   2 

(3.51)

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It is to be noted here that the lateral earth pressure induced by cohesive strength is negative, so a sign (-) is added to pac. θ is the same as eq.(3.41). Similar to the previous process in deriving Kac, substituting the above boundary conditions into eq.(3.16) gives:

 cos  2 sin  2   2 sin 1  K ac  1  exp   2  1   1   2   2  tan   cot  (3.52)  cos  1 sin 1   1 sin  2  By comparing eq.(3.43) with eq.(3.49), it is clear that the relation between Nq and Kaq for general rough interface is given as:

N q  K aq  1

(3.53)

Combining eq.(3.46), (3.49) and (3.52) together, it is noted that the following relation among Nc, Kaq and Kac can be developed

N c  K ac K aq

(3.54)

Eq.(3.54) can also be viewed from another point of view. The active pressure along OD induced from the uniform surcharge q1 along AO will be negative for the cohesive strength. If q1 is chosen such that the lateral earth pressure on OD is exactly 0, this condition will correspond to the ultimate bearing capacity induced by cohesive strength. Using this concept, eq.(3.54) is also determined from eqs.(3.46), (3.49) and (3.52). The bearing capacity factors Nc and Nq are hence demonstrated to be related to the active lateral earth pressure coefficients Kac and Kaq by eqs.(3.49) and (3.52). For the terms Nγ and Kaγ related to self weight of soil, analytical expression is not possible and numerical computation will be adopted for the comparisons which will be explained in the later section. For Nγ, Sokolovskii (1965) and Cheng and Au (2005) and others determined it by solving eq.(3.5) and (3.6) from right to left. For the ultimate pressure at the bottom of the footing on a level ground, the slip line program SLIP by Cheng and Au (2005) starts at with a uniform distributed load (very small and tends to zero, similar to that by Martin 2004) along OD (which is horizontal now) and the construction of the slip-line field begins from the right hand side to the left hand side. The ultimate bearing stress along OA due to the weight of soil and friction angle of the soil will be a triangular pressure along OA, which are given by Sokolovskii (1965), Booker and Zheng (2000) and Cheng and Au (2005). However, the image passive pressure problem can be determined by applying a small uniform distributed load (tends to zero) on line OA as shown in Figure 3.42 and solving the slip line equations from left to right. The passive pressure along line OD from passive pressure determination is virtually the same as the bearing stress on line OA from bearing capacity determination from the program output from SLIP and KP. This approach is conceptually the reverse of the classical method of characteristic for solving the bearing capacity problem, but it is not explicitly considered in the past.

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Figure 3.42. Typical slip line pattern for lateral earth pressure problem with triangular surcharge.

A typical slip line system for the lateral pressure problem is shown in Figure 3.42, which is actually the same as the slip lines obtained by Cheng and Au (2005) or Martin (2004a,b) for bearing capacity problem. Just from Figure 3.42 alone, it is actually not possible to tell whether it is a bearing capacity problem or a lateral earth pressure problem (even though the ways of solution are the inverse). The mean pressure acting on OA is the ultimate bearing pressure due to the self weight of soil and N is then determined accordingly. The present results are actually equal to that from the Prandtl’s mechanism (Prandtl, 1920), and the results are compared with the classical solution by Sokolovskii (1966), Hansen (1970), Vesic (1973), Meyerhof (1963) and Chen (1975). Based on the slip line program ABC developed by Martin (2004a,b) for bearing capacity determination and the slip line lateral earth pressure program KP developed by Cheng (2003), the equivalent Nγ for level ground and sloping ground are given in Tables 3.9 and 3.10. Table 3.9. Comparison of N on level ground for smooth foundation by various methods of analysis (* based on asymmetric failure), and results by KP will be equal to that by ABC if asymmetric failure mechanism is adopted \N

ABC

10 20 30 40 50

0.281* 1.579* 7.653* 43.19* 372*

Sokolovskii (1965) 0.56 3.16 15.3 86.5 --

Hansen (1970) 0.39 2.95 15.07 79.54 568.33

Bolton & Lau (1993) 0.29 1.6 7.74 44 389

Vesic (1973) 1.22 5.39 22.4 109.41 762.7

Chen (1975) 0.72 3.45 15.2 81.79 --

KP 0.56 3.158 15.3 86.5 748.1

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Advanced Topics in Slope Stability Analysis Table 3.10. Comparison of N for different methods of analysis (=35) for sloping ground Vesic (1973) Hansen \N 180 48.03 33.93 170 32.59 21.38 160 19.43 12.42 150 8.58 6.18 145 4.32 3.9 Note: Correction factors by Vesic 1973 and Hansen 1970 are used.

KP 17.59 11.25 7.44 4.86 0

Since Nγ from SLIP is defined by the average value of the bearing stress along OA from bearing capacity determination (after Sokolovskii 1966) from the Prandtl’s mechanism (Prandtl, 1920), numerically, it is the same as the passive pressure Kpγ from KP, or Nγ = Kpγ

(3.55)

Nγ can be considered as the corresponding passive earth pressure coefficient, and the classical bearing capacity equation γBNγ/2 can be considered as equivalent to KpγH2/2. The classical definition of bearing capacity γBNγ/2 is a hence better definition as compared with using γBNγ, as it actually reflects the fact that bearing capacity and passive pressure problems are just image problem of each other. The author should point out a major issue for the results as given by Martin (2004a,b) and some other researchers in Table 3.9, for which the failure mechanism is based on an asymmetric failure mode. Although most of the available solutions based on the method of characteristic use an asymmetric failure mechanism, Graham et al. (1988) actually adopted a near symmetric failure mechanism for problem with a sloping ground with the method of characteristic. Cheng and Au (2005) pointed out that the results by Graham et al. (1988) were not good as compared with experimental results due to the use of symmetric failure mechanism. However, the results by Sokolovskii (1965) and other famous classical results (used by many engineers and many design codes) as given in Table 3.9 are actually based on a symmetric failure mechanism. In this respect, the author present the result from KP with a symmetric failure mechanism in Table 3.9. If the author adopt asymmetric failure similar to that by Martin (2004a,b), the results from program SLIP will be equal to that by ABC. From Tables 3.9 and 3.10, it is clear that Nγ can also be determined from the lateral earth pressure program KP by the author. It is interesting to note that when  equals to 145, N should equal to 0 when c=0. This result is correctly reflected in the active pressure program KA or program ABC but is not correct in the Vesic’s (1973) and Hansen’s (1970) bearing capacity factors for purely cohesionless soil. It should also be noted that the results by program SLIP and KP for bearing capacity and lateral earth pressure determination by Cheng and Au (2005) and Cheng (2003) and are virtually the same in Tables 3.9, 3.10 and 3.11. While Cheng developed these programs for different purposes, the extension of these programs will give similar results if they are properly used.

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Program SLIP Sokolovksii (1965) Chen (1975) Present (Automatic search)  () 10 0.56 0.56 0.72 0.57 20 3.16 3.16 3.45 3.18 30 15.31 15.3 15.2 15.3 40 86.8 86.5 81.79 86.6 Note: Solution from Slip is based on Cheng and Au (2005) using slip line solution, solution from Sokolovskii is also based on slip line solution, solution from Chen is based on limit analysis, present solution (automatic) is based on extremum principle with search for critical slip surface.

Based on the results from above, it can be concluded that the bearing capacity problem and lateral earth pressure problem are inverse image problems which can be unified under the same formulation. It is true that it is more convenient to obtain the three bearing capacity factors in the traditional way by solving eqs.(3.5) and (3.6) from right to left, but it is also completely possible to solve the bearing capacity problem by solving the problem from left to right as a lateral earth pressure problem at the expense of slightly more effort in the computation. In fact, for problem as shown in Figure 3.42, if there is loading along AO but no loading along OD, this kind of problem has been considered as bearing capacity problem as well as slope stability problem in different research papers in the past (Cheng et al. 2013a). In this respect, the classification of such a problem is sometimes arbitrary in the past, and this also illustrates that there is no difference between a bearing capacity problem and a slope stability problem by nature.

3.11.3. Relations between Slip Line and Limit Equilibrium Extremum Principle Cheng et al. (2010) demonstrated that the extremum principle provides a good approximation to the ultimate condition, and the results by extremum principle are very close to (usually equal to) the slip line solutions in general. In fact, Cheng et al. (2013b) demonstrated that the extremum principle is equivalent to the solution of partial differential equations, and the slope stability problem and bearing capacity problem can be viewed as equivalent problem under the extremum condition (Cheng et al. 2013a). In this section, the previous work by Cheng et al. (2013a) is further extended to the problem of lateral earth pressure. To determine the lateral earth pressure from the extremum principle, a prescribed slip surface is first defined and the maximum factor of safety is determined with the extremum principle by adjusting f(x) in accordance with the approach by Cheng et al. (2010). The critical slip line is then determined by the minimum factor of safety from different trial slip line which is the typical upper bound approach for slope stability analysis. The lateral force between the retaining wall and the soil is varied until the minimum factor of safety is 1.0. Under this condition, the frictional force and the lateral earth pressure arising from the extremum limit equilibrium principle should be a good approximation (usually exactly equal to) of the results from slip line analysis. Refer to Figure 3.43, the  line corresponding to AB is obtained for a specific height from the extremum limit equilibrium method. Different  lines are constructed by the extremum limit equilibrium method, then the  lines can be

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constructed by drawing lines at an angle of (/2-) to the  lines. In this way, the slip line system is obtained by the classical limit equilibrium method using the extremum principle (Cheng et al. 2010). Similarly, the same approach can be used to determine the bearing capacity factors from slip line solutions.

Figure 3.43. Construction of slip line from limit equilibrium method (=30, =20).

To determine Nc factor, a surcharge is applied underneath a foundation with a value equal to cNc without any external surcharge while the self weight is maintained at zero. For factor Nq, the self weight and cohesive strength are maintained at zero while a surcharge of 1 unit is applied outside the foundation and the foundation load is maintained at Nq. If the slip surface based on the classical Prandtl’s or Hill’s mechanism (Chen 1975) is used, the factors of safety as determined for different  are exactly 1.0 if the extremum principle is used and f(x) is kept on varying until the maximum factor of safety is found. For the term Nr, a triangular pressure is applied underneath the foundation, and the maximum pressure is adjusted until the critical factor of safety from extremum limit equilibrium method is 1.0. The critical slip surface is allowed to be automatically determined from the extremum principle in this analysis, and the resulting critical slip surface with a factor of safety 1.0 from the extremum principle will be very close to that by slip line analysis. The bearing capacity factors Nr obtained from different approaches are given in Table 3.11. For example, for =10, the slip line system using a very fine grid for the case of Nr is given in Figure 3.44. The failure zone at the right of Figure 3.44 is a typical triangular zone underneath the footing with a curved narrow transition zone, and this slip surface is far from the Hill’s or Prandtl’s mechanism as adopted by Chen (1975). The slip surface from the extremum principle is close to that as shown in Figure 3.44, and the Nr from extremum principle limit equilibrium slope stability analysis is also very close to that by the slip line method. For Nc and Nq, if the slip surfaces are not specified but is searched from the extremum principle in the way as suggested by Cheng et al. (2007a, 2010), the critical slip surfaces from extremum limit equilibrium solution are exactly equal to that from the slip line solution as shown by Cheng et al. (2013a).

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Figure 3.44. Slip line for the bearing capacity problem when =10.

It is also interesting to note that the lower bound principle has been clearly illustrated in the present analysis. For the case of factor Nc using the Prandtl’s mechanism, the author has analyzed the intermediate results during the extremum analysis. f(x) is kept on changing during the analysis, with an initial value of 1.0 for all xi during the simulated annealing optimization analysis. With f(x)=1.0, the factor of safety is actually 0.925 which is far from 1.0. As f(x) is changed, the factor of safety converges towards 1.0 during the simulated annealing analysis, but no factor of safety greater than 1.0 can be obtained. The same results are also obtained for the case of Nq as well. The results as shown in Figure 3.45 are actually good illustration of the lower bound principle. 1600 1400

Number

1200 1000 800 600 400 200

0. 93 -0 .9 4 0. 95 -0 .9 6 0. 96 -0 .9 7 0. 97 -0 .9 8 0. 98 -0 .9 9 0. 99 -0 .9 91 0. 99 10. 99 2 0. 99 20. 99 3 0. 99 30. 99 4 0. 99 40. 99 5 0. 99 50. 99 6 0. 99 60. 99 6 0. 3 99 63 -0 .9 96 5 0. 99 65 -0 .9 97 0. 99 70. 99 76 0. 99 76 -0 .9 97 0. 7 99 77 -0 .9 97 0. 8 99 78 -0 .9 97 0. 85 99 78 50. 99 78 0. 99 8 78 80. 99 79 0. 99 79 -0 .9 98

0

Interval

Figure 3.45. Distribution of acceptable factor of safety during simulated annealing analysis for Nc (same results for Nq) with =30 using f(x) as the variable.

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For the lateral earth pressure, consider the case of a vertical wall with a wall friction  and the ground is inclined at an angle  above the horizontal direction. The active pressure coefficient can be determined from slope stability method in the following way. Refer to Figure 3.43, a normal force FORCE on the retaining wall AC is assumed, then the wall friction is given by FORCExtan. The normal and tangential forces are applied at the vertical boundary AC, and soil mass ABC is considered as a slope stability problem. For any prescribed failure surface, the extremum of the factor of safety (by treating f(x) as variable) from limit equilibrium analysis is obtained. Upper bound analysis is then carried out to determine the minimum factor of safety. If the minimum factor of safety from the extremum is not equal to 1.0, the value of FORCE will be adjusted until a minimum factor of safety 1.0 is achieved. The lateral earth force and hence lateral pressure coefficient is then determined. From the results as shown in Table 3.12, it is clear that there is no practical difference between a lateral earth pressure problem and a slope stability problem, provided that the critical extremum solution from slope stability analysis is used for comparison. Table 3.12. Active pressure coefficients from slip line analysis and extremum principle Case =10, =10

=15, =10

 () 20 30 40 20 30 40

Slip line 0.531 0.35 0.225 0.61 0.379 0.23

Extremum Principle 0.54 0.353 0.228 0.615 0.382 0.232

3.11.4. Unification of Lateral Earth Pressure, Bearing Capacity and Slope Stability Problems under Ultimate Condition In previous sections, the author has applied the method of characteristics to study two classical and important geotechnical problems: active lateral earth pressure problem and ultimate bearing capacity problem. The active lateral earth pressure and bearing capacity problems are demonstrated to be equivalent, except for the ease of mathematically manipulation. Based on the slip line theory, the coefficients Nq, Nc, Kaq and Kac are derived and the relations among them are established. Two finite difference programs ABC and KA are further used to confirm the validity of the results as derived in the present study. Cheng and Au (2005) found that iterative analysis is required for normal bearing capacity problem but not for lateral active earth pressure (Cheng 2003). When the retaining wall gradually becomes horizontal and behaves as a bearing capacity problem, it is found that iterative analysis will be necessary for the active pressure program to achieve a good result (similar to slip line analysis of bearing capability problem). Other than the ways in solving eqs.(3.5) and (3.6), there are no practical difference between the two classical geotechnical problems, and the two problems can actually be viewed as equivalent problems under the ultimate condition. It is demonstrated that classical bearing capacity and active lateral earth pressure problems are actually the same, as they are controlled by both the yield and equilibrium

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equations. Based on the present study, for a normal shallow foundation problem, it can be viewed as a lateral earth pressure problem and vice versa. The surcharge (ultimate bearing capacity) behind an imaginary retaining wall which generate a net upward stresses outside the foundation (or equivalently the retaining wall) will be the ultimate limit state of the system. A bearing capacity problem can hence be viewed as a lateral earth pressure problem in this manner. The author has demonstrated the lower bound principle in Figure 3.45 using the extremum analysis. Based on the minimum of the extremum from limit equilibrium method, the author has also demonstrated that there are no practical differences between the slope stability problem and the lateral earth pressure and bearing capacity problems, provided that the extremum from the slope stability analysis is used in the comparisons. Based on the present study, it can be concluded that the three geotechnical problems are practically the same problem – the ultimate condition of the system where the maximum resistance of the system is fully mobilized. In this respect, the three geotechnical problems can be considered as equivalent under the ultimate condition. It is interesting to note that there are many previous laboratory tests conducted for slopes with surcharge on top as well as bearing capacity problem with inclined ground, as reported in various literature. Actually, the two types of tests are exactly the same, but different researchers prefer to consider the same problem as either slope stability problem or bearing capacity problem. For example, Vesic (1973), Shield et al. (1977) and Graham et al. (1988), Cheng and Au (2005) considered a bearing capacity problem with an inclined ground by laboratory tests as well as numerical analysis. Empirical correction factors have also been put forward for bearing capacity with inclined ground by Vesic (1973), Meyerhof (1963), Hansen (1970) which are well received by the engineers. On the other hand, the same problem is also termed as slope with a top surcharge by many researchers. For example, Kusakabe et al. (1981), Buhan and Garnier (1988) termed the problem as a slope stability problem but are actually determining the bearing capacity factors in their works. It is also interesting to note that the laboratory slope tests with a top surcharge by Manna et al. (2014) and Li and Cheng (2015) are practically the same as the bearing capacity laboratory bearing capacity tests by Shield et al. (1977). In the limit analysis or slip line analysis of this problem, Graham et al. (1988) and Cheng et al. (2015) considered only failure mechanism without actually classifying the problem as a bearing capacity or slope stability problem. In the laboratory bearing capacity/slope stability tests by Yoo (2001), the author also does not have a clear distinction between slope stability and bearing capacity problem. Based on these previous works, it is clear that there is not a real difference between the slope stability and bearing capacity problem in literature, which is also illustrated clearly in the present work. Overall, it can be said that there is no physical difference between slope stability, lateral earth pressure or bearing capacity problems. Each problem can be viewed as the other problem from a slightly different point of view. Such unification is not surprising, as all the three problems are controlled by the same yield and ultimate condition criterion. The author’s program can also be used for all the three problems, with only minor changes in the procedures of operation, but no change to the computer coding is actually needed. Hence, all these three important geotechnical problems can be considered as unified if the ultimate condition is considered. Actually, the slope stability problem is also a determinate problem under this condition.

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3.12. SOME NEW METHODS OF ANALYSIS In recent years, some new slope stability methods have evolved. These methods are still green at present, and have not been studied in great details by different researchers. The author also views that there may be various limitations similar to that in other numerical methods, nevertheless, such methods are worth to be discussed. Out of the various new methods, the author has chosen two methods for discussion: Smoothed-particle hydrodynamics (SPH) method and the Spectral Element Method.

3.12.1. Smoothed-Particle Hydrodynamics Method SPH is a new computational method used for simulating the dynamics of continuum media, such as solid mechanics and fluid flows, and its application has been extended to slope stability analysis in recent years. SPH is developed by Gingold and Monaghan (1977) and Lucy (1977) initially for astrophysical problems. Details of the SPH can be found in Liu & Liu (2003) and Monoghan (2005). Since the strain and strain rate can be large in SPH, the use of Eulerian Jauman stress rate is commonly adopted. This method is actually a mesh-free non-local Lagrangian method (where the coordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to various control variables. The SPH method works by dividing the fluid into a set of discrete elements, referred to as particles. These particles have a spatial distance (known as the "smoothing length"), over which their properties are "smoothed" by a kernel function. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. The contributions of each particle to a property are weighted according to their distance from the particle of interest, and their density. Kernel functions commonly used include the Gaussian function and the cubic spline. The latter function is exactly zero for particles further away than two smoothing lengths (unlike the Gaussian, where there is a small contribution at any finite distance away). This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles. The advantages of SPH over classical finite element method (FEM) is that SPH can tolerate very large movement which is not possible for the FEM. There are basically two approaches to model the geomaterials in SPH: computational fluid dynamic (CFD) approach and the plasticity/damage mechanics approach. In the fluid dynamics modeling, it is not necessary to consider mesh deformation as the mesh is fixed in space. However, we must assume that the geomaterials equivalent type of fluid (Moriguchi, 2005; Moriguchi et al., 2005). In the SPH method, the motion of a continuum is modelled using a set of moving particles; and each particle is assigned a constant mass and carries field variables at the corresponding location. The continuous fields are then interpolated from the various particles by a weighted summation, in which the weights fall off with distance according to an assumed kernel function. The range of ‘interparticle interaction’ then corresponds to the radius of the ‘support domain’ of the kernel. Similarly, spatial derivatives are calculated through an interpolation process over the support domain. This approach is effective for flow problems, but is difficult to use to solve static deformation problems. In addition, it is difficult to use classical constitutive models, because the approach cannot easily

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handle the stress/strain history of a material during deformation. Discrete modeling uses an assembly of discrete elements, and is inappropriate for dealing with constitutive models of geomaterials based on a continuum approximation. The shear strength of soil which is governed by the Mohr-Coulomb criterion can be introduced into a Bingham fluid model by this analogy: .

  0   f

for Bingham fluid and  f  n tan   c ' for geomaterials

.

(3.56)

  0  (n tan   c ' ) Viscosity of the Newtonian fluid can then be related to the shear strength by .

 '  0  (n tan   c ' ) /  for  '  max  max for  '  max

(3.57)

SPH model is well suited to landslide or debris flow analysis, as well as erosion, mud flow, sediment transport or similar approach. Compared with the DEM which is discontinuum based, SPH is continuum based with the advantages of able to model very large displacement with faster computation time while the classical constitutive model can be defined easily. No micro-parameters as appear in the DEM is required for the analysis, which is an important advantage to engineering application. For elastic analysis, the results from SPH can be compared directly with that by elasticity theory or finite element analysis, hence the results are quantitative as compared with the qualitative in the DEM. Bui et al. (2007, 2008, 2010, 2011) demonstrated the application of SPH for varieties of applications. Bui et al. (2007) also demonstrated the use of SPH for two and three-dimensional slope stability analysis, and the results from SPH can agree well with the classical limit equilibrium method. Various researchers have applied SPH in slope stability problems. Ono (2012) and Bui et al. (2010) applied SPH for earthquake induced slope failures. Nonoyama et al. (2013, 2015) demonstrated that the results from SPH and Fellenius method agree reasonably well for both frictional and frictionless soil. The shear strain distribution and critical failure surface appear to be similar to that by the SRM (from visually observation by the author). Similar works have also be accomplished by Wu et al. (2015a, b) for slope stability involving soil-rock interaction and soil-water interaction. Bui et al. (2008) considered rainfall induced slope failure by SPH. Bui et al. (2011) investigated the numerical instability problem for the toe of a slope in SPH, for which similar problem will not happen for SRM. SRM may sometimes face difficulty in identifying the precise factor of safety, as unconverged solution is difficult to be identified for nail/pile reinforced slope. Since SPH is practically similar to the SRM, it bears the same problem of difficulty in identifying the factor of safety in some cases as the SRM. Bui et al. (2011) proposed to plot the relations between the maximum displacement of soil particles and the number of user-specified iterations for each trial factor of safety. The number of user-specified iterations depends on each specific problem, and normally corresponds to about 1.0 s of physical time. Based on this relation, curves that remain convex within the specified number of iterations correspond to convergent SPH solutions. On the other hand, a curve that becomes concave represents an unconverged SPH solution, and the

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lowest corresponding SRF is considered to be the safety factor of the slope. Such approach is actually practically equivalent to that in SRM. Since SPH is not tested for reinforced slopes, the problem for factor of safety determination in SRM will possibly be present in the SPH analysis. In general, most of the slope stability problem that can be considered by SPH can be better analyzed by the SRM at present, at SRM is much mature than SPH at present. SPH does offer some advantages in that very large displacement analysis can be modelled with reduced computer time as compared with SRM. In the future, when the numerical instability problems in SPH are fully explored and solved, SPH can be a competitive alternative to the SRM for solving practical engineering problems.

3.12.2. Spectral Element Method (SEM) Spectral method is a form of discretization methods for the approximate solution of partial-differential equations which can be expressed in a weak form. Higher order Lagrangian interpolants are used in conjunction with particular quadrature rules. For the spectral element method, it is assumed that the solution can be expressed as a series of polynomial basis functions, which can approximate the solution well in some norm when the polynomial degree tends to infinity. These smooth basis-functions usually form an L2complete basis. For the sake of computational efficiency this basis is typically chosen to be orthogonal in a weighted inner-product. The spectral element method is a high-order finite element technique that combines the geometric flexibility of finite elements with the high accuracy of spectral methods. This method originated in the mid 1980's by Patera (1984). The advantages of this method include the natural diagonal mass matrices which is highly efficient in numerical computation, adequacy to implementations in a parallel computer system. Due to these advantages, the spectral element method is a viable alternative to currently popular methods such as finite volumes and finite elements, if accurate solutions of regular problems are sought. FEM uses a separate nodes/points for the numerical integration and interpolation, and it will produce a relatively dense matrix and increased computing time. The integrals in SEM are evaluated based on the Gauss–Lobatto–Legendre (GLL) quadrature points, since both integration point and interpolation point are the same, it will lead to a diagonal matrix and therefore simplifies the analysis. The method adopts tensor-product Lagrange interpolants within each element, where the nodes of these shape functions are placed at the zeros of Legendre polynomials (Gauss-Lobatto points), mapped from the reference domain [-1, 1] x [1, 1] to each element. For smooth functions, it can be shown that the resulting interpolants converge exponentially fast as the order of the interpolant is increased. The efficiency in numerical computation is achieved by using the Gauss-Lobatto quadrature for evaluating elemental integrals. The quadrature points reside at the nodal points, which enables fast tensor-product techniques to be used for iterative matrix solution methods. Gauss-Lobatto quadrature hence naturally results in diagonal mass matrices which is computational light as compared with the classical finite element method. Depending on the size of the resulting matrix, two approaches are commonly used to solve the matrix equation. When matrix size is large but the degree of polynomial is small, the use of preconditioned conjugate gradient method or similar is popular. For the reverse condition, the Schur complement is commonly

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adopted. The Spectral element method is originally developed for solving the computational fluid dynamic problems, but is later extended to the solution of many problems in solid mechanics. Gharti et al. (2012, 2017) considered elasto-plastic 3D SEM analysis with parallel implementation. Similar to SRM and SPH, the factor of safety is determined from plotting the maximum displacement against the trial factor of safety, for which the author has found it to be less reliable when there is soil reinforcement in a slope. Gharti et al. (2012) also found that the results from SRM and SEM agree very well in general. Tiwari et al. (2013, 2014, 2015) carried out series of works in applying SEM in slope stability problems. For interested readers, the open source program Specfem3D-Slope (version 1.2 at present) by Gahrti et al. (2017) is a good reference for trying the SEM slope stability analysis. Detailed documentation is provided so that the users can operate the program to solve many slope stability problems. Comparing the SEM and SPH, the most distinct difference is the ability of SPH to handle very large deformation which is not possible with the SEM. The SEM and SPH are two important new methods, for which the author view them to be still not mature for practical routine applications. The most important limitation of both methods is the inherent numerical problems which may come out. For the rigorous LEM, it takes about 50 years before all the numerical problems are identified and solved (by double QR method, global optimization methods and extremum principle). For the SRM which has about 40 years of history, the internal tests by the author has shown that even for the most current versions of the SRM programs, numerical problems are found for all the commercial SRM programs that have been tested by the author. No single commercial program can avoid numerical problem under all cases. It is true that many updated SRM programs have avoided some major numerical problems, however, the author is able to produce numerical problems for all commercial SRM programs under certain combination of soil parameters, geometry or soil reinforcement. For the SPH and SEM, the author expects that it will also takes a long time before these two methods are mature and stable enough for normal applications.

Chapter 4

THREE-DIMENSIONAL SLOPE STABILITY ANALYSIS As discussed in previous chapters, all failures are practically three-dimensional in nature, but most of the analyses are carried out on two-dimensional basis. Such discrepancy is not surprising, as discussed previously. For three-dimensional slope stability analysis, the twodimensional methods that have been discussed in previous chapters can be extended to threedimensional analysis with minor modifications. There are however some basic differences between two-dimensional and three-dimensional analysis which will be discussed in this chapter. This chapter will emphasize on the three-dimensional limit equilibrium method, as this is the most commonly used method in practice.

4.1. LIMITATIONS OF THE CLASSICAL THREE-DIMENSIONAL LIMIT EQUILIBRIUM METHODS All slope failures are practically three-dimensional (3D) in nature, and near twodimensional failures are scarce. Two-dimensional analysis is however usually carried out in practice, as this will greatly simplify the analysis with the limited site investigation that are available for normal engineering projects. There are many drawbacks in most of the existing 3D limit equilibrium slope stability methods which include: 1) Direction of slide is not considered in most of the existing slope stability formulation so that the problems under consideration must be symmetrical in geometry, subsurface condition and loading. Some formulation vary the direction of slide using a symmetric formulation until the minimum factor of safety is found for a given prescribed failure surface, but such approach is highly inefficient due to the various geometry determination required for the axes rotation. This approach is particularly unacceptable if location of the critical 3D failure surface is required. 2) Location of the critical non-spherical 3D failure surface under general conditions is a difficult N-P hard type global optimization problem which has not been solved effectively and efficiently. For 3D problems, this is particularly important, as much more variables are required to define a general 3D surface than a 2D curve. 3) Existing 3D limit equilibrium methods of analyses are numerically unstable under transverse horizontal forces.

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Because of the above mentioned limitations, 3D analysis based on limit equilibrium is actually less convenient as compared with the 3D SRM analysis, which is contrary to the 2D situation. Nevertheless, many engineers still prefer 3D limit equilibrium analysis, as they can get a feeling of the problem easily from their past 2D experience. Cavounidis (1987) demonstrated that the factor of safety for a normal slope under three-dimensional analysis was greater than that under two-dimensional analysis, but this result was based on the same soil parameters for 3D and plane strain cases which was actually not correct. 3D factor of safety is usually several percent higher than the corresponding 2D analysis, however, after the adjustment of the soil parameters for plane strain condition which is about 10% higher than the 3D condition, the factor of safety for 3D analysis may actually be lower than the corresponding 2D analysis, and this is one of the reason for the actual 3D failures observed in practice. Another critical factor will be the different ground conditions for 2D and 3D consideration. Baligh and Azzouz (1975), Azzouz and Baligh (1983) presented a method that extended the concepts of the 2-D circular arc shear failure method to 3-D slope stability problems. The method was just appropriate for a slope in cohesive soil. The results obtained by the method showed that the 3-D effects could lead to 4 to 40 percent increase in the factor of safety. Hovland (1977) proposed a general 3-D method for cohesion-frictional soils. The method was an extension of the 2-D Ordinary Method of Slices (Fellenius, 1927). The inter-column forces and pore-water pressure were not considered in this formulation. Two special cases have been analyzed: (a) a cone-shaped slip surface on a vertical slope, and (b) a wedge-shaped slip surface. It was shown that the 3-D factors of safety were generally higher than the 2-D ones, and the ratio of factor of safety in 3-D to that in 2-D was quite sensitive to the magnitudes of cohesion and friction angles and to the shape of the slip surface in 3-D. Chen and Chameau (1982) extended Spencer 2-D method to 3-D. The sliding mass was assumed to be symmetrical and divided into several vertical columns. The inter-column forces directions were assumed to be constant throughout the mass (which were obviously not possible), and the shear forces were parallel to the base of the column, and such assumptions actually came from the corresponding 2D Spencer formulation. It was shown that: (a) the configuration of a sliding mass in 3-D had significant effects on the factor of safety when the length of the sliding mass was small; (b) for gentle slopes, the dimensional effects were significant for soils with high cohesion and low friction angles;(c) under certain circumstances the 3-D factor of safety for cohesionless soils might be slightly less than the 2D one. Hungr (1987) directly extended Bishop simplified 2-D method of slices (1955) to analyze the slope stability in 3-D. The method was derived based on the two key assumptions: (a) the vertical forces of each column were neglected; (b) both the lateral and the longitudinal horizontal force equilibrium conditions were neglected. Hungr et al. (1989) presented a comparison of 3-D Bishop and Janbu simplified methods (1957) with other published limit equilibrium solutions. It was concluded that the Bishop simplified method might be conservative for some slopes with non-rotational and asymmetric slip surfaces. Zhang (1988) proposed a simple and practical method 3-D stability analysis for concave slopes using equilibrium concepts. The sliding mass was symmetrical and divided into many vertical columns. The slip surface was approximately considered as the surface of an elliptic revolution. To render the problem statically determinate, the forces acting on the sides and ends of each column which were perpendicular to the potential direction of movement of the

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sliding mass, were neglected in the equilibrium conditions. The investigations using this method showed that: (a) the stability of concave slopes in a plane view increased with the decreases in their relative curvature; (b) the effect of a plane curvature on the stability of concave slopes increased with the increase in the lateral pressure coefficient. However, the lateral pressure coefficient had only a small effect on the stability of the straight plane. By using the method of columns, Lam and Fredlund (1993) extended the 2-D general limit equilibrium formulation (Fredlund and Krahn, 1977) to analyze a 3-D slope stability problem. The inter-column force functions of an arbitrary shape to simulate various directions for the inter-column resultant forces were proposed. All the inter-column shear forces acting on the various faces of the column were assumed to be related to their respective normal forces by the inter-column force functions. A geostatistical procedure (i.e., the Kriging technique) was used to model the geometry of a slope, the stratigraphy, the potential slip surface, and the pore-water pressure conditions. It was found that the 3-D factors of safety determined by the method (Lam and Fredlund, 1993) were relatively insensitive to the form of the inter-column force functions used in the method. Lam and Fredlund (1993, 1994) however had not given a clear and systematic way for solving a general 3D problem. Chang (2002) developed a 3-D method of analysis of the slope stability based on the sliding mechanism observed in the 1988 failure of the Kettleman Hills Landfill slope and the associated model studies. Using a limit equilibrium concept, the method assumed the sliding mass as a block system in which the contacts between the blocks were inclined. The lines of intersection of the block contacts were assumed to be parallel, which enabled the sliding kinematics. In consideration of the differential straining between blocks, the shear stresses on the slip surface and the block contacts were evaluated based on the degree of shear strength mobilization on those contacts. The overall factor of safety was calculated based on the force equilibrium of the individual block and the entire block system as well. Due to the assumed inter-block boundary pattern, the method was not fully applicable for dense sands or overly consolidated materials under drained conditions. Zheng (2012) developed a mesh/column free method where the stability of the global mass was considered without the use of inter-column force functions. Under such formulation, the internal acceptability of the system will not be checked. This method can give factor of safety similar to other methods for normal problem with smooth slip surface, and this is a reflection that the internal forces will not have great effect on the analysis for smooth slip surfaces. The method by Zheng (2012) can give thrust line which is outside the soil mass, and the Mohr-Coulomb criterion can be violated within the soil mass, but this method appears to be acceptability for normal slip surfaces, while the results may need further consideration for irregular slip surfaces. For the internal forces of non-smooth failure surface from this approach, the author views that the results have to be accepted with care. 3-D stability formulations based on the limit equilibrium method and variational calculus have been proposed by Leshchinsky et al. (1985), Ugai (1985, 1988), and Leshchinsky and Baker (1986). The functionals are the force and/or moment equations where the factor of safety can be minimized while satisfying several other conditions. The shape of a slip surface can be determined analytically. In such approaches, the minimum factor of safety and the associated failure surface can be obtained at the same time. These methods were however limited to homogeneous and symmetrical problems only. In the follow-up studies, Leshchinsky and Huang (1992) developed a generalized approach which is appropriate for symmetrical slope stability problems only. The analytical solutions approach based on the

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variational analysis are difficult to obtain for practical problems with complicated geometric forms and loading conditions. Cheng et al. (2013c) proposed to use modern optimization method to replace the tedious variational principle, and have shown that for a partial differential equation system which can be written as a control functional, the use of modern optimization can be a practical alternative to the classical variational principle. Most of the existing 3D methods rely on an assumption of a plane of symmetry in the analysis, and the commonly used methods are summarized in Table 4.1. For complicated ground conditions, this assumption is no longer valid, and the failure mass will fail along a direction with least resistance so that the sliding direction will also control the factor of safety of a slope. Stark and Eid (1998) have also demonstrated that the factor of safety of a 3D slope is controlled by the direction of slide and a symmetric failure may not be suitable for general slope. Yamagami and Jiang (1996, 1997) and Jiang and Yamagami (1999) developed the first method for asymmetric problems where the classical stability equations (without direction of slide/direction of slide is zero) are used while the direction of slide is considered by a minimization of the factor of safety with respect to rotation of axes. Yamagami and Jiang formulation (1996, 1997) could be very time consuming even for a single failure surface, as the formation of columns and the determination of geometry information with respect to rotation of axes was the most time-consuming computation in the stability analysis. Huang and Tsai (2000) proposed the first method for 3D asymmetrical Bishop method where sliding direction enters directly into the determination of the safety factor. The generalized 3D slope stability method by Huang et al. (2002) is practically equivalent to the Janbu rigorous method with some simplifications on the transverse shear forces. Since it is difficult to satisfy completely the line of thrust constraints in Janbu rigorous method which is well known in 2D analysis, the generalized 3D method by Huang et al. (2002) will also face converge problem so that this method is less useful to practical problems. At the verge of failure, the soil mass can be considered as a rigid body. The direction of slide can take three possibilities: 1) Soil columns are moving in the same direction with unique sliding direction – adopted by Cheng and Yip (2007) and many other researchers in the present formulation. 2) Soil columns are moving towards each other – violate the assumption of rigid failure mass and is not considered. 3) Soil columns are moving away from each other – adopted by Huang and Tsai (2000), Huang et al. (2002). Such condition can also be studied by the use of DEM. Since the sliding directions of soil columns are not unique in Huang and Tsai formulation (2000) and some columns are moving apart, the summation process in determining the factor of safety may not be applicable as some of the columns may be separating from the others. Cheng and Yip (2007) demonstrated that under transverse load, the requirement of different sliding directions for different soil columns may leads to failure to converge. For soil columns moving away from each other, distinct element method is the recommended method of analysis. Since the parameters required for distinct element analysis is different from the classical soil strength parameters, it is not easy to adopt the results from distinct element analysis directly, and the results should be considered as the qualitative analysis of the slope stability problem.

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The assumption of a unique sliding direction may be an acceptable formulation for the analysis of the ultimate limit state, and the present formulation is based on this assumption. It is not a bad assumption to assume that all soil columns slide in one unique direction at the verge of failure. From the observations of many 3D slope failures in Hong Kong and other countries, the author view that this assumption is reasonable and realistic, at least up to the verge of failure. After failure has initiated, the soil columns may separate from each other and sliding directions can be different among different columns. In this respect, the use of a unique sliding direction is sufficient to determine the factor of safety for engineering analysis and design. Table 4.1. Summary of Some 3-D Limit Equilibrium Methods Method Hovland (1977) Chen and Chameau (1982) Hungr (1987) Lam and Fredlund (1993) Huang and Tsai (2000) Cheng and Yip (2007)

Related 2-D method Ordinary Method of Slice Spencer method Bishop simplified method General limit equilibrium Bishop simplified method Bishop and Janbu simplified, Morgenstern-Price

Assumptions No inter-column force Constant inclination Vertical equilibrium Inter-column force function Consider direction of slide Consider direction of slide

Equilibrium Overall moment equilibrium Overall moment equilibrium Overall force equilibrium Overall moment equilibrium Vertical force equilibrium Overall moment equilibrium Overall force equilibrium Overall moment equilibrium Vertical force equilibrium Overall and local force equilibrium, and overall moment equilibrium for M-P

4.2. NEW FORMULATION FOR 3D SLOPE STABILITY ANALYSIS BY AUTHOR – BISHOP, JANBU SIMPLIFIED, MORGENSTERN-PRICE METHOD There are many major issues in 3D limit equilibrium analysis, as compared with the corresponding 2D analysis. In this section, the formulation by the author is discussed, which has advantages over other existing formulations. The readers are however strongly advised to read the other existing formulations to gain sufficient depth about 3D slope stability analysis.

4.2.1. Basic Formulation with Consideration of Sliding Direction For three-dimensional analysis, the potential failure mass of a slope is divided into a number of columns, which is a simple extension of the corresponding 2D analysis. At the equilibrium condition, the internal and external forces acting on each soil column are shown in Figure 4.1. Weight of soil and vertical load are assumed to act at the centre of each column for simplicity. This assumption is not exactly true, but is good enough if the width of each column is small enough, and the resulting governing equations will be greatly simplified and

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should be sufficiently good for practical purposes. For practical purposes, the author uses more than 10000 columns for many 3D slope analysis. The assumptions required in the present 3D formulation are: 1) Mohr-Coulomb failure criterion is valid. 2) For Morgenstern-Price method, the factor of safety is determined based on the sliding angle where factors of safety with respect to force and moment are equal. 3) Sliding angle is the same for all soil columns (Figure 4.2). By Mohr-Coulomb Criteria, the global factor of safety, F, is defined as F

S fi Si



Ci  N i'  tan  i Si

(4.1)

where F is the factor of safety, Sfi is the ultimate resultant shear force available at the base of column i, N’i is the effective base normal force and Ci is c’Ai and Ai is the base area of the column. The base shear force S and base normal force N with respect to x, y and z directions for column i are expressed as the components of forces by Huang and Tsai (2000, 2002):

S xi  f1  S i ; S yi  f 2  S i ; and S zi  f 3  S i N xi  g1  N i ; N yi  g 2  N i ; and N zi  g 3  N i

(4.2)

in which  f1  f 2  f 3  and g1  g 2  g 3 are unit vectors for Si and Ni (see Figure 4.1). The projected shear angle a’ (individual sliding direction) is the same for all the columns in the xy plane in the present formulation, and by using this angle, the space shear angle ai (see Figure 4.3) can be found for each column and is given by Huang and Tsai (2000) as eq.(4.3):

ai  tan 1{sin i /[cosi  (cos ayi / tan a ' cos axi )]}

Figure 4.1. External and Internal forces acting on a typical soil column.

(4.3)

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where: ai space sliding angle for sliding direction with respect to the direction of slide projected to x-y plane (see also a’ in Figure 4.2 and eq.(4.3)) ax, ay base inclination along x and y directions measure at center of each column (shown at the edge of column for clarity) Exi, Eyi Intercolumn normal forces in x and y directions, respectively Hxi, Hyi Lateral intercolumn shear forces in x and y directions, respectively Ni’, Ui Effective normal force and base pore water force, respectively Pvi, Si Vertical external force and base mobilized shear force, respectively Xxi, Xyi Vertical intercolumn shear force in plane perpendicular to x and y directions Figure 4.1 External and Internal forces acting on a typical soil column

  tan axi  tan a yi 1  ni   , ,    g1 , g 2 , g3  [-ve adopted by Huang and Tsai (2000) J J  J and +ve adopted by Cheng and Yip (2007)]  sin( i  ai )  cos a xi sin ai  cos a yi sin( i  ai )  sin a xi  sin ai  sin a yi  si   , ,   f 1 , f 2 , f 3  sin  i sin  i sin  i  

in which

J  tan 2 a xi  tan 2 a yi  1

Figure 4.2. Unique sliding direction for all Columns (on plan view).

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Figure 4.3. Relationship between projected and space shear angle for the base of column i.

An arbitrary intercolumn shear force function f(x,y) is assumed in the present analysis, and the relationships between the intercolumn shear and normal forces in the x and ydirection are given as:

Xxi  = Exi  f ( x, y)  x ; Xyi  = Eyi  f ( x, y)  y

(4.4)

Hxi  = Eyi  f ( x, y)  xy ; Hyi  = Exi  f ( x, y)  yx

(4.5)

where x and y = intercolumn shear force X mobilization factors in x and y directions respectively xy and yx = intercolumn shear force H mobilization factors in xy and yx planes respectively Taking moment about z-axis at the center of the ith column, the relations between lateralintercolumn shear forces can be expressed as:

yi  ( Hxi 1  Hxi ) = xi  ( Hyi 1  Hyi )

(4.6)

From eq.(4.6), Hxi 1 

xi  ( Hyi 1  Hyi )  Hxi yi

(4.7)

From eq.(4.6), Hyi 1 

yi  ( Hxi 1  Hxi )  Hyi xi

(4.8)

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where xi and yi are the widths of the column defined in Figure 4.4. Hxi and Hyi for the exterior columns should be zero in most cases or equal to the applied horizontal forces if defined. By using the property of complementary shear (or moment equilibrium in the xyplane), Hyi+1 or Hxi+1 can then be determined from eq.(4.5) and (4.7) or (4.8) accordingly, so only xy or yx is required to be determined but not both. The important concept of complementary shear force which is similar to the complementary shear stress (xy=yx) in elasticity has not been used in any 3D slope stability analysis method in the past but is crucial in the present formulation. It should be noted that Huang et al. (2002) actually assumed Hyi to be 0 for asymmetric problem in order to render the problem determinate which is valid for symmetric failure only. Although the concept of complimentary shear is applicable only in infinitesimal sense, if the size of column is not great, this assumption will greatly simplify the equations. More importantly, Cheng and Yip (2007) demonstrated that the effect of xy or yx is small in the later section of this chapter and the error in this assumption is actually not important.

Figure 4.4. Force Equilibrium in x-y plane.

Figure 4.5. Horizontal Force Equilibrium in x-direction for a typical column (Hxi = Net lateral intercolumn shear force).

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Figure 4.6. Horizontal Force Equilibrium in y-direction for a typical column (Hyi = Net lateral intercolumn shear force).

Figure 4.7. Moment equilibrium in x- and y-direction.

(Earthquake Loads and Net external moments are not shown for clarity).

4.2.1.1. Force Equilibrium in X, Y and Z Directions Considering the vertical and horizontal forces equilibrium for ith column (Figure 4.5 and 4.6) in z, x and y directions give:

Fz  0  Ni  g3i  Si  f3i  (Wi  Pvi )  ( Xxi 1  Xxi )  ( Xyi 1  Xyi )

(4.9)

Fx  0  Si  f1i  Ni  g1i  Phxi  Hxi  Hxi 1  Exi 1  Exi

(4.10)

Fy  0  Si  f 2i  Ni  g2i  Phyi  Hyi  Hyi 1  Eyi 1  Eyi

(4.11)

Solving (4.1), (4.4) and (4.9), the base normal and shear forces can be expressed as:

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Ni  Ai  Bi  Si ;

Si 

Ai 

Ci  ( Ai  U i )  tan i B  tan i F (1  i ) F

Wi  Pvi  Exi  x  Eyi  y g3i

Bi  

(4.12)

;

f 3i ; Ui=uiAi (ui = average pore pressure at i-th column) g 3i

4.2.1.2. Overall Force and Moment Equilibrium in X and Y Directions Considering the overall force equilibrium in x-direction:

 Hxi   N i  g1i   Si  f1i  0

(4.13)

Let Fx = F in equation (4.1), using (4.12) and rearranging equation (4.13), the directional safety factor Fx can be determined as:

Fx 

[( Ni  U i )  tan i  Ci )]  f1i ,  Ni  g1i   Hxi

0 < Fx