Modeling in Geotechnical Engineering [1 ed.] 0128212055, 9780128212059

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MODELING IN GEOTECHNICAL ENGINEERING

MODELING IN GEOTECHNICAL ENGINEERING

Edited by

PIJUSH SAMUI Associate Professor, Department of Civil Engineering, NIT Patna, Patna, Bihar, India

SUNITA KUMARI Associate Professor, Department of Civil Engineering, NIT Patna, Patna, Bihar, India

VLADIMIR MAKAROV Professor, Head of Laboratory on “Geomechanics of Highly Compressed Rock & Rock Mass” Far-Eastern Federal University (FEFU), Vladivostok, Russia

PRADEEP KURUP Professor and Chair, Civil and Environmental Engineering, University of Massachusetts Lowell, Lowell, MA, United States

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-821205-9 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Brian Guerin Editorial Project Manager: Gabriela D. Capille Production Project Manager: Sojan P. Pazhayattil Cover Designer: Miles Hitchen Typeset by SPi Global, India

Contributors

Nicholas A. Alexander United Kingdom

Structural Dynamics, University of Bristol, Bristol,

M. Aleem SAGE Laboratory, University of Surrey, Guildford, United Kingdom Sadra Amani Kingdom

SAGE Laboratory, University of Surrey, Guildford, United

Daniel Barreto Edinburgh Napier University, Edinburgh, United Kingdom Subhamoy Bhattacharya Geotechnical Engineering, University of Surrey, Guildford, United Kingdom Neeraj Chaudhary Jharkhand, India

Department of Civil Engineering, NIT Jamshedpur,

Gabriele Chiaro Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand Wu Chongzhi Key Laboratory of New Technology for Construction of Cities in Mountain Area; National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas; School of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China Manas Ranjan Das Civil Engineering Department, ITER, SOA University, Bhubaneswar, Odisha, India Sarat Kumar Das Civil Engineering Department, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, India Hasan Emre Demirci Izmir, Turkey

Geotechnical Engineering, Izmir Katip Celebi University,

Anthony Goh (Teck Chee) School of Civil and Environmental Engineering, Nanyang Technological University, Singapore, Singapore Mikhail A. Guzev Institute for Applied Mathematics FEBRAS, Vladivostok, Russia Raoul H€ olter Department of Civil and Environmental Engineering, RuhrUniversit€at Bochum, Bochum, Germany Li Hong School of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China Yu Huang Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai, People’s Republic of China Rennie B. Kaunda Department of Mining Engineering, Colorado School of Mines, Golden, CO, United States

xi

xii

Contributors

Junichi Koseki Department of Civil Engineering, University of Tokyo, Tokyo, Japan Sunita Kumari Department of Civil Engineering, National Institute of Technology Patna, Patna, Bihar; Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India James Leak

Edinburgh Napier University, Edinburgh, United Kingdom

Wang Lin Key Laboratory of New Technology for Construction of Cities in Mountain Area; National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas; School of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China G.R. Liu Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH, United States Domenico Lombardi Kingdom

The University of Manchester, Manchester, United

Elham Mahmoudi Department of Civil and Environmental Engineering, RuhrUniversit€at Bochum, Bochum, Germany Vladimir Makarov Far-Eastern Federal University, Vladivostok, Russia Zirui Mao Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH; Department of Material Science and Engineering, Texas A&M University, College Station, TX, United States Subhadeep Metya Department of Civil Engineering, NIT Jamshedpur, Jharkhand, India Madhumita Mohanty Civil Engineering Department, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, India George Mylonakis University of Bristol, Bristol, United Kingdom; University California Los Angeles, Los Angeles, CA, United States; Khalifa University, Abu Dhabi, United Arab Emirates George Nikitas Kingdom

SAGE Laboratory, University of Surrey, Guildford, United

Vladimir Odintsev Mahesh Pal India

Institute for Mining IPCON, Moscow, Russia

Department of Civil Engineering, NIT Kurukshetra, Kurukshetra,

Ganga Kasi V. Prakhya Kingdom

Sir Robert McAlpine Ltd, Hemel Hempstead, United

Neelima Satyam Discipline of Civil Engineering, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India V.A. Sawant Department of Civil Engineering, National Institute of Technology Patna, Patna, Bihar; Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India Nalin L.I. De Silva Department of Civil Engineering, University of Moratuwa, Moratuwa, Sri Lanka Tanvi Singh Department of Civil Engineering, PIET, Panipat, India

Contributors

Boris Tarasov

xiii

Far-Eastern Federal University, Vladivostok, Russia

Sai Vanapalli Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada Xiuhan Yang Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada Liu Yunlong Department of Civil Engineering, Zhengzhou University, Zhengzhou, China Runhong Zhang National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas, Chongqing University, Chongqing, People’s Republic of China Wengang Zhang Key Laboratory of New Technology for Construction of Cities in Mountain Area; National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas; School of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China

C H A P T E R

1 A model adaptation framework for mechanized tunneling: Subsoil uncertainty consideration from observation to construction Elham Mahmoudi and Raoul H€ olter Department of Civil and Environmental Engineering, Ruhr-Universit€at Bochum, Bochum, Germany

Introduction The range of applications for mechanized tunneling extends from shallow tunnels in urban areas under buildings to deep tunnels under mountains with a large overburden. A safe and stable tunnel construction plan demands reliable knowledge of the expected effects of tunneling on the surroundings. The prerequisite for this is insights into the interactions between the encountered geological conditions or the existing infrastructure and the tunnel boring process. The heterogeneous geological conditions that change during the course of the tunneling operation and the numerous interactions of the excavation with the surroundings enforce specific requirements for planning instruments and tunneling technologies. Moreover, to predict the future behavior of similar situations in nature, one should consider that the natural material deals with spatial and temporal uncertain phenomena, which may affect decisions in planning tremendously. The adequate representation of the constitutive and geometrical properties of the subsoil for numerical simulations in mechanized tunneling is a difficult task due to the complex interactions between the ground, the boring machine, and surface constructions. The more precise these properties can be determined, the more accurate the system

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00002-2

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© 2021 Elsevier Inc. All rights reserved.

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1. A model adaptation framework for mechanized tunneling

response such as surface and structural deformations will be forecasted in both the design and execution phases. Thereupon, this chapter is dedicated to the investigation of all essential processes affecting the accuracy of mechanized tunneling simulation procedure. This concerns both the planning and the execution phase, through an adaptive computeroriented simulation and an observational campaign. To achieve this, the numerical tunnel simulation model is validated by means of measurement data, parallel to the progress of the tunnel boring machine (TBM) advancement. For this purpose, back analysis methods through optimization are utilized to minimize the differences between the model outcomes and recorded system behavior (Mahmoudi, H€ olter, Zhao, Datcheva, & K€ onig, 2020). Moreover, the concept of sensitivity analysis is introduced to assess the relevance of the different model inputs, including design and constitutive parameters. However, for sensitivity analyses and system identification, a large amount of time-consuming numerical simulation must be carried out. Therefore, the development of adequate surrogate models is of essential importance (Khaledi, Miro, K€ onig, & Schanz, 2014). The evaluation of the different approaches is made depending on the complexity and dimension of the mathematical problem. In addition to common uncertainties due to approximation in simulating physical systems, many natural phenomena cannot be easily monitored at irregular locations as underground. Therefore, characteristic estimations of these phenomena require probabilistic techniques along with a proper measurement campaign. Given the inherent heterogeneity of natural soils, the use of laboratory data allows only a very rough estimation. Hence, it is recommended to use the optimal experimental design (OED) approach in the context of which in situ measurements should be used to develop an existing model of construction or to validate them. The purpose of OED is to increase the effectiveness of measurements concerning the quality of the forecasts derived from them. In this chapter, some methodology and concepts to achieve an optimum arrangement are introduced. First, measuring concepts were optimized by means of sensitivity analyses and inverse methods (H€ olter, Mahmoudi, Datcheva, & Schanz, 2018). The performance of the proposed techniques can be optimized with respect to their computational efficiency utilizing sigma points and Bayesian analysis method (H€ olter, Schoen, Lavasan, & Mahmoudi, 2019). Even after performing above-mentioned techniques to reduce the uncertainty level of tunneling simulations, the model still includes some degree of subsoil parameter uncertainty which one should evaluate their impact on the reliability of the tunneling process. It can be addressed by calculating the probability of system failure against various limit-state functions, which, for example, dictate the admissible ground surface settlement, the tilt of surface structures or internal deformation of tunnel tube, etc. Various researches evaluated the influences of soil parameter uncertainties on tunnel-induced ground movements (for instance, see

Introduction

3

Miro, K€ onig, Hartmann, & Schanz, 2015; Mollon, Dias, & Soubra, 2013). However, they did not consider spatial variation in the subsoil and just considered random variables in a homogeneous medium. In this chapter, the concept of the random field is also incorporated into the uncertainty quantification process to deliver more realistic results. In this regard, different approaches to represent spatial variability including random field and kriging are reviewed. In the following, the random field theory is utilized to vary material properties in a spatial range according to the correlation structures of the soil. Recent comparative studies show considerable differences in the reliability characteristics depending on whether the spatial variability of geomaterials is taken into account or not (Mahmoudi, K€ onig, & Schanz, 2017). On the other hand, the influence of uncertainties on the reliability measures can be accurately evaluated using the concept of model adaptation, which is mainly concerned about including recorded in situ measurements in the further prediction. The excavation-induced settlement in the surface structures in tunneling projects is frequently measured during different excavation phases. Therefore, one may apply the Bayesian approach to update the spatial distribution of soil parameters based on the recorded data. In this stage, updating random field distribution is conducted based on MetropolisHastings algorithm (Hastings, 1970; Straub, 2011)) as an adaptive sampling technique with a rejection criterion to consider the measured data in midterm construction. Later, the failure probability is evaluated by utilizing an advanced Monte-Carlo simulation method, named as subset simulation (Au & Beck, 2001). It has been shown that subset simulation can calculate very small failure probabilities using a sequence of nested samples generated based on Markov chains in the framework of conditional probabilities (Mahmoudi, Khaledi, von Blumenthal, K€ onig, & Schanz, 2016; Mahmoudi, Khaledi, Miro, K€ onig, & Schanz, 2017). Early recognition of geological anomalies through pattern recognition can be used to identify both complex soil behavior on the one hand (e.g., temporally hydro-mechanically coupled processes in the context of consolidation) and the geometric properties of the defined geological scenarios on the other hand. Pattern recognition can be defined as the classification of unknown data based on statistical information extracted from patterns and/or their representation. It involves classification and clustering of patterns to recognize unfamiliar objects. In the field of geotechnics, so far only a few works related to pattern recognition have been carried out. In Yiqiang and Berrill (1993), an approach was presented, which uses pattern recognition to analyze the liquefaction potential of a soil based on pressure probe measurements. Duda, Hart, and Stork (2000) proposed different methods for pattern classification and clustering using, among others, stochastic and Bayesian decision theories. The first step in pattern recognition via machine learning methods is time series feature extraction. This process includes exploring different features and extracting relevant

4

1. A model adaptation framework for mechanized tunneling

ones. Later, the selected features should be evaluated by their statistical significance to predict the proper label. In recent decades, many researchers in the field of computer science dedicated their studies to developing new methodologies and techniques in supervised learning. For instance, K-nearest neighbor (KNN) is a straightforward model for classification and regression purposes (Cover & Hart, 1967). As a supervised learning method, it relies on labeled data to learn a function that evaluates an appropriate class for the unlabeled data. The support vector machine (SVM) is another supervised learning method used widely for classification and regression introduced by Cortes and Vapnik (1995). SVM is an effective method in case of high-dimensional spaces. Application of SVM in structural engineering is reviewed in C ¸ evik, Kurtog˘lu, Bilgehan, G€ uls¸ an, and Albegmprli (2015). Using the machine learning methodologies, the applicability of pattern recognition procedures is evaluated. Different geological conditions from homogeneous soil medium to existing rock boulders are to be examined. Here, the main goal is a successful classification of the soil situation to answer the following questions: Which geometrical properties of the soil layers can be classified? Which properties does the interface have (inclination, interlayers, or layer change)? Which patterns arise with which interfering bodies concerning size, position, and material? Can the methods also be transferred to other types of soil (e.g., solid rock with fissures)? In this chapter, the methodologies and concepts of model adaptation with an application on mechanized tunneling advanced simulation are presented. A graphical depiction of model identification steps is illustrated in Fig. 1. Every concept is presented in an individual section, including an introduction to theory and exemplary cases to illustrate the intended procedure. In the following section, the basic concept of uncertainty quantification via sensitivity and back analysis methods is introduced. In section of optimal experimental design, a framework for designing an optimal observation design is presented. It follows with section of reliability analysis, where the concept of reliability analysis and its updating based on measured data is illustrated. Here, random field theory is integrated into a Bayesian analysis to update the reliability measures of a twin tunnel excavation model. The concept of geological model prediction using machine learning approach is presented in section entitles as machine learning.

Uncertainty quantification Sensitivity analysis For the identification of input model parameters which have a decisive influence on the output of the simulations, the sensitivity analysis concept can be utilized. The model parameters are varied within physically

FIG. 1 Overall concept of model identification and validation in mechanized tunneling. SFB837, Subproject C2.

6

1. A model adaptation framework for mechanized tunneling

motivated limits and the effects on displacements and stresses at defined observation points are evaluated. Sensitivity analysis can quantify the relative importance of input factors related to the the system behavior. This analysis is substantial for identifying those input variables that have the least and those that have the most effect on model responses. Ranking of parameters’ importance can reduce the dimension of the calibration problem, as well as uncertainty reduction by further investigations and subsoil exploration on the most effective variables, respectively. Also, in the context of model development in general and model calibration in particular, sensitivity analysis determines how the input variations affect model outputs. There are many deterministic or statistical methods to perform sensitivity analysis. Two of the main categories, namely, are the local and the global sensitivity analyses (GSAs). The derivative-based methodologies as examples of local sensitivity analyses evaluate the effect of a small variation of a single input factor on the model behavior, while all the other factors are fixed. Calculating derivatives around a base point in the space of just one individual input factor makes this methodology computationally efficient, but also model dependent. It means when the model is nonlinear/nonmonotonic, the local approach may not deliver a reliable importance ranking of the input factors. In this regard, the second category of sensitivity analysis methods, that is, “global” techniques have been developed, which explore the entire input parameter space. GSA provides independent information from the model features including linearity, monotonicity, and additivity (Saltelli, Andres, & Ratto, 2008). The performance of different local and global sensitivity analysis techniques was reviewed in Mahmoudi, H€ olter, Georgieva, K€ onig, and Schanz (2019). The variance-based sensitivity analysis method has proven to be particularly suitable for various geotechnical systems, while it ranks the parameter’s importance and indicates any interactions between them. The main idea of the variance-based method is to evaluate how the variance of inputs contributes to the variance of the model output. The total effect sensitivity index STi is a comprehensive index, which takes the interaction between parameters into account (Saltelli et al., 2008). For a generic model with n factors, first, two randomly sampled (K, 2n) matrices C1 and C2 are generated. Where, K is the number of samples, and n is the number of input parameters. Afterward, a new matrix Ri is defined with resampling all arrays from C2, but its ith column which is identical with C1 matrix. Then, model outputs for C1 and C2 are evaluated as yC1 ¼ f ðC1 Þ yC2 ¼ f ðC2 Þ yR ¼ f ðRi Þ: Finally, the variance-based indices for model inputs are evaluated as

(1)

7

Uncertainty quantification

STi ¼ 1 

yC2 :yRi  f02 yC1 :yC1  f02

,

(2)

where yC1 , yC2 , and yRi are vectors containing model evaluations for matrices C1, C2, and Ri, respectively. While the symbol (.) denotes the scalar product of two vectors, the mean value f0 is defined as 0 12 K X 1 ðjÞ (3) f0 ¼ @ y A: K j¼1 C1 Variance-based sensitivity method is not merely used to identify the effectiveness of input factors, but also later in the context of optimal experimental design is employed to optimize sensor localization. However, performing these methods leads to a very large calculation effort, since numerical simulation s must be carried out for numerous variations, making it impracticable for large systems with many parameters. In particular, the “metamodel or surrogate model” concept as an adjoint model is introduced to substitute the original computationally expensive model with a mathematical formulation. The metamodels can be constructed based on the approaches of polynomial regression, moving least-squares method, and proper orthogonal decomposition with radial basis functions. Khaledi et al. (2014) introduced an extended version of the later approach, which uses extended radial basis functions. This approach combines radial basis functions with nonradial basis functions. Development of a metamodel demands a relatively affordable number of original model’s run. Mahmoudi et al. (2019) reviewed various metamodel techniques and evaluated their performance in approximating a special geotechnical structure, namely energy storage mined in rock salt.

Back analysis As mentioned earlier, considering the uncertainties involved in soil stratigraphy and material properties which can be attributed to insufficient in situ or laboratory tests, uncertainty of the model responses in geotechnical problems is inevitable. In general practice, the engineering design is based on deterministic analysis. It can deliver either a conservative design or lead to a high risk of instability. With the significant development of computer technology and advanced algorithms, the concept of back analysis via in situ measurement through optimization procedure is illustrated in Fig. 2. On the basis of the real measurements (for instance, tunnel-induced displacement), back analysis is applied to obtain the optimal values of the related input parameters, which provide a good

concepts to explore mechanised tunnelling in urban areas. In Aktuelle Forschung in der Bodenmechanik 2018.

FIG. 2 The applied iterative approach via back analysis. From Mahmoudi, E., H€olter, R., Zhao, C., Datcheva, M., & K€onig, M. (2020). System identification

Uncertainty quantification

9

agreement between the model predictions and recorded field data. The following two aspects form the mathematical procedure of optimization, (i) the formulation of an error function measuring the difference between model response and real measurements, and (ii) the selection of an optimization strategy to enable the search for the minimum of this error function. This can be achieved using different optimization techniques. There are mainly two groups of optimization techniques, namely, deterministic or stochastic optimization techniques. Gradient-based algorithms are one of the oldest and most widely used approaches to solve the optimization problems. They examine the gradient of the objective function to find the function minimum moving along the function’s gradient descent. A rapid convergence is the primary advantage of these methods, their main disadvantage is being dependent on the initial trial solution, an improper choose of initial guess leads the results to an immediate local minimum instead of a better global minimum. In order to obtain the global optimal solution of an optimization problem in a multidimensional space, evolutionary algorithms have found to be efficient and reliable. The genetic algorithm (GA) as a simulation mechanism of Darwinian natural selection proceeds by initializing a population of solutions and then improving it through repetitive applications of selections, crossover, and mutation operators. Compared with deterministic optimization algorithms, the optimal solution for GA does not depend on the initial trial solutions. Therefore, they are considered to be more robust. Khaledi, Mahmoudi, Datcheva, and Schanz (2016) and M€ uthing, Zhao, H€ olter, and Schanz (2018) applied GA to identify parameters of time-dependent constitutive model. Particle swarm optimization (PSO) is an evolutionary optimization algorithm that was introduced in Kennedy and Eberhart (1995) working with a population (called a swarm) of candidate solutions. In this algorithm, a number of “particles” are randomly placed in the search space of a given function, and evaluates the objective function at particular locations. Then, each particle determines its movement through the search space by combining some aspect of the history of its own actual and best locations with those of one or more numbers of the swarms, with some random perturbations. Zhao et al. (2015) conducted GSA to identify the most influencing parameters for a real tunneling project (Western Scheldt tunnel). By doing so, the dimension of the back analysis space decreased. After that, PSO algorithm was applied to conduct parameter identification, and the optimized parameters are able to well capture the tunneling-induced ground movements. In addition, after obtaining the optimized parameters, assessment of the quality of the obtained optimal set of model parameters (e.g., residual plots, confidence intervals, etc.) is necessary. This is to check the efficacy and credibility of the solution of the optimization problem.

10

1. A model adaptation framework for mechanized tunneling

Optimal experimental design Within the last section, the idea of minimizing the discrepancy between model responses and measurement data collected on site was discussed. Even though performing back analysis is still often neglected in practice, an in situ measurement is commonly performed on large construction sites. Also the design codes, such as EN 1997-1 (2004) which explicitly requests such measurements, do not provide recommendations how these measurements should be performed. In common practice, obtained measurement data is simply set into relation to a certain threshold that must not be exceeded. However, the largest observed value in the context of model adaptation is not inevitably the best one to validate the model, but rather the value that allows the greatest reduction of uncertainties of the considered parameters. This concept of identifying which measurements allow most uncertainty reduction in a model is known in several research field as OED. The term “experimental design” might be unexpected in the context of geotechnical structure measurements, but uncoupled from the specific application, the methods presented in the following correspond to those encountered in other fields of research. In Fisher (1935), fundamentals of this concepts were formulated initially with theoretical applications in biology. In this monograph, the Fisher information matrix (FIM) is introduced that corresponds to a certain extent to local sensitivity analysis and allows the comparison of different experimental designs. To evaluate such a matrix w.r.t. the quality of the corresponding experimental design, optimality criteria like the trace ΦA or the determinant ΦD are applied to this matrix. This FIM is still employed in the recent publications such as in Ucinski (2005) that investigated where to place sensors to survey air pollution. In Lahmer (2011), a more statistical investigation is performed on sensor placements on dams, wherein the mean square error of parameter identification results is employed to define the quality of an experimental design. This approach was further developed by the employment of the bootstrap method in Schenkendorf, Kremling, and Mangold (2009) where again repeated parameter identification is performed to obtain a covariance matrix Cθ of all considered parameters θ. The variation in the parameter identification runs rises from an artificial error that is added with defined properties to the model results. To the covariance matrix Cθ, the aforementioned optimality criteria can be applied to find the best possible experimental design. As obtaining reliable measures for the entries of Cθ requires a large number of parameter identification runs, the so-called sigma-point method allows substituting the randomly generated parameter sets of probabilistic parameter distributions by few specific samples. An alternative approach to improve the efficiency of this concept is to

Optimal experimental design

11

employ so-called Bayesian OED (Huan & Marzouk, 2013). Hereby, it is investigated which experimental design is most likely to reduce the bandwidth of an a Priori parameter distribution that is supposed to reflect the uncertainty of soil properties. Each time when possible new measurement data is obtained, a Bayesian updating is performed, accounting for a parameter uncertainty and measurement error (as another type of uncertainty) as well. An explicit investigation on the impact of different error types in the context of OED can be found in Reichert, Olney, and Lahmer (2019) where the monitoring of a tower-like structure is considered.

Methods Geotechnical engineering could be called predestined for applications of the OED methods mentioned earlier as large material uncertainties may be observed and the vast dimensions of construction sites (i.e., experimental setups) allow numerous different measurement arrangements. In Schanz and Meier (2008), a concept close to the FIM was employed by performing LSAs in various positions all over a defined model domain. This concept follows the idea that when a model response is sensitive to a certain parameter, this is the parameter that will be identified most accurately. The further development of using GSA in this context with its benefits of accurate and reliable results but drawbacks of high computational requirements are introduced in H€ olter, Mahmoudi, and Schanz (2015). In this regard, the modified sensitivity index STi, j is introduced: STi, j ¼

STi  σ j : σ j,max

(4)

Herein, the sensitivity index STi is obtained from variance-based GSA, and the index i referring to the individually considered parameters. σ j is the standard deviation of the considered output, whereby j refers to different positions, or time steps. Multiplying the sensitivity index with deviation and then normalizing it by the maximum value of standard deviation σ j, max of the considered model response allows to account for the size of a measured value. This is relevant as for instance in border areas of the simulation, where very small outputs can be obtained that might not be even measurable in situ. Still they may indicate a high sensitivity, which is compensated by the very small value of σ j. In case j refers to possible locations in a model domain, the values of STi, j can be displayed as contour plots allowing a demonstrative visualization of the results where placing a measurement device is the most informative in model identification process. This was performed in H€ olter et al. (2015) and H€ olter, Mahmoudi, et al. (2019) for the application examples of a laboratory setup

12

1. A model adaptation framework for mechanized tunneling

and a dike under rapid drawdown, respectively. In H€ olter, Zhao, et al. (2018), where the excavation process of a mechanized tunnel construction is simulated, time and location both play a role as depending on the current position of TBM; therefore, different possible measurement positions become relevant. After evaluating sensitivity distributions of different areas of the model domain at several time steps, a custom-tailored measurement arrangement was suggested for the specific problem. To evaluate the quality of the identified design, tests performed by back analysis are required. To do so, random noisy data are generated in those points selected for the design and then back calculated. The properties of the noise, like distribution type and standard deviation should reflect the accuracy level of intended type of measurement instruments. The accuracy by which the parameter is identified reflects the quality of the selected design. To include this aspect since the beginning of the investigation, the alternative statistical method using bootstrap resampling is suggested as employed in H€ olter, Mahmoudi, et al. (2019). Hereby, a defined number of positions, or any other variable of the experimental design, is defined as well as realistic ranges of the parameters of interest. Within these ranges, random samples θ are generated and run in the existing model to obtain corresponding model responses y. To each of the model responses, some random noise is added to obtain artificial noisy data. Most important is that the error terms of the actually intended measurement devices are reflected by same probability distribution. For each possible sensor  arrangement, a set of B possible artificial measurements y is generated according to the bootstrap resampling approach. For each considered arrangement, all B possible results are back calculated to obtain a matrix of identified parameters θ. From the B identified parameter sets, the covariance matrix Cθ is generated: Cθ ¼

B   1 X ðθ l  θÞðθ l  θÞT , B  1 l¼1

(5)

where the parameter mean value θ corresponds to θ¼ 

B  1X θl B l¼1

(6)

and θ l is the vector of identified parameters in one of the back analysis run. Doing so, one individual covariance matrix Cθ corresponds to each experimental design to which one optimality criteria is applied. A detailed discussion on these criteria can be found in Nishii (1993). By ranking the different designs according to these criteria, the best one can be identified. In this manner, comparing designs that include, for example, a few sensors of high accuracy with designs containing many sensors with

13

Optimal experimental design

low accuracy is now possible. However, the drawback of this method is its high computational demands. Even by substituting the initial simulation model by a metamodel, each possible measurement setup needs to be back calculated hundreds of time in the optimization process. The amount of possible measurement setups t increases rapidly by the possible locations p and dimensions n (time, space, etc.), leading to unbearable calculation costs:  n p (7) t¼ : s Therefore, as developed in H€ olter, Mahmoudi, et al. (2019), it is useful performing the spatial sensitivity analysis on the considered problem in advance of a bootstrap investigation to restrict the search area and so reduce the number of possible measurement point p and so the combinations t. Besides, the so-called sigma points can be employed to substitute the distributed parameter samples θ by few equivalent points as described in H€ olter, Mahmoudi, et al. (2018). An alternative to overcome high computational demands of the large statistical evaluations is represented by the so-called concept of Bayesian OED that intends to iteratively retrace the experimental design space. Here, the well-established concept of Bayesian learning of obtaining a posterior distribution based on a prior distribution and additional information is extended by the aspect of how this information was gained: 



Pðθj y , δÞ ¼

Pðy jθ, δÞ  PðθjδÞ 

Pðy jδÞ

,

(8)



where the posterior probability function Pðθj y Þ is an update of the prior probability function P(θ) that is multiplied by the likelihood function  Pðy jθÞ. These multiplied probabilities are “normalized” by the probability  Pðy Þ that a certain measurement will occur. In case of Bayesian OED, the specific experimental design δ is included as second condition in the terms of Eq. (8) whereby the posterior probability depends on the selected experimental design. Depending on the employed measurement data, that is, according to the employed experimental design δ, different posterior distributions are obtained and can be compared. The objective of OED and the intention of any engineer is logically to reduce most the parameter uncertainty, expressed by the standard deviation of the parameter distribution. Therefore, a utility function is formulated that can be employed to perform an optimization of the experimental design: δ* ¼ max UðδÞ, dD

(9)

where the optimal design δ* is the design, which allows to maximize the utility U, that is, to decrease most the parameter uncertainty. Herein, as

14

1. A model adaptation framework for mechanized tunneling

being a probabilistic concept, one should consider U(δ) as integral over the possible spaces of parameters and results: Z Z B X UðδÞ ¼ uðδ, y, θÞpðθ, yjδÞdθdy  uðδ, yl ,δl Þ=B: (10) y

Θ

l¼1

As solving these integrals might be time consuming, Overstall and Woods (2017) suggest to approximate them using a Monte-Carlo simulation using B samples, as suggested in the second part of Eq. (10). The employment of this approach is presented in the following by application to an example of mechanized excavation of a twin tunnel in urban areas.

Examples Some of the aforementioned OED concepts were applied to the simulation of the excavation of the line 5 of the Milan Metro project that is introduced in Fargnoli, Boldini, and Amorosi (2015). Within this project, one building that is crossed below by the two tubes of the tunnel was monitored by recording the vertical settlements at 13 positions during the excavation time. In Fig. 3A, the position of this building is shown in a top view. This figure also shows the measurement instruments, which are located on the edges of the building to record the settlements. In Fig. 3B, the geometry of the employed FE model is displayed, showing the three simulated soil layers, the location of the building, and the two tunnels. The simulation of the tunnel excavation via TBM advancement is effectuated by predefining the two tunnels by soil clusters that are removed step by step. The tunnel tubes are simulated by plate elements with properties corresponding to the employed concrete lining segments. However, the most complex part of the simulation of mechanized tunneling is the advancement of the TBM itself. Its shield is also simulated using plate elements, but with the corresponding stiffness properties. To account for the conicity and the overcutting of the TBM, a contraction factor is applied to the surrounding soil elements. The face pressure is considered as a distributed load that increases over depth. The grouting pressure is supposed to ensure contact between the tunnel and the adjacent soil and to fill the annular gap that arises from the thickness of the TBM shield. Moreover, to involve the increase in stiffness of grout overtime, it is applied as distributed load instead of the first segment ring behind the TBM. More details on FE simulation of mechanized tunneling can be found in Zhao et al. (2015). To this example, the introduced OED concepts shall be applied to illustrate their procedure and results. As a first step, the GSA is considered here to identify which model parameters are most relevant for the tilt and settlement of the building induced by tunnel construction. These parameters are those on which the OED should be

Boldini, & Lavasan, 2020).

FIG. 3 (A) Location of the building and its measurement points w.r.t. tunnel course and (B) view and dimension on the FE model (Schoen, H€olter,

16

1. A model adaptation framework for mechanized tunneling

focused. Including all constitutive, geometrical and steering parameters employed in the FE model would be out of scope and highly inefficient. Therefore, using engineering judgment, a preselection is performed to consider only those parameters that might be of relevance. In the present case, these are the stiffness of the building’s footing plate EF, the friction angle φ0 , the secant stiffness E50, and the small strain shear modulus G0 of the upper soil layer, the face pressure ps and grouting pressure pv of the TBM, and its volume loss coefficient VL. An overview on all the employed parameters and their values, corresponding to gravelly and silty sands, is given in Schoen et al. (2020). It should be mentioned that the further stiffness parameters of the employed hardening soil small strain constitutive model (Benz, 2008), the oedometric stiffness Eoed and unloading-reloading stiffness Eur, are assumed to be correlated to E50, making a total of seven independent parameters of interest. After defining an adequate range of variation for these parameters, a set of 150 parameter samples is generated to be run in the numerical model. From each model run, the vertical displacements in the 13 sensor positions shown in Fig. 3B are extracted to generate a metamodel and run a variance-based sensitivity analysis. An exemplary extract of the obtained results are shown in Fig. 4. This illustration depicts that, depending on the position of the sensors, the sensitivity plots varies strongly. However, it can be determined that according to their overall relevance, E50, G0, and VL move into focus of the OED considerations, whereas the remaining parameters remain fixed based on a preliminary parameter identification described in Schoen et al. (2020). The objective of this example is to investigate which positions outside the building limits are most appropriate to record settlement measurements. Following the approach of spatial sensitivity introduced in H€ olter, Mahmoudi, et al. (2019), a grid of artificial measurement points is

FIG. 4 STi bar plots showing the sensitivity of the displacements obtained in different position w.r.t. to the seven considered parameters (Schoen et al., 2020).

Optimal experimental design

17

generated over the model surface shown in Fig. 5D, where the area of the building is left out, leading to a total of 263 spots. For these points, the previously described procedure for generating a metamodel is repeated, but only considering the three relevant parameters mentioned above as input variables. Afterward, Eq. (4) is applied to obtain the modified sensitivity index STi, j for the three parameters in all the marked locations. The outcome of these evaluations are displayed as contour plots where the distances between the grid points are quadratically interpolated. In ref ref Fig. 5A–C, the results are shown for E50 , G0 , and VL, respectively. As the  STi, j values are uniformly normalized, it can be seen that VL is overall the most relevant parameter, followed by E50 and G0. The obtained results suggest avoiding the immediate vicinity of the building. Rather, it is more favourable to measure in front or behind the building (in the case of VL). This is probably caused by the enhanced stress level below the building that does not allow larger incremental settlements due to the tunnel excavation, compared the area around it. This proposed method of spatial sensitivity only indicates the relevance level of sensor locations when providing measurement data, but does not account for any measurement error and does not include the interaction between the sensors. Now the question arises, does it make sense to place all the sensors at the same position, because the sensitivity is the highest there? Or is it more promising to have a certain distance between the sensors, even if the individual sensitivity is lower in those spots? Using the approach of Bayesian OED addresses these issues. For the present case, a utility function U(δ, y, θ) must be defined that rewards experimental designs δ, that reduce the uncertainty of the input parameters θ. To this end, Eq. (11) is employed, that approximate the integral U by the ˜ where B refers to the amount of generated Monte-Carlo approximation U Monte-Carlo samples.  B  X 1 ~ ð δÞ ¼ U =B: (11) det Cθ,l l¼1 Herein, the covariance matrix Cθ,l is obtained according to Eq. (5) in which  θ l is the vector of parameters that is obtained when back calculating the lth  sample of outputs y l . The concept of Cθ is to assume measurement uncertainties for each possible measurement value that is obtained. When generating the B samples first mentioned in Eq. (11), they are falsified with some Gauβian white noise according to Eq. (12): 

y l ðθÞ ¼ yðθÞ + yðθÞ  ωsys  e + ωran  e, ω  N ½0, 1,

(12)

where y(θ) denotes the vector of model responses as obtained from the model, ωsys and ωran are normally distributed random variables. e is the error term that should account for the properties of the intended

(C)

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(A)

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–10

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–10

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ref

60

ref

TBM 1

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(D)

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(B)

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FIG. 5 Contour plots of GSA with respect to (A) E50 , (B) G0 , (C) VL, and (D) location of the artificial measurement points employed for generation of sensitivity plots and Bayesian OED.

STi* :

Reliability analysis

19

measurement devices. In addition, higher-order error types could be added, like bias or skewness. Afterward, these falsified outputs are back calculated to obtain a vector  of model parameters θ l . By repeating this B times, the obtained vector of mean values θ will be accurate, but depending on the employed design, the variation of the parameters will differ, expressed by the covariance matrix Cθ. In Eq. (11), applying the determinant is selected as the utility functions. Accordingly, one can see that the more accurate the parameters are identified, that is, the more their uncertainty is reduced, the larger the ~ utility UðδÞ will become. In this formulation, different experimental designs can be evaluated by varying the values of m and e that can considered relevant when defining the costs of the possible designs. In the present case, the number of sensors m is set to six. An initial random design is generated that distributes the sensors within the area of Fig. 5D. To vary the positions of the sensors, that is, the parameters of the experimental design, a metamodel is generated that allows to vary not only the model parameters θ, but also the geometric parameters, namely, the positions of the different sensors δ. The variation of δ is performed, iteratively, using the approximate coordinate exchange algorithm (Overstall & Woods, 2017). Herein, the entries of δ are varied one by one until the highest utility is obtained for each of them. After each entry of δ is varied, the parameter distributions of θ are updated following the concept of Bayesian updating Eq. (8). The results of this procedure are exemplary shown for the case of six sensors in Fig. 6, where the black marks represent the initially generated random design and the white marks identify the final identified locations. It can be observed that some of the sensors should be located in the middle of the settlement draft, but a few should also be placed on the boundary to fully capture the underlying soil parameters. This is in accordance with outcomes of previous investigations (for instance, H€ olter, Mahmoudi, et al., 2019; H€ olter et al., 2015) where often not those positions of maximum model response are most relevant, but those locations that generate the highest gradient of the model response. Further insight to the current OED problem can now be brought in by varying the number of sensors and their accuracy. Doing so it could be shown whether it is more economical to use many sensors of low accuracy or few ones with high accuracy.

Reliability analysis Significant measures of uncertainties can be observed in the design process of geotechnical structures. The nature of uncertainties may be divided into two main categories, namely aleatoric and epistemic. While the first

20

1. A model adaptation framework for mechanized tunneling

0.000 [m] [m] 80 –0.001

60

–0.002

40

–0.003

–0.004 20 –0.005 –40 –30 –20 –10

0

10

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30 [m]

FIG. 6 Positions of the initial and identified designs using Bayesian OED, underlied by displacement distribution.

one arises from intrinsic randomness of a phenomenon, the latter results from lack of knowledge. The presented methodology in this chapter intends to reduce the involved uncertainties by implying the in situ measurements, thereupon the epistemic uncertainties and their impact on the reliability of the system behavior are here targeted. With a given performance function Gx, the failure event is defined as Z PF ¼ P½Gx  0 ¼ fðxÞ dx, (13) where f(x)dx is the joint probability density function of X (X is the vector of uncertain input variables). In order to improve the reliability of designed infrastructure projects as mechanized tunnels, one may utilize inverse analysis techniques, by employing recorded measurements during the construction phase to expand the knowledge about the uncertainty propagation of soil properties. For this purpose, the Bayesian back analysis technique (Bayesian theorem) is employed. Bayesian updating has been performed previously in many geotechnical problems in order to modify the system evaluations (H€ olter, Schoen, et al., 2019; Miro et al., 2015), though they merely considered the heterogeneity of subsoil. In the present chapter, the main challenge is to update the subsoil parameters, while they have been spatially varied over the problem domain as as occurs in nature. In the following, the applied methodologies are described more in detail.

Reliability analysis

21

We consider the limit state of Gx ¼ Su  Sx(x), where Su and Sx(x) are the admissible and the calculated (given a random set of variables x) intended system response, respectively. According to Straub (2011), we may consider the likelihood function as   ðSðx,mÞ  Sx Þ (14) LðBjAÞ ¼ Φ σm, σm where S(x, m) is the measurement recorded in midterm construction phase and σ m is the standard deviation of the measurement error that might change due to different human or environmental influences. In general, mechanized tunneling can be considered as a spacious geotechnical project involving quite a large area. Thereupon, design and analysis of such projects are performed in engineering practice by averaging the characteristics of geomaterials causing uncertainties in evaluations. Due to the different characteristics of lithological composition, stress history and sedimentation, weathering and erosion, probable faults and fractures, almost all properties of subsoil shall be treated as heterogeneous materials. Thereupon, characteristics of the spatial distribution of geotechnical parameters should be quantified and taken into account for every stochastic analysis. Various statistical techniques to estimate spatial variability have been developed that can be classified in three main categories: regression analysis, geostatistics, and random field theory. Regression analyses assume that all sample values within a medium show an equal and independent likelihood, that is, they are not autocorrelated, which contradicts the made observation on the formation of the microscopic structural units within the soil mass (Yong, 1984). Field observations also show a stronger correlation between samples taken at a closer distant compared to samples further apart. Therefore, regression methods are better studied to be utilized in an initial exploration, when only a limited number of samples with large distances between them are taken from the site (Rendu, 1978). However, the other two categories do consider the autocorrelation concept in geomaterials. Geostatistical methods mainly include estimation and simulation methods. The aim of simulation methods is the computational generation of further realizations. These methods mainly reproduce random structures in nature through variography using the value distribution and autocorrelation properties of collected datasets. The subsequent estimation process is called Kriging in memory of Engineer Danie Gerhardus Krige, who first introduced a distance-weighted interpolation method in gold mining practice. The term “kriging” covers a multitude of different types of interpolation methods, including, simple, ordinary, universal and indicator kriging as well as cokriging. The common feature of all types of kriging is the determination of weighted mean values on basis of the autocorrelation structure (Cressie, 1993).

22

1. A model adaptation framework for mechanized tunneling

The other well-known category of methods to threat spatial uncertainties in geotechnics is an extension of the time series analysis methodology called as random field analysis (Vanmarcke, 1977). A time series is a chronological sequence of a particular variable observations at specific time intervals. The variable itself is constant at each instant of time and space. Each of these individual space-time intervals are considered as a field where the variable value may vary randomly. Time series analysis considers the aspect of autocorrelation, that is, the observed properties at adjacent time steps are more related than those at greater time intervals. Since geomaterials show no significant temporal variation in human life span, one may substitute the concept of time in data series by location. However, the spatial structure of variation is also called autocorrelation, because it refers to correlations of an individual variable with itself over space, instead of over time as in time series analyses (Baecher & Christian, 2003). In random field analysis, the stochastic process involves a subset of location-based random variables generated by specific spatial functions, known as random fields. A random field H(x, θ) illustrates the value of random variables θ in the location of x. Here, both location and the input variable can be defined as vectors to define multivariate and multidimensional random fields. Different methods and concepts have been presented to generate a random field to indicate the variation of material properties in space. A widely used method in random field theory entitled as the Karhunen-Loe`ve expansion method (Ghanem & Spanos, 1991) is utilized in this chapter. The Karhunen-Loe`ve expansion involves a spectral decomposition of the covariance function and can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ θÞ ¼ μH ðxÞ + σ H ðxÞΣM ðλi ÞΦi ðxÞξi ðθÞ, (15) Hðx, i¼1 where μH(x) and σ H(x) are the mean and standard deviation, respectively. λi and Φi(x) are the eigenvalues and eigenfunctions, respectively, of a Fredholm integral equation, where the auto-covariance function is defined as a kernel. M is a finite number of series terms and ξi(θ) is a vector of uncorrelated random variables, which represent the random nature of the uncertain parameters. Later in the model adaption context, this vector will face variations in a Bayesian manner. A governing value in an autocorrelation function is the correlation length or scale of fluctuation. Previous studies indicate the impact of this value on the probabilistic evaluations, though its identification is not simple due to lack of data (Mahmoudi, K€ onig, & Schanz, 2017). In this chapter, the related soil strength parameters that are subjected to the following random field study and their statistical characteristics are obtained from CPT investigation described in Phoon and Kulhawy (1999).

Reliability analysis

23

Case study The methods described in the previous sections are applied to the problem of a shallow twin tunnel excavation using TBMs. This case study is addressed by several researchers (Avgerinos, Potts, Standing, & Wan, 2018). The complex constitutive behavior of the soil causes effects on ground settlement that cannot be captured by simple superposition of the individual tubes, but can be calculated by the Peck’s equation (Peck, 1969). The complex stress paths that take place and cause settlements at the ground surface, induced by the tunnel construction, demand advanced modeling techniques such as the FE model presented in Fargnoli et al. (2015). The case of a tunnel construction shown in Fig. 3 as a three-dimensional (3D) FE simulation is considered again as background for this study where similar geometries and ground conditions are employed in a two-dimensional (2D) simulation. The model consists of two tunnel tubes that are constructed consecutively next to each other spaced 10 m apart. The diameter of the circular tunnel tubes is 6.7 m with a lining thickness of 0.3 m. A total number of 3000 six-nodal elements is employed to generate the mesh discretization of this problem where the bottom boundaries are fixed in the horizontal and vertical direction, while the side boundaries are fixed only in the vertical direction. Before installing the tunnels, an initial phase is applied in which the deadweight of the soil is considered to obtain a uniform stress field. Afterward, the initial displacements caused by these stress conditions are reset to zero to only consider the settlements induced by the tunnel construction. To simulate the process of stepwise segmental lining installation, the construction of each tunnel is executed within two steps. First, an excavation step in which the soil medium within the first tunnel area is removed, and a support pressure is applied to imitate the transient process after installing the lining segments, but before the grout pressure has reached its final stiffness. The relaxation factor λ is set to 0.25, that is, the internal support pressure is reduced to a fraction of 0.25 of the previous in situ stress level (Potts & Zdravkovic, 2001). In the next step, the internal pressure is removed, and the lining segments are installed without any further support. They are modeled as linear-elastic material with a normal stiffness of 10,389 kN/m and a bending stiffness of 77,918 kNm2/m. After the installation of the left tunnel is completed, the procedure is repeated for the right one. The surrounding soil material is modeled using the wellknown Mohr-Coulomb constitutive model that assumes a linear elastic and perfect plastic behavior by applying the Coulomb failure criterion. The subsoil is defined with properties typical for dense silty sand following lognormal distribution (μE ¼ 70, 000 kPa, σ E ¼ 7,000 kPa, μΦ ¼ 35 degrees, μC ¼ 5 kPa, σ C ¼ 0.25 kPa).

24

1. A model adaptation framework for mechanized tunneling

First simulation outcomes on the homogeneous case show that the settlements at the ground surface are at a maximum located right above the two tunnel centerlines. Due to the homogeneity of the soil parameters and the tunnel tubes being identical, the displacement field is perfectly symmetric. These results should be considered as the reference value for evaluations later on. Figs. 7 and 8 show an adapting process of random field discretization by application of Bayesian updating. The ground measurement in the depicted point in Fig. 7 is assumed to be recorded as a settlement of 15 mm at the middle of the tunnels’ axis after the excavation of the first tunnel is completed. To identify that position that provides the most informative data, the OED concept can be employed. Since now not a specific parameter value is of interest, but rather the correlation between settlements after the first and second tunnel passage are of focus, the realization of the concept of spatial GSA must be rethought. To do so, a series of points is generated at the ground surface. By performing a sensitivity analysis, it is identified how sensitive the results of the excavation of the

55

70

85 A random field discretization example of elastic modulus.

55

70

85

Stiffness modulus Es [MN/m²]

FIG. 7

Stiffness modulus Es [MN/m²]

Measurement spot

(identified by OED analysis)

FIG. 8 The updated random field discretization of Fig. 7 based on the recorded settlement equal to 15 mm.

25

Machine learning

second tunnel are w.r.t. to those of the first phase. The point that is most relevant for the settlements concerning the point of interest (here near to the foundation itself ) is selected as measurement point. The depicted random field discretization in Fig. 7 is just random. After five iterations by the Metropolis-Hastings technique (Hastings, 1970), the model adapted to the discretization depicted in Fig. 8. The material with a lower elastic modulus (yellow color [light gray in print version] in the contour plots) is now located on the top of the domain where more settlement happened regarding the observed data. Therefore, model updating based on measurements apparently took place correctly. We also presented the updated reliability index β for different measurements after the first excavation. The represented data in Table 1 show that the higher the measured settlement are, the higher will be the chance of observing failure in context of ground settlements. Thereupon, by carrying out the Bayesian updating approach, the spatial arrangement of the model is updated according to the recorded data. Here, it should be mentioned that even changing the accuracy of the measurement process itself impacts the reliability of the system. This has been shown by considering different values of standard deviation for the same measurement. Increasing the measuring error leads to higher failure probability.

Machine learning The control of ground movements is critical in tunneling projects due to the potential damage to existing infrastructures. The specification of tunnel control parameters requires a good knowledge of the surrounding soil stratum. As boreholes are discretely distributed, interpolations based on obtained information may conduce to approximate the entire overall TABLE 1 Reliability measures updated by measurements. S(x,

σ m[mm]

PF

β index

5

1

0.001

3.09

8

1

0.002

2.87

12

1

0.012

2.25

15

1

0.168

0.96

15

2

0.274

0.60

without meas.

–

0.011

2.29

m)

[mm]

26

1. A model adaptation framework for mechanized tunneling

stratigraphy, which results in inevitable uncertainty in the subsoil identification process. For instance, high stiffness blocks, thin weak layers, or faults between two adjacent boreholes may not be properly identified. Due to this reason, a pattern recognition approach is proposed in the present chapter to predict the soil stratum variations through tunnel advancement. Pattern recognition methods can be used to evaluate measurements to predict which subsoil can be expected in the near field of the TBM. The pattern recognition is carried out by means of substeps. For data reduction and quality improvement, a preprocessing is developed to smooth noisy signals or measurements on the one hand, and to map the information to a uniform value range on the other hand. Features may be represented as continuous, discrete, or discrete binary variables. A feature is a function of one or more measurements, computed so that it quantifies some significant characteristics of the object. By extracting features, the patterns are transformed into the feature space. The final classification is carried out by a classifier, which assigns the characteristics to different classes. In classification, an appropriate class label is assigned to a pattern based on an abstraction that is generated using a set of training patterns or domain knowledge. The training data are generated depending on variations of the input variables. In principle, the pattern recognition should be valid for all common soil types. However, cohesive and noncohesive soils can be distinguished, as these show different consolidation behavior depending on their permeability. It has to be investigated systematically, which changes (here the individual classes) can be classified at all with a sufficient detection rate and thus can be predicted. The classification of strength and friction angle in noncohesive soils is done, for example, on the basis of changed settlement patterns. Differences in pore water pressure allow the classification of soils with different permeability. Also, the classification of a layer transition from cohesive (undrained) to noncohesive (quasidrained) soils can be carried out by changes in the pore water pressure. However, the robust measurement program and pattern recognition cannot be considered independently. The classification or pattern recognition is only successful if the corresponding measured values, which are the basis for the characteristics of the classification, are recorded at all. The recognized patterns are also only to be understood as a prognosis that a certain unforeseen ground situation may exist. The reliability of this prognosis also depends on the accuracy of the assumed soil model and the recorded measurements. The sequence of the work packages is shown in Fig. 9. The behavior of a soil domain subjected to mechanized tunneling is represented in a multidimensional feature space and the unexpected soil layer in front of tunnel face is assumed to be classified into one of the four

FIG. 9 Based on an optimized measurement program, measurement data are recorded and processed within the framework of pattern recognition, so that these can be assigned to specific subsoil scenarios. SFB 837, Subproject C2.

28

1. A model adaptation framework for mechanized tunneling

classes: a homogeneous soil medium, a layer change, an interlayer, or a block with higher stiffness. The training set of data is generated by distributed computing using Python scripts via PLAXIS 3D software. Synthetic data are derived from the numerical results of simulations, which are enhanced with noisy data to conduct the machine learning process. Afterward, different techniques are investigated to achieve an accurate estimate of the soil stratum in front of the tunnel face. Different combination of supervised learning methods are examined to predict the geological patterns, including, SVM, Kriging and KNN. In the following section, these methods will be reviewed.

Methods As a well-known methodology in artificial intelligence and machine learning, one may mention the support vector machine. SVM maximizes the width of a separating hyperplane based on a maximum margin method, which results in a unique, optimal separating hyperplane. A hyperplane is a decision boundary to classify the samples, its location depends upon support vectors, that is, the closest data points to the hyperplane. The dimension of a hyperplane is governed by the feature space’s dimension. The challenge of finding the hyperplane in nonlinear spaces also can be overcome by transferring the data space via kernels. KNN is a supervised learning method that evaluates an appropriate label when given new unlabeled data points based on a labeled training dataset. KNN is known as a powerful machine learning algorithm for classification and regression, which includes global and local methods. The first type, for instance, finds a separating hyperplane by linear models. Such a plane separates the whole sample space at once, that is, a global boundary. On the other hand, when the classification of an example takes place based just on the similarities with the of adjacent samples, we are here dealing with a local model. KNN is a common classification method in practice due to it being a simple algorithm but very efficient in delivering competitive results. Using local KNN, the class identification for an unknown object is often determined by the majority criterion, but other schemes are also conceivable. The number of the nearest neighbors K should be odd to avoid even votes. Furthermore, it should be mentioned that a large K may provide a poor classification when the training samples are not well distributed in feature space, thereupon, it should be kept as small as possible. However, it should be stated that performing KNN demands a large training dataset, which may not always be available in actual geotechnical problems. The performance of every machine learning model depends on the underlying dataset, and every machine learning model has own

Machine learning

29

limitation. Hence, each method should be carefully verified for solving specific problems on an individual basis. In this chapter, a hybrid method which preprocesses data via semivariograms and performs KNN is utilized.

Model configuration A 3D tunnel simulation is generated using the FE method, considering the stepwise excavation of the soil and accordingly the advancement of the TBM. In this case, however, undrained material properties are assumed, meaning excess pore water pressures are generated and dissipated over time, depending on the soil permeability and the assigned boundary conditions of the soil. In the model, the hydraulic boundaries of the model are open with the exception of the symmetry axis, where it is closed. It should be stated that the boundaries of the model are selected wide enough to not influence the pore water pressure dissipation. In this context, the length of the model is set to 200 m, while width and height are 63.75 and 75 m, respectively. In each excavation step that lasts one hour to be executed, the TBM advances by 1.5 m, whereby the obstacle (layer change, boulder, or interlayer) is reached after 80 excavation steps. The first layer is defined with properties typical for dense sand (E50 ¼ [34, 000, 54, 000] kPa, k ¼ [3.6, 5.6] 104 m/s), while the second layer has properties of normally consolidated clay (E50 ¼ [16, 000, 23, 000] kPa, k ¼ [3.6, 5.6] 107 m/s), causing a remarkable decrease in stiffness and permeability. Four rows of sensors are considered at different levels to record tunneling-induced ground settlement and excess pore water pressure. Fig. 10 shows the model configuration for the layer change scenario, where the sensor arrangement is considered the same for other scenarios, as well. In the interlayer scenario, a cohesive layer of 10 m is modeled. The size of the boulder is varied in different simulations from 0.5D to 1.5D in the boulder scenario, D being the tunnel diameter. The closest side of the boulder is located at 130 m. The material properties of the boulder are assumed to remain constant, being impermeable, and considerably much stiffer than the surrounding sand medium that is varied within each model run. For the following process of machine learning, 100 models for each of four scenarios are simulated using PLAXIS 3D. In any individual simulation, the soil properties or size of the boulder is changed to cover a wide range of possible situations. The computational costs of this large amount of 3D calculations are managed using distributed computing technique. Here, a Python script managed the communications and coordinates the actions of 25 networked computers to decrease the required computational time. In such a message-passing mechanism, whenever a simulation task is ended on one PC, the outputs are saved on a cloud, and a new task

30

1. A model adaptation framework for mechanized tunneling 40 Timesteps

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Sensors to measure pore water pressure and settlements

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FIG. 10

Numerical model configuration for the layer change scenario.

is assigned to that computer. After gathering the entire calculation results, the output data of interest on the predefined location of sensors are extracted to be later fed to the machine learning algorithms. Also, it should be stated that the obtained synthetic data are falsified by adding Gaussian noise with a standard deviation of 5%. Afterward, 80 cases are randomly chosen to be employed as training data the rest serve as test data. Feature extraction is conducted using Time Series FeatuRe Extraction based on Scalable Hypothesis algorithm (TSFRESH) (Christ, Braun, Neuffer, & Kempa-Liehr, 2018). This algorithm extracts a comprehensive meaningful set of features from time series. Each extracted feature vector is individually evaluated concerning its significance for predicting the target under investigation. The first 150 most relevant features are employed to classify the data by SVM and KNN methods. To perform SVM analysis, various kernels, such as linear, RBF and polynomial kernels along with their corresponding parameters, (a misclassification factor c for example) were explored by a grid search technique. The most accurate results are obtained using a linear kernel and c ¼ 0.1. Moreover, various K values to run the KNN method are examined, a K value of seven delivered the best results. The obtained results show high accuracy for time steps 60 and 80, that is, when TBM is located 30 m and 9 m far from the geological anomalies, respectively. To increase the prediction’s certainty even further away from the TBM face, namely in 40th drilling phase, the pore water pressure and settlement data are transferred by semivariograms. A number of various models have been introduced in the literature that are commonly used in different practice.

Machine learning

31

We used one of the most widely applied in our study that is the Gaussian model:    1 x  μ2 WðxÞ ¼ exp  , 2 σ

(16)

where x is the location of TBM or distance it has passed since model boundary. μ is the x-coordinate of the sensor and σ is the distance between sensors. In our case, this is equal to 4.75 m and for instance, when TBM is in 40th time step, this means x ¼ 60 m. In this way, all weighting factors for each time step and all sensor are calculated and resulted in a matrix. This weighting matrix is multiplied to sensor data. Fig. 11 illustrates the semivariograms calculated for ground settlement sensors. One may find different patterns for various geological scenarios, even visually. In the following, these datasets are treated as input samples for training KNN and improved the classifier accuracy, significantly (see Fig. 12). Therefore, hybridizing KNN with kriging approach raised the certainty in the model identification process. It should be stated that the illustrated case study is a synthetic one and not based on a real-world tunnel. Therefore, its output cannot accurately represent in

FIG. 11 distance.

Illustration of calculated semivariograms of ground settlement versus lag

32

1. A model adaptation framework for mechanized tunneling

40

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80

100

Accuracy [%]

80

60

40

20

0

FIG. 12

SVM

KNN

KNN and Kriging

Accuracy of different machine learning applied methods.

situ sensors that may be affected by many other parameters in the field. Nevertheless, an ongoing project by the authors applies the complete framework to a real case mechanized tunneling project with the aim of obtaining the optimum sensor localization to make pattern recognition process as accurate as possible.

Final remarks Nowadays, we observe an ever-growing accuracy of deterministic simulations in the field of geotechnics to model extensively coupled behavior of different subsystems in spacious projects as mechanized tunneling. However, these models are not able to consider the natural randomness of soil in its behavior or to involve measured data to update the system. In this chapter, we illustrated the necessity of probabilistic analysis for adapting numerical simulations of mechanized tunneling, considering field measurements. A consistent framework including different statistical methods is presented to simulate and predict the behavior of surrounding soil in mechanized tunneling. Different methodologies and techniques in different levels as back analysis, reliability analysis, Bayesian analysis, random fields, etc. are shown which enables designers to evaluate and minimize uncertainties, and also to calculate and update

References

33

the regarding reliability measures. A brief review of various sensitivity analysis methods is presented to evaluate the importance level of different input characteristics. These methods can lead designers to optimize the distribution of their sources in gathering knowledge about the input data to minimize the uncertainty at most. Since the model adaptation process rigorously relies on the observed field performance data, we also proposed methods to obtain an optimum measurement design. The proposed OED concept can reveal an optimum sensor arrangement (i.e., type, location, accuracy level, and numbers) considering different design criteria. At the last section, the research in the field of system identification methods for ground models in tunneling by machine learning techniques is introduced. In the last decade, machine learning and artificial intelligence showed promising performance in almost every field. We introduced a concept which applies these techniques to cope with a great deal of existing uncertainties in geological alternation ahead of a TBM. In such a supervised learning process, different subsoil arrangements are defined as classes, and recorded ground reaction to excavation as settlement and pore water pressure are treated as learning features. Afterward, the performance of various machine learning methods is evaluated through an exemplar comparison. The presented concepts in this chapter show promising results in mechanized tunneling case studies for model adaptation. Nevertheless, they might be applied analogously on other spacious geotechnical projects.

Acknowledgments The authors would like to gratefully acknowledge the support of the German Research Foundation (DFG) through the Collaborative Research Center (SFB 837) in subproject C2. Also, the authors would like to acknowledge the effort of Mr. Milosˇ Marjanovic in developing distributed computing code. Last but not the least, the authors would like to thank Ms. Melanie Breyer for her kind assistant to produce and modify the illustrations and graphs.

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Vanmarcke, E. H. (1977). Probabilistic modeling of soil profiles. ASCE, Journal of the Geotechnical Engineering Division, 103(11), 1227–1246. Yiqiang, D., & Berrill, J. B. (1993). A pattern recognition approach to evaluation of soil liquefaction potential using piezocone data. Soil Dynamics and Earthquake Engineering, 12 (2), 91–101. Yong, R. N. (1984). Probabilistic nature of soil properties. In D. S. Bowles, & H. Y. Ko (Eds.), Probabilistic characterization of soil properties: Bridge between theory and practice. (Eds.). Zhao, C., Lavasan, A. A., Barciaga, T., Zarev, V., Datcheva, M., & Schanz, T. (2015). Model validation and calibration via back analysis for mechanized tunnel simulations—The Western Scheldt tunnel case. Computers and Geotechnics, 69, 601–614.

C H A P T E R

2 Prediction of lateral and oblique load for batter pile group using GRNN, NN, and ANFIS Tanvi Singha and Mahesh Palb a

b

Department of Civil Engineering, PIET, Panipat, India Department of Civil Engineering, NIT Kurukshetra, Kurukshetra, India

Introduction Tall buildings, transmission towers, offshore structures, bridge pier, and earth retaining structure or structures constructed in earthquakeprone areas are usually subjected to a large amount of overturning moment caused by wave pressure, wind and earthquake forces, etc. These overturning moments are transferred to the foundation in the form of horizontal and vertical forces. To resist these forces, different types of pile foundations are used. The use of vertical piles allows resisting only a small amount of lateral load, thus necessitating the use of batter pile in combination with vertical piles. Batter pile have distinct advantage over vertical pile because of their capabilities to convert lateral load in tension and compression forces. In the past, several theoretical and laboratory works have been reported by various researchers to study the behavior of batter pile under lateral loading. Tshebotarioff (1953), Reese and Matlock (1956), Murthy (1964), Prakash and Subramanyam (1965), Alizadeh and Davisson (1970), Poulos and Madhav (1971), Meyerhof and Ranjan (1973), Ranjan, Ramasamy, and Tyagi (1980), Lu (1981), Veeresh (1996), Ming (2018), Kang and Kim (2018), and Toma Sabbagh, Al-Salih, and Al-Abboodi (2019) carried out load tests on batter piles and concluded that negative batter piles have more load-bearing capacity than positive batter piles. Only a few studies are reported in the literature to study the behavior

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00008-3

37

© 2021 Elsevier Inc. All rights reserved.

38

2. Prediction of lateral and oblique load for batter pile group

of batter pile under oblique loading. Meyerhof and Ranjan (1973) suggested that with an increase in inclination of loading angle, pullout capacity of negative batter pile is found to increase, whereas it decreases up to 60 degree for positive batter pile and increasing afterward when the angle is more than 60 degree. Al-Shakarchi, Fattah, and Kashat (2004) carried out model studies on vertical and batter piles (at angles
30 degree) in sandy soil and concluded that with an increase in loading angle, ultimate pullout capacity of vertical and batter pile increases and performance of negative batter and vertical piles was superior to positive batter piles at all loading angles. Mroueh and Shahrour (2009) concluded that ultimate pullout capacity continuously decreases with an increase in inclination in loading angle for vertical, negative, and positive batter piles, and the decrease was significant from 0 to 10 degrees. The reported results so far suggest a contradictory information about inclination in loading angle and the capacity of batter piles, thus necessitating the need for further investigation. The use of laboratory and theoretical analysis are labor and computationally intensive, hence the need for alternate approaches to model pile capacity. Within the last two decades, many researchers adopted machine learning techniques for various civil engineering problems and found these techniques performing well and requiring less computational resources. Chan, Chow, and Liu (1995), Chow, Chan, Liu, and Lee (1995), Goh (1996), Teh, Wong, Goh, and Jaritngam (1997), Lok and Che (2004), Mosallanezhad and Moayedi (2017), Wang, Moayedi, Nguyen, Foong, and Rashid (2019) and Wengang, Goh, Runhong, Yongqin, and Ning (2020) used artificial neural network (NN) with both static and dynamic datasets to predict the pile capacity and compared the results with available empirical formulae and found neural network performing equally well or better than the available empirical equations. Abu-Kiefa (1998), Nawari, Liang, and Nusairat (1999) Juang, Chen, and Jiang (2001), Cigizoglu (2005), Kurup and Griffin (2006), Pal and Deswal (2008), Alzoubi and Ibrahim (2018), Kaloop, Hu, and Elbeltagi (2018) and Alzoubi and Ibrahim (2019) used generalized regression neural network (GRNN) with different civil engineering problems. Nayak et al. (2004), Najafzadeh (2015), Harandizadeh, Toufigh, and Toufigh (2019), Srokosz and Bagi nska (2020) used ANFIS (neuro-fuzzy (ANFIS), a combination of NN and fuzzy logic), which has been widely used in civil engineering problems due to its reasoning and learning capabilities, and these techniques show better results in comparison with empirical relations. Keeping in view the usefulness of various machine learning approaches, the present study aims to investigate the performance of NN, GRNN, and ANFIS for batter pile group subjected to lateral and oblique load.

39

Introduction

Generalized regression neural network The generalized regression neural network (GRNN) approach was proposed by Specht (1991). Unlike backpropagation approach, this method does not use an iterative training procedure. GRNN is composed of four layers. The first layer consists of input units, followed by the pattern unit in the second layer. The third layer has the summation unit, and the fourth layer has the output units. The first layer is completely linked to the second layer, where each unit signifies a training pattern, and output is given as distance of the input from the saved patterns. The optimal value of spread parameters, a user-defined parameter of GRNN, is found experimentally and depends on the dataset used. For details of generalized regression neural network (GRNN), readers are recommended to refer to Specht (1991).

Adaptive neuro-fuzzy inference system (ANFIS) Fig. 1 shows the structural design of first-order Sugeno fuzzy model of ANFIS having two input (assumed to have two member functions (MFs)), four rule, and an output. First-order Sugeno fuzzy model (Sugeno, 1985) has four fuzzy if-then rule, given as: Rule 1: if a is X1 and b is Y1, then f11 ¼ m11 ðaÞ + n11 ðbÞ + q11 Rule 2: if a is X1 and b is Y2, then f12 ¼ m12 ðaÞ + n12 ðbÞ + q12

Layer 1

Layer 2

Layer 3

Layer 4 a

X1

Π

W11 W12

N

W11

W11 f11 a

a X2

Π

N

Π

W12 f12

Y2

Π

W22

W21 f21 a

N

W22

Σ

b

W21

W21

b

FIG. 1

N

b

W12 a

Y1

Layer 5

b

b W22 f22

ANFIS structure for a two-input Sugeno model with four rules.

c

40

2. Prediction of lateral and oblique load for batter pile group

Rule 3: if a is X2 and b is Y1, then f21 ¼ m21 ðaÞ + n21 ðbÞ + q21 Rule 4: if a is X2 and b is Y2, then f22 ¼ m22 ðaÞ + n22 ðbÞ + q22 where X1, X2, Y1 andY2 are MFs for input a and b, mij, nij and rij (i,j ¼ 1,2) are consequent parameters ( Jang, 1993). Layer 1: All nodes are adaptive nodes and produce membership grade of input and output layer and defined by: O1Xi ¼ μXi ðaÞ, ði ¼ 1, 2Þ O1Yj ¼ μYj ðbÞ, ðj ¼ 1, 2Þ where a and b are crisp inputs, and Xi and Yjare fuzzy set such as low, medium, and high categorized by appropriate MFs, which could be of any shape such as triangular, trapezoidal, Gaussian function, etc. Layer 2: All nodes in this layer are fixed nodes (labeled Π), function as simple multiplier, and output is given as below and represents firing strength (the degree to which the predecessor part of rule is satisfied) of each rule: O2ij ¼ Wij ¼ μXi ðaÞμYj ðbÞ, ði, j ¼ 1, 2Þ Layer 3: All nodes are again fixed node (labeled as N) and play a normalization role in the network, and the output is given as below which represents normalized firing strength: O3ij ¼ Wij ¼

Wij , ði, j ¼ 1, 2Þ, W11 + W12 + W21 + W22

Layer 4: In this layer, all nodes are adaptive nodes whose output is the product of normalized firing strength and first-order polynomial. The output obtained in this layer is given below. Parameters in this layer are called consequent parameters.
 O4ij ¼ Wij fij ¼ Wij mij ðaÞ + nij ðbÞ + qij , ði, j ¼ 1, 2Þ Layer 5: the only node in this layer is a fixed node and labeled overall output is given as summation of all incoming signals z ¼ O5ij ¼

P

where

X2 X2 1


 W f ¼ W m ð a Þ + n ð b Þ + q ij ij ij ij ij ij 1

ANFIS has two adaptive layers. Layer 1 has parameters {xi, yi, zi} and {xy, yy, zy} and Layer 4 has modifiable parameter {mij, nij, qij}. The

41

Introduction

function of the learning algorithm for ANFIS is to tune all the modifiable parameters to match the ANFIS output with that of training data. Hybrid learning algorithm used with ANFIS is a two-step process which involves adjusting to its modifiable parameter. In a hybrid learning algorithm, the principle parameter was held fixed in forward pass and outputs from node go forward up to Layer 4. Least square method was used to identify the consequent parameter, whereas consequent parameters were held fixed in a backward pass and error signal propagates backward. Gradient descent method was adopted to identify the principal parameters. The details of a hybrid learning algorithm can be found in Jang (1993).

Choice of membership function Membership function may be of many shapes, for example, trapezoidal, triangular, generalized bell-shaped, Gaussian functions, etc. In the present study, performances of triangular, generalized bell-shaped, and Gaussian functions have been compared (Bateni & Jeng, 2007; Lo, 2002; Najafzadeh, 2015). The major reason for using these membership functions is their popularity, and these functions are smooth and nonlinear, and their derivatives are continuous. Triangular: μ X ðaÞ ¼

ð a  xÞ , xay ð y  xÞ

μX ðaÞ ¼

ðz  aÞ , yaz ðz  yÞ

Gaussian: μX ðaÞ5 1+

1  a  z 2 x

Bell-shaped: μXi ðaÞ ¼

1   a  zi 2xi 1+ xi

i ¼ 1,2

1   b  zj 2yj 1+ xj

j ¼ 1,2

μYj ðbÞ, ¼

42

2. Prediction of lateral and oblique load for batter pile group

Neural network (NN) The neural network is a machine learning technique widely used for numerical prediction of pile capacity (Ismail, Jeng, & Zhang, 2013). It is inspired by the functioning of nervous system and brain architecture. NN has one input layer, one or more hidden layers, and one output layer. Each layer consists of a number of nodes, and the weighted connection between these layers represents the link between nodes. Input layer having nodes equal to the number of input parameters distributes the data presented to the network and doesn’t help in processing. This layer follows one or more hidden layers that help in the processing of data. The output layer is the final processing unit. When an input layer is subjected to an input value which passes through the interconnections between nodes, these values are multiplied by the corresponding weight and summed to obtain the net output (zj) to the unit X zj ¼ Wij  yi i

where Wij is weighted interconnection from unit i to j, yi is the input value at input layer, zj is output obtained by activation function to produce an output for unit j. A detailed discussion about NN is provided by Haykin, (2000). In the present analysis, a three-layer NN based on the backpropagation algorithm is used.

Methodology and data set Data used in this study were obtained from experimental investigations carried out in the soil mechanics laboratory of the National Institute of Technology Kurukshetra (India). Two types of tests were carried out viz. 1. Lateral load test and 2. Oblique load test. Description of the experimental setup is given below:

Model tank This setup consists of a steel tank with 5 mm of wall thickness and has a dimension of 1  2  1 m. To apply an oblique load, a pulley whose height can be adjusted so that when the rope passes over it, the required loading angle (0, 10, 20, 30, 45 degrees) can be obtained from horizontal. Fig. 2 shows the model setup.

43

Methodology and data set

Pulley

α

Pile cap

Batter pile

Vertical pile

1.0 m Load

Testing tank

1.0 m

FIG. 2

Model setup.

Soil used Soil used for all tests was poorly graded sand (Table 1). Tank was filled using a rainfall method of sand filling with two densities of sand, 15.79 kN/m3 (when the height of fall was 0.30 m) and 16.28 KN/m3 (when the height of fall was 0.40 m).

TABLE 1

Properties of sand.

S.No.

Properties

Values

1

Soil type

SP

2

Effective size (D10)in mm

0.175

3

Uniformity coefficient (Cu)

2

4

Coefficient of curvature (Cc)

3.84

5

Specific gravity (G)

6

2.63 3

Minimum dry density (γ dmin) in KN/m

3

14.3

7

Maximum dry density (γ dmax) in KN/m

17.3

8

Maximum void ratio, emax

0.84

9

Minimum void ratio, emin

0.52

44

2. Prediction of lateral and oblique load for batter pile group

Model pile Aluminum pipes of 0.90, 0.60, and 0.40 m length, with outer diameter 20 mm and wall thickness of 1 mm was used as a model pile. Pile surface has surface roughness Ra ¼ 0.254 μm.

Pile caps To keep all piles in a group for equal distribution of loads and to guide batter piles at proper angle pile caps were required. Vertical and inclined holes with 20 mm diameter were drilled in perpex plates. The 20 mm extra dimension was provided for attaching hooks to both sides of plates so that wire can be fixed to hooks to both sides and load can be applied through wires. For lateral load test, altogether 36 pile caps were used out of which 8 on each of four batter angle (20, 25, 30, 35 degrees), whereas for oblique load test, only 5 pile caps on 25 degree (first five in Fig. 3) batter angle were used. Fig. 3 gives a plan of the pile cap.

Testing procedure Pile cap was placed at the center of steel tank, and then piles are placed by a gentle tapping at the top. The height of pulley was adjusted to attain the desired angle with a horizontal axis. Deflection in pile cap after application of load was measured using dial gauge attached to pile caps. Loads were applied until the deflection value reaches the limiting value of 5%–6% of pile diameter (Shirato, Nonomura, Fukui, & Nakatani, 2008) (equal to 10–15 mm in the present case). The unloading and reloading of load were done in each experiment in the same manner and deflection value was measured. Tables 3A and 3B provide a summary of lateral load test and oblique load test on pile groups, respectively. The notation for pile caps used is given as follows: 1B: one batter pile, 1B1V:groups of one vertical and one batter pile, 2B: groups of two batter piles, 2V2B: groups of two vertical and two batter pile, 4B: four batter piles, 1B3V: one batter and three vertical, 3B6V: three batter and six vertical, and 4B12V: four batter and 12 vertical. Table 2 gives the details of different parameters used in the lateral and oblique load tests.

Testing procedure

FIG. 3

Plan of pile caps.

45

46

2. Prediction of lateral and oblique load for batter pile group

TABLE 2 Details of parameter. Parameter

Lateral load test

Oblique load test

Density

16.28 KN/m

16.28 KN/m3, 15.79 KN/m3

Pile length

0.9 m

0.9 m, 0.6 m, 0.4 m

Batter angle (θ)

20°, 25°, 30°, 35°

25°

Load inclination (α)

0°

0°, 10°, 20°, 30°, 45°

No. of pile caps

36

5

No. of tests

64

147

TABLE 3A

3

Summary of Lateral load test performed.

Batter angle with vertical (θ)

Type of batter pile

No. of test performed

20

Positive Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

20

Negative Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

25

Positive Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

25

Negative Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

30

Positive Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

30

Negative Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

35

Positive Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

35

Negative Batter

8

1B, 1B1V, 2B, 2B2V, 4B, 1B3V, 3B6V, 4B12V

Pile groups

Analysis and detail of GRNN and NN ANFIS, GRNN, and NN were used to predict lateral as well as the oblique load capacity of batter piles. For lateral load test, lateral load resisted (Q) in Newton (N) was used as an output parameter and input parameter that consists of number of negative batter pile ( B), number of positive

Analysis and detail of GRNN and NN

TABLE 3B

47

Summary of Qblique load test performed. Constant parameter

ρ (kN/m )

α

L (m)

Variable parameter

16.28

0

0.40

1B, 1V1B, 2B, 2V2B, 4B

16.28

0

0.60

1B, 1V1B, 2B, 2V2B, 4B

16.28

0

0.90

1B, 1V1B, 2B, 2V2B, 4B

16.28

10

0.40

1B, 1V1B, 2B, 2V2B, 4B

16.28

10

0.60

1B, 1V1B, 2B, 2V2B, 4B

16.28

10

0.90

1B, 1V1B, 2B, 2V2B, 4B

16.28

20

0.40

1B, 1V1B, 2B, 2V2B, 4B

16.28

20

0.60

1B, 1V1B, 2B, 2V2B, 4B

16.28

20

0.90

1B, 1V1B, 2B, 2V2B, 4B

16.28

30

0.40

1B, 1V1B, 2B, 2V2B, 4B

16.28

30

0.60

1B, 1V1B, 2B, 2V2B, 4B

16.28

30

0.90

1B, 1V1B, 2B, 2V2B, 4B

16.28

45

0.40

1B, 1V1B, 2B, 2V2B, 4B

16.28

45

0.60

1B, 1V1B, 2B, 2V2B, 4B

16.28

45

0.90

1B, 1V1B, 2B, 2V2B, 4B

15.79

0

0.40

1B, 1V1B, 2B, 2V2B, 4B

15.79

0

0.60

1B, 1V1B, 2B, 2V2B, 4B

15.79

0

0.90

1B, 1V1B, 2B, 2V2B, 4B

15.79

10

0.40

1B, 1V1B, 2B, 2V2B, 4B

15.79

10

0.60

1B, 1V1B, 2B, 2V2B, 4B

15.79

10

0.90

1B, 1V1B, 2B, 2V2B, 4B

15.79

20

0.40

1B, 1V1B, 2B, 2V2B, 4B

15.79

20

0.60

1B, 1V1B, 2B, 2V2B, 4B

15.79

20

0.90

1B, 1V1B, 2B, 2V2B, 4B

15.79

30

0.40

1B, 1V1B, 2B, 2V2B, 4B

15.79

30

0.60

1B, 1V1B, 2B, 2V2B, 4B

15.79

30

0.90

1B, 1V1B, 2B, 2V2B, 4B

15.79

45

0.40

1B, 1V1B, 2B, 2V2B, 4B

15.79

45

0.60

1B, 1V1B, 2B, 2V2B, 4B

15.79

45

0.90

1B, 1V1B, 2B, 2V2B, 4B

3

48

2. Prediction of lateral and oblique load for batter pile group

batter pile (B), number of vertical pile (V), and batter angle (θ). Out of 64 experimental results on lateral load test, 42 randomly selected samples were used for training, whereas the remaining 22 samples were used for testing the models. Summary of training and testing data for lateral load test is given in Table 4A. Similarly for oblique load test, oblique load Qα in Newton (N) was used as output parameter and input parameter that consist of the angle of oblique load (α) in degree, pile length (L) in meters (m), sand density (ρ) in KN/m3, number of vertical pile (V), and number of batter pile ( B). Out of 147 experimental results on the oblique load test, 105 randomly selected test results were used for training and the remaining 42 for testing different algorithms. The summary of training and testing data set for the oblique load test is given in Table 4B. A trial-and-error method was used to find optimal values of userdefined parameters of various modeling approaches. Optimal values of user-defined parameters for ANFIS, GRNN, and NN in case of lateral and oblique load tests are given in Table 5. ANFIS and GRNN model was implemented using MATLAB (2013), whereas the WEKA software was used for NN modeling. The performance of all modeling approaches was compared using coefficient of correlation (CC) and root mean square error (RMSE) for both lateral and oblique load tests. Parametric analysis was also carried out using an approach that performs best with given data to know the effect of batter angle (θ) on lateral load capacity of batter pile and to determine the effect of angle of oblique load (α) on batter pile group. Sensitivity analysis was carried out to know the importance of each parameter in both cases.

TABLE 4A

Summary of training and testing data set for lateral load test. Training data set

Input parameter

Training data set

Min

Max

Mean

Std dev

Min

Max

(B)

0

4

1.19

1.486

0

4

0.955

1.253

(+B)

0

4

1.19

1.486

0

4

1

1.309

V

0

12

3.381

4.078

0

12

2.273

3.628

Θ

20

35

27.738

5.653

20

35

27.045

5.703

Mean

Std dev

0.00

0.40

0.00

1.00

15.79

L

V

(B)

Ρ

Min

Α

Input parameters

16.28

4.00

2.00

0.90

45.00

Max

16.032

2.019

0.60

0.638

20.905

Mean

Training data set

0.246

1.118

0.804

0.207

15.793

Std. dev.

TABLE 4B Summary of training and testing data set for oblique load test.

15.79

1.00

0.00

0.40

0.00

Min

16.28

4.00

2.00

0.90

45.00

Max

16.058

2.024

0.643

0.638

22.024

Mean

Testing data set

0.246

1.07

0.821

0.205

15.659

Std. dev.

50

2. Prediction of lateral and oblique load for batter pile group

TABLE 5 User-defined parameter for GRNN and NN. Classifier used

Types of test

User-defined parameters

NN

lateral load

Learning rate ¼ 0.2, momentum¼0.1, hidden nodes¼8, number of iterations¼200

NN

oblique load

Learning rate ¼ 0.3, momentum¼0.2, hidden nodes¼8, number of iterations¼200

GRNN

lateral load

spread ¼ 0.7

GRNN

oblique load

spread ¼ 0.3

ANFIS

Lateral load

Membership function ¼ 3, Epoch ¼ 10

ANFIS

Oblique load

Membership function ¼ 3, Epoch ¼ 10

Results and discussion Effect of batter angle (u) on batter pile group under lateral load test A plot of experimental results and predicted values obtained using all modeling techniques is provided in Fig. 4. It can be observed from Fig. 4 that predicted values from GRNN and NN are quite close to experimental values. To determine which model performs best for given data, two performance evaluation parameters, coefficient of correlation (CC) and Root mean square error (RMSE) have been calculated and tabulated (Table 6). Results (Table 6) suggest slightly better performance by NN (CC ¼ 0.9581 and RMSE ¼ 77.778 N) in comparison with other models. Among all three membership functions used with ANFIS, the triangular membership function performs best for used data. Fig. 5A–C (surface diagram for smooth pile group from ANFIS for Gaussian MFs) focuses on three-dimensional surface graph of two input and an output form smooth pile group. It can be concluded from Fig. 5 that there is a nonlinear relationship among the input and output parameters. ANOVA single factor test is a hypothesis testing technique to examine whether statistically significant differences occur between the mean of two or more groups. If F-value is less than F-critical obtained from ANOVA single factor test, the difference among groups is said to be statically insignificant and vice versa. Similarly, if the p-value obtained is greater than 0.05, then also a difference among groups is statically insignificant and vice versa. Result obtained from ANOVA single factor test indicates that F-value (0.5726) is less than F-critical (2.2862) and p-value

51

Results and discussion

Predicted lateral load (N)

1000 GRNN NN ANFIS(triangular) ANFIS(bell shape) ANFIS(gaussian)

800

600

400

200

0 0

200

400

600

800

1000

Actual lateral load (N)

FIG. 4

Actual versus predicted lateral load by GRNN NN and ANFIS.

TABLE 6

Comparison of CC, RMSE, and MAE for GRNN and NN.

Modeling approach

CC

RMSE (N)

GRNN

0.9341

79.6292

NN

0.9581

77.7780

ANFIS (triangular MF)

0.8083

235.2345

ANFIS (bell-shaped MF)

0.6540

355.3453

ANFIS (Gaussian MF)

0.7567

269.961

(0.7208) is greater than 0.05, hence it can be concluded that differences between predicted and actual values from all models are statistically insignificant suggesting that any approach can be used to predict lateral load for batter pile group. Sensitivity analysis Sensitivity analysis was carried out to know the relative importance of each parameter in resisting lateral load. For this, only NN was used because of its better performance in comparison with GRNN and ANFIS with used data. Different training sets were generated by removing one parameter at a time, and models were created. Table 7 suggests that the number of vertical pile is important in batter pile group, and batter angle is the important factor in resisting lateral load by batter pile groups. Further, it is also clear that negative batter piles are more efficient than positive batter piles.

52

2. Prediction of lateral and oblique load for batter pile group

80 Predicted lateral load (N)

Predicted lateral load (N)

150

100 50 0 35 Ba tt

Predicted lateral load (N)

(A)

30 er a

ng le

60 40 20 0 35

10 25

5 20

0

e (V al pil

Ba tt

)

tic f ver

(B)

No o

4

30 er an gle

3 25

2 20

0

1 ative f neg

ile (–

rp batte

B)

No o

150 100 50 0 35 Ba tt

(C)

30 er an gle

4 3 e (B er pil

2

25

1 20

0

ative f neg

)

batt

No o

FIG. 5 (A) 3D-surface graph of two input viz batter angle and no. of vertical pile verses predicted lateral load (output). (B) 3D-surface graph of two input viz batter angle and no. of negative batter pile verses predicted lateral load (output). (C) 3D-surface graph of two input viz batter angle and no. of positive batter pile verses predicted lateral load (output).

TABLE 7

Sensitivity analysis using NN on lateral load test data.

Input combination

Parameter removed

(B), V, (+B), θ

cc

RMSE

MAE

0.9793

58.6335

46.5295

V, (+B), θ

(B)

0.9567

61.118

47.3991

(B), (+B), θ

V

0.6456

161.6991

131.7288

(B), V, θ

(+B)

0.9599

67.0312

46.0022

(B), V, (+B)

θ

0.9281

77.576

54.084

Results and discussion

53

Predicted lateral load (N)

300 250 200 150 100 50 0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Angle of pile in clination (q°)

FIG. 6

Variation of lateral load with varying angle of pile inclination keeping other parameters constant.

Parametric analysis Parametric analysis was carried out to study the effects of batter angle (within the range of input) on the lateral load capacity of batter pile groups. Again, only NN was used for parametric analysis. In this analysis, a hypothetical testing data set was generated by varying value of one input parameter (angle of pile inclination in this case) within input range and keeping other parameter constant and testing this data set for model generated with training data. The value of angle of pile inclination was varied (20 to 35 degrees) and other parameters were kept constant. Fig. 6 provides the plot between batter angle and predicted lateral load. From the plot it can be inferred that the lateral load capacity of batter pile group increases with an increase in angle of pile inclination and reaches the maximum value at 25 degree and then decreases. Hence, the most efficient batter angle comes out to be 25 degree. Results were close to the finding of Chattopadhyay and Pise (1986) and Nazir and Nasr (2013).

Effect of angle of oblique load (a) on batter pile group under oblique load test As discussed earlier, batter pile groups are also subjected to vertical load (dead and live load) in addition to lateral load, resulting in oblique load on the batter pile group. Results from Section “Parametric analysis” suggest that the most efficient batter angle is 25 degree, and sensitivity analysis indicates a negative pile to be more efficient than a positive batter pile (Section “Sensitivity analysis”). Therefore, for determining the effect of angle of oblique load, batter angle of pile groups was kept at 25 degree, and all batter piles in the group are negative batter pile. ANFIS GRNN, and NN were

54

2. Prediction of lateral and oblique load for batter pile group

used to model the data obtained by the oblique load test. A comparison of experimental results and predicted values from all models is presented in Fig. 7. It can be observed from Fig. 7 that predicted value from GRNN and NN is quite close to experimental values. Comparison of performance evaluation parameter suggests slightly better performance of NN (CC ¼ 0.8369, RMSE ¼ 112.0335 N) Table 8. Among all three types of membership functions, Gaussian MFs provide slightly better performance than triangular and bell-shaped MFs with ANFIS. Fig. 8A–D (surface diagram for smooth pile group from ANFIS for Gaussian MFs) focuses on a three-dimensional surface graph of two input and an output parameters from the smooth pile group, and it can be concluded that there is a nonlinear relationship among the parameter of oblique load test and the resisted oblique load.

Predicted oblique load (N)

800 NN GRNN ANFIS(triangular) ANFIS(bell shape) ANFIS(gaussian)

600

400

200

0 0

100

200

300

400

500

600

700

800

Actual lateral load (N)

FIG. 7

Actual versus predicted oblique load by GRNN NN and ANFIS.

TABLE 8 Comparison of CC, RMSE for GRNN, NN, and ANFIS. Modeling approach

CC

RMSE (N)

GRNN

0.7708

107.7035

NN

0.8369

112.0335

ANFIS (triangular MF)

0.5343

563.5217

ANFIS (bell-shaped MF)

0.6366

158.3147

ANFIS (Gaussian MF)

0.7695

123.8708

55

Predicted oblique load (N)

60 40 20 4

No

of

ver tic

Predicted oblique load (N)

al

2 pile

(V)

1

0

10

20

Load

60 40 20 2 0.8 el en gth

60 40 20

fb a

30 (α) angle

80

80

2 No o

40

3

Predicted oblique load (N)

Predicted oblique load (N)

Results and discussion

1 tte

rp

ile

(V)

0

0

10

30

20

Load

ang

150 100 50

1.62

Pil

30

0.6 (L)

0.4

0

10

20

Load

40

(α) angle

40

) le (α

x 104

De

1.6 ity (ρ)

ns

1.58

10 0

20

Load

30

40

(α) angle

FIG. 8 (A) 3D-surface graph of two input viz no. of vertical pile and load angle verses predicted oblique load (output). (B) 3D-surface graph of two input viz no. of batter pile and load angle verses predicted oblique load (output). (C) 3D-surface graph of two input viz pile length and load angle verses predicted oblique load (output). (D) 3D-surface graph of two input viz density and load angle verses predicted oblique load (output).

Result obtained from ANOVA single factor test indicates that F-value (2.2925) was greater than F-critical (2.2862), and p-value (0.0494) was lesser than 0.05, hence it can be concluded that difference between predicted and actual values from all model was statistically significant. Sensitivity analysis To know the important factor in resisting oblique load for batter pile groups with oblique load tests, only NN-based sensitivity analysis was carried out. Results from Table 9 indicate that number of batter piles (B) is the most important parameter in resisting oblique load on batter piles. A possible reason for this may be that lateral load component of the oblique load was resisted by the batter piles only. The next important parameter was found to be pile length (L) and angle of oblique load (α), respectively.

56

2. Prediction of lateral and oblique load for batter pile group

TABLE 9 Senstivity analysis using NN on oblique load test data. Input combination

NN

Parameter removed

α, L, V, B, ρ

CC

RMSE

0.8916

119.1799

L, V, B, ρ

α

0.7951

120.1095

α, V, B, ρ

L

0.7675

123.5943

α, L, B, ρ

V

0.8921

87.4531

α, L, V, ρ

B

0.6172

165.2342

α, L, V, B

ρ

0.8294

121.1945

Parametric analysis In this section, parametric analysis was carried to study the effect of angle of oblique load (α) on load capacity of batter pile using NN because of its suitability with this data. Procedure of doing this is the same as in Section “Parametric analysis.” Value of angle of oblique load was varied (0 to 45 degrees), whereas other parameters were kept constant. Fig. 9 provides a plot between the angle of oblique load and predicted oblique load., From the plots it can be concluded that as angle of oblique load increases, oblique load-carrying capacity increases.

Predicted oblique load (N)

600 500 400 300 200 100 0 0

5

10

15

20

25

30

35

40

45

Angle of oblique load (a°)

FIG. 9 Variation of oblique load with varying angle of oblique load keeping other parameters constant.

References

57

Conclusions This chapter attempts to predict lateral and oblique load capacity of batter pile group using GRNN, NN, and ANFIS with three types of membership functions. Major conclusions of this study are listed as follows: 1. Neural network (NN) performs better than the other two modeling approaches to predict lateral load and oblique load for batter pile group. 2. Among three types of membership functions used with ANFIS, triangular-shaped membership function gives the best results for prediction of lateral load test for batter pile group. 3. Sensitivity analysis indicates the number of vertical piles in batter pile group and batter angle are important parameters in resisting lateral load. Negative batter piles are found to be more efficient than positive batter piles. 4. Result of parametric analysis indicates that 25 degree angle is the most efficient batter angle. 5. Result of ANOVA single factor test indicates the difference between actual and predicted values from all three models is statically insignificant, hence any approach can be used to predict lateral load for batter pile group. 6. Among three types of membership functions, Gaussian-shaped membership function gives the best results for the prediction of oblique load test for batter pile group. 7. Sensitivity analysis indicates that the number of batter piles, pile length, and angle of oblique load are important parameters in resisting oblique load. 8. Parametric analysis concludes as angle of oblique load increases, loadcarrying capacity of batter pile increases in case of oblique load test. 9. Result of ANOVA single factor test indicates the difference between actual and predicted values from all three models is statically significant for the oblique load test.

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C H A P T E R

3 Non-Euclidean model of rock masses Mikhail A. Guzev Institute for Applied Mathematics FEBRAS, Vladivostok, Russia

Introduction The last decades of the 20th century have been marked by the discoveries of new geomechanical phenomena, which seem anomalous from the traditional viewpoints on the deformation and destruction of rocks. Among them, it is worth highlighting the zonal disintegration of rocks around underground workings. It is observed when the internal stresses in the medium exceed the value characterizing the rock strength; therefore, the stress state arising in this case is naturally to call a strong compression of the rock. It should be noted that researchers in different countries recorded the zonal, or rather, the periodic oscillatory nature of the stresses acting in the rock mass, deformations of the mass around the workings, and changes in various physical properties of the rocks in this area. The existence of periodic variability of the physical properties of rocks around underground workings was first established in Borisovets (1972), but the fracture structure of the massif was not investigated. Somewhat later, physical properties such as electrical resistance and density were studied by geophysical methods for rocks surrounded the permanent mine workings at a depth of up to 1000 m in Norilsk, Russia (Oparin & Tapsiev, 1979). The periodic distribution of these physical properties was established, and it was also indicated that the crack zones follow the contour of the workings, and the intermediate zones have a densified character. In Zborschik, Malyarchuk, and Morozov (1980), the study of the electrical resistance of rocks around a single preparatory mine working was supplemented by a direct study of the fracture structure by the periscopic method.

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00020-4

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62

3. Non-Euclidean model of rock masses

Among foreign works of this period should be highlighted (Adams & Jager, 1980), which had established the periodic distribution of technogenic cracks in the massif ahead of the active face at a depth of 2300 m at a gold mine in South Africa. By the 1990s of the 20th century, a research team led by academician E.I. Shemyakin completed a set of theoretical and experimental studies on the phenomenon of zonal disintegration (Shemyakin et al., 1986, 1987). This phenomenon consists of the alternation of destroyed and nondestroyed zones of rock in massifs, for example, deep tunnels created by drilling or blasting. In the above period, studies were also conducted in China. In Xu (1989), the zonal nature of the deformation of the mountain environment was established, characterized by alternating radial deformations along the length of the wells. The results were recorded during long-term observations of rock deformations in drift passed through coal at the Changuangba field (China). In the first decades of the 21st century, studies of the zonal nature of the destruction of the massif by the periscopic method were presented in Majcherczyk, Malkowski, and Niedbalski (2005), where it was indicated that the spread of the destruction zones naturally decreases with distance from the excavation, while the spread of intermediate zones grows. Studies on zonal disintegration in China were obtained by studying the in situ velocity for ultrasonic waves (Li, Wang, Qian, et al., 2008) and by measuring multipoint displacements in a nickel mine (Pan, Li, Tang, & Zhang, 2007). Although the fact of the existence of the above-mentioned anomalous phenomena is reliably established experimentally, it still raises a debate among specialists in the construction of geomechanical models. This forces us to study the observed patterns as well as develop new mathematical models to bridge the gap between scientific research in geomechanics and engineering applications. From a macroscopic point of view, the description of the phenomenon of zonal disintegration of rocks should naturally be based on equations of continuum mechanics and on the principles of nonequilibrium thermodynamics. The detailing degree of this approach depends on the need to focus on certain features of the phenomenon under consideration. In this regard, it should be noted that a characteristic feature of the stress-strain state of highly compressed rocks is the presence of periodic oscillatory structures. Based on classical mechanics, many attempts were made to describe the zonal character of rock mass failure near openings ( Jia & Zhu, 2015; Li et al., 2008; Li, Pan, & Li, 2006; Li, Pan, & Zhang, 2006; Metlov, Morozov, & Zborshchik, 2002; Shemyakin et al., 1987; Tan, Ning, & Li, 2012; Tropp, Rozenbaum, Reva, & Glushikhin, 1985; Wu, Fang, Zhang, & Gong, 2009a; Wu, Fang, Zhang, & Gong, 2009b)

Introduction

63

However, none of the theories could explain all properties of the zonal failure structures. The reason is that the mechanics of a deformable solid as a macroscopic theory does not take into account the presence of defects in a real massif, whose presence leads to the appearance of the discontinuous and incompatible deformation of rock masses, resulting in the formation of fracture zones (macrocracks). Thus, novel ideas and methods must be introduced to gain a better understanding of the zonal disintegration mechanism in deep rock mass engineering. The study of the microcharacteristics of various materials by physical methods led to the introduction of concepts such as dislocation, disclination, vacancy, and other characteristics of the crystal structure defects into the materials science literature (Sih & Tang, 2005). From the point of view of classical continuum mechanics and nonequilibrium thermodynamics, the characteristics introduced by various researchers to describe the internal structure of materials explicitly required an extension of the kinematic foundations of the theory. As early as the 1950s, Kondo (1952) and Bilby, Bullough, and Smith (1955) concluded that it is necessary to use nonEuclidean geometric objects, forbidden in the classical theory of elasticity, in their description. The general idea of constructing non-Euclidean geometric models of a continuous medium is presented in Myasnikov and Guzev (1999). The thermodynamically correct derivation of the basic relations of the nonEuclidean model of the phenomenon of zonal behavior is presented in Guzev and Paroshin (2001). The detailed description of the non-Euclidean model and further developments are collected in Guzev (2010) and Guzev and Makarov (2007). Subsequently, the non-Euclidean model was used in the analysis of various effects of the mechanism of the zonal behavior of rocks (Dorovsky, Romensky, & Sinev, 2015; Qian & Zhou, 2011; Zhou & Qian, 2013; Zhou, Song, & Qian, 2011). The verification of the results of the non-Euclidean model based on the gradient model is given in (Chen, Qi, Wang, & Wang, 2019; Zhang, Pan, Xiao, & Wang, 2019). Thus, the phenomenon of zonal disintegration of highly compressed rocks and the effects associated with them are investigated based on the non-Euclidean model of a continuous medium. Although the Guzev (2010) and Dorovsky et al. (2015) use various kinematic parameters in a non-Euclidean model of a continuous medium to obtain defining relations, the general idea is to use the nonequilibrium thermodynamics formalism, which can be attributed to the folklore of continuum mechanics. Based on the thermodynamic approach, a wide class of models can be formed to describe the various effects of the physicomechanical behavior of materials. However, in rock mechanics, the use of nonequilibrium thermodynamics methods is not generally accepted practice, which inhibits the active use of the non-Euclidean model for solving engineering problems. Therefore, in this work, when describing a non-Euclidean rock

64

3. Non-Euclidean model of rock masses

model, an attempt was made to present the classical and non-Euclidean models of a continuous medium from a unified point of view, within which it is possible to formulate the direction of expansion of the classical theory and the transition to non-Euclidean in terms of traditional for the classical approach. In this section, we provide a narrative description of the above point of view. Recall that in the classical theory of elasticity, the Airy stress function method is well known Gurtin (1963) and Jaeger, Cook, and Zimmerman (2007). Since the equilibrium equation (1) does not change for the non-Euclidean model of a continuous medium, the stress function can also be introduced for the non-Euclidean model in accordance with the well-known classical relations (2), identically satisfying (1). The difference between the models will be manifested in the formulation of the equation for the stress function. The corresponding homogeneous biharmonic equation for it in the classical theory follows from the Saint-Venant compatibility condition. The expansion of the classical model is associated with the rejection of this condition and the introduction of the incompatibility function as an additional parameter of non-Euclidean theory. On this path, a modification of the classical theory leads to the inhomogeneous biharmonic equation for the stress function, the right side of which is determined by the incompatibility function. Moreover, the classical theory is a limiting case of non-Euclidean one and corresponds to the vanishing of the incompatibility function, i.e. transition to a homogeneous biharmonic equation for the classical stress function. Consequently, in the non-Euclidean model, the introduction of the stress function by the generally accepted relations (2) leads to the differential relations used in the classical theory, which allows the use of traditional approaches of the theory of elasticity to study the non-Euclidean model of a continuous medium. The mathematical formalism of this approach is presented in “Detailed theory” section. To illustrate the possibilities of the obtained formulas of the non-Euclidean model, we consider their application for the analysis of the phenomenon of zonal disintegration of rocks around underground workings in “Description of rock mass damage at great depths” section.

Detailed theory Transition from the classical model to the non-Euclidean one We will consider the massif as a weightless medium in the conditions of a plane stress state. The equations of mechanical equilibrium have the form ∂σ 11 ∂σ 12 ∂σ 21 ∂σ 22 + 2 ¼ 0, 1 + 2 ¼ 0 ∂x1 ∂x ∂x ∂x

(1)

65

Detailed theory

Relations (1) are identically satisfied if we introduce the stress function Φ according to the equations: σ 11 ¼

∂2 Φ ∂2 Φ ∂2 Φ ,σ 22 ¼ 1 1 , σ 12 ¼  1 2 2 2 ∂x ∂x ∂x ∂x ∂x ∂x

(2)

In the classical theory of elasticity, the Airy stress function method is applied for investigating the plane-deformed state (Gurtin, 1963; Jaeger et al., 2007). A further definition of the Airy stress function Φclas is based on the use of additional assumptions. For the plane state, Saint-Venant’s compatibility condition is given (Godunov & Romensky, 2003; Gurtin, 1963; Jaeger et al., 2007) by the equation: ∂2 ε11 ∂2 ε22 ∂2 ε12 +  2 ¼0 ∂x2 ∂x2 ∂x1 ∂x1 ∂x1 ∂x2

(3)

Let the rheological relationship between the components of the stress and strain field be linear (Hooke’s law): X σ ij ¼ λδij ε + 2μεij , ε ¼ εkk (4) k

where λ, μ are the phenomenological parameters of the Lam
e (Lam
e’s coefficients), δij is the symbol of Kronecker. Components εij are determined from (4):   X 1 λσ σ ij  δij εij ¼ σ jj (5) ,σ ¼ 2μ 3λ + 2μ j Combination of Eqs. (1), (3), (5) results in the equation for the classical stress function Φclas: Δσ clas ¼ Δ2 Φclas ¼ 0

(6)

If we consider the massif as a weightless medium, weakened by a cylindrical cavity and simulating a circular fixed working in conditions of comprehensive compression, then, in the classical theory, σ clas ¼ const and the distribution of the stress field is described by a monotonic function ( Jaeger et al., 2007; Shemyakin et al., 1987). Now back to the question that is formulated in the introduction: what methods and ideas should be used to describe the wave-like behavior of the stress tensor components for the phenomenon of zonal disintegration? A problem can be more pragmatically formulated as follows: how to change minimally the classical solution to get the right result? What is the physical meaning of these changes? Turning to the task of modifying the classical model, we recall that the main differential relations, on which the classical solution is constructed, are the equilibrium equation (1) and the compatibility equation (3). Since we consider phenomena in the framework of the laws of classical mechanics, Eq. (1) does not change. Therefore, the representation of the solution in the form (3) remains valid.

66

3. Non-Euclidean model of rock masses

Considering compatibility conditions (3), we note that not every arbitrary tensor εij for the plane case satisfies them. In the general case, for the tensor εij, which is calculated in accordance with (5) for arbitrary values σ ij, the compatibility conditions (3) may not be satisfied. Therefore, the incompatibility function characterizing the nonfulfillment of condition (3) can be considered as an additional kinematic characteristic of the material, depending on its internal mechanical state, parameterized through σ ij. Thus, a possible way to modify the classical model is associated with the rejection of the compatibility condition (3). Let’s introduce the incompatibility function R by setting R ∂2 ε11 ∂2 ε22 ∂2 ε12 ¼ 2 2 + 1 1 2 1 2: 2 ∂x ∂x ∂x ∂x ∂x ∂x

(7)

In a classical theory, we have R ¼ 0 (3) and this condition is necessary and sufficient (Godunov & Romensky, 2003) for representing the tensor εij through components ui of the displacement vector as 2εij ¼ ∂ ui/∂ xj + ∂ uj/∂ xi. The difference of function R from zero means that when rheological relation (4) is fulfilled, the quantity Δσ 6¼ 0, i.e., the first invariant of the stress tensor is not a harmonic function. Substituting (5) into (7) and using (1), we obtain Δσ ¼

μ λ R, ν ¼ 1ν 2ð λ + μ Þ

(8)

Eq. (8) contains two unknown functions—σ, R, so it arises the natural question of constructing a closed system of equations for them.

Heuristic recipe of getting the equation for the incompatibility function Since for a massif weakened by a cylindrical cavity, in the framework of the classical model of a continuous medium, the first invariant of the stress tensor σ clas ¼ const, it is natural from the point of view of physics to study the deviations σ  σ clas ¼ g from the classical solution σ clas. From here and from (6), Δσ ¼ Δg follows. Then, it is natural to assume that Δg is a function of g and its first derivatives: Δg ¼ Φðrg, gÞ

(9)

Since a linear rheological relationship (4) is considered, for further analysis, we restrict ourselves to small deviations | g | ≪ | σ кл |. In this case, we can expand the function on the right-hand side of (9) in a neighborhood of zero and leave only linear contributions: Δg ¼ ðΦ1 , rgÞ + Φ0 g + …,

(10)

Detailed theory

67

where Φ1 is an arbitrary constant vector, Φ0 is a scalar. Eq. (10) admits a reduction in which we exclude the first derivatives on the right-hand side of (10). For this, we assume g ¼ fG,rg ¼ f rG + Grf ,Δg ¼ f ΔG + GΔf + 2ðrG, rf Þ Substitution of those expressions into (10) gives f ΔG + GΔf + 2ðrG, rf Þ ¼ ðΦ1 , rGÞf + ðΦ1 , rf ÞG + Φ0 Gf We will require the fulfillment of the condition 2(r G, r f) ¼ (Φ1, r G)f. To do this, choose f in the form   1 f ¼ f0 exp ðx, Φ1 Þ 2 The functions g and G differ by a factor of the monotonic function f. Since the modification of the classical model is aimed at obtaining a solution that differs from monotonic, then, without loss of generality, we can assume Φ1 ¼ 0. Then Eq. (10) is reduced to the form Δg ¼ γg,

(11)

where γ is the phenomenological parameter of the model, and its sign can be any. Then from (8), (11), it follows γg ¼ 

μ R, ΔR ¼ γR 1ν

(12)

Consequently, a modification of the classical model associated with the rejection of the harmonicity property of the first invariant of the stress tensor σ leads to the simplest representation for σ in the form σ ¼ σ clas + g, where g is determined, as can be seen from (12), through the incompatibility function R. It is a characteristic of the internal structure of the medium, and the parameter γ determines the scale of the inhomogeneities, since its dimension is [γ] ¼ 1/l2, where l has the dimension of length. From the viewpoint of physics, the classical model corresponds to the case of the absence of heterogeneities, i.e. the limit l ! 0 and γ ! ∞, which means, as follows from (12), the incompatibility function R ! 0. To take into account the different signs of the parameter γ, we can write Eq. (12) for R as follows: Δ2 R ¼ γ 2 R

(13)

It should be noted that for the first time Eq. (13) for the incompatibility function was presented in Guzev and Paroshin (2001). The object R was used in Myasnikov, Guzev, and Ushakov (2004) to describe self-balanced stresses as well as in models of continuous media with an internal

68

3. Non-Euclidean model of rock masses

structure (Guzev, 2011). The presented a heuristic method of obtaining the equation for R gives a result that coincides with the thermodynamically correct one.

Stress function of the non-Euclidean model and the internal stresses If we substitute (2) in (8), then we obtain the equation for the stress function: μ R (14) Δ2 Φ ¼ 1ν A comparison of (14) with (6) shows that the transition from the classical model to the non-Euclidean one leads to the inhomogeneous biharmonic equation for the stress function. A similar equation was obtained in Kr€ oner (1985), and on page 753, the author notes: “The simplicity of eqns … [our comment: inhomogeneous biharmonic equation] makes the stress potential … a most useful tool for incompatible strain problems.” Because Eq. (14) is linear, then its solution can be represented as the sum of the classical stress function Φclas and the additional contribution of Φnon-Eucl: Φ ¼ Φclas + ΦnonEucl

(15)

Since Φclas satisfies the homogeneous biharmonic equation (14), then Φnon-Eucl is a particular solution of (14). Performing a renormalization μ for R in (14), i.e., assuming 1ν R ! R, we rewrite (14) in the form Δ2 ΦnonEucl ¼ R From here and (16), we obtain   R Δ2 ΦnonEucl  2 ¼ 0 γ

(16)

Since we assigned the solutions of the homogeneous biharmonic equation to the function Φclas (6), it follows from (16) ΦnonEucl ¼

R γ2

(17)

The complete stress function is given by Eq. (15): Φ ¼ Φclas +

R γ2

Then, the combination of (2), (18) results in

(18)

Description of rock mass damage at great depths

σ 11 ¼ σ 11,clas + σ 11,nonEucl , σ 11,clas ¼

69

∂2 Φclas 1 ∂2 R ,σ ¼ 11,nonEucl γ 2 ∂x2 ∂x2 ∂x2 ∂x2

σ 22 ¼ σ 11,clas + σ 11,nonEucl ¼ σ 22,clas ¼

∂2 Φclas 1 ∂2 R , σ 22,nonEucl ¼ 2 1 1 1 1 γ ∂x ∂x ∂x ∂x

σ 12 ¼ σ 12,clas + σ 12,nonEucl , σ 12,clas ¼ 

(19)

∂2 Φclas 1 ∂2 R , σ 12,nonEucl ¼  2 1 2 1 2 γ ∂x ∂x ∂x ∂x

It can be seen from (19) that the structure of the field of internal stresses consists of the classical field of stresses and the stress field parameterized via the incompatibility function R.

Description of rock mass damage at great depths The stress field Returning to the boundary-value problem of describing the state of highly compressed rocks, we formulate it for a weightless solid body under conditions of a plane stress state. The plate contains a round hole that simulates an unsupported underground opening, and the stresses given at infinity simulate the gravitational field. Due to the polar symmetry of the problem, the equilibrium equation (1) are written as follows: ∂σ rr σ rr  σ φφ + ¼ 0, σ rφ ¼ 0 ∂r r

(20)

Then, at the hole boundary and at infinity, the following boundary conditions are assumed σ rr ¼ 0 at r ¼ r0 , σ rr , σ φφ ! σ ∞ at r ! ∞, where σ ∞ ¼ γ rH, γ r is the specific gravity of the rock, and H is the depth of disclosure. In polar coordinates, the stress components (19), see e.g., ( Jaeger et al., 2007), are equal to σ rr ¼ σ rr,clas + σ rr,nonEucl , σ rr, clas ¼

1 ∂Φclas 1 ∂ΦnonEucl ,σ nonEucl ¼ r ∂r r ∂r

σ φφ ¼ σ φφ,clas + σ φφ,nonEucl , σ φφ,clas ¼

∂2 Φclas ∂2 ΦnonEucl ,σ ¼ ;σ rφ ¼ 0 φφ,nonEucl ∂r2 ∂r2 (21)

It is seen from here that relations (20) are identically satisfied. pffiffiffiffiffi Since the parameter γ > 0 or γ < 0, we can introduce variable s ¼ |γ|r and consider the incompatibility function R(s) for both positive (R+ ¼ R+(s)) and negative (R ¼ R(s)) values of γ:

70

3. Non-Euclidean model of rock masses

∂2 R + 1 ∂R + ∂2 R 1 ∂R + R  R ¼ 0 + ¼ 0, + + s ∂s s ∂s ∂s2 ∂s2 The solutions of these equations decreasing at r ! ∞ are determined through cylindrical functions: R + ðsÞ ¼ aJ 0 ðsÞ + bY0 ðsÞ, R ðsÞ ¼ cK0 ðsÞ where J0(s) and Y0(s) are the zeroth-order Bessel and Neumann functions, respectively, K0(s) is the zeroth-order MacDonald function. From here and from (17), we obtain RðsÞ ¼ R + ðsÞ + R ðsÞ,ΦnonEucl ¼

1 ½aJ ðsÞ + bY0 ðsÞ + cK0 ðsÞ γ2 0

(22)

Substituting (22) into (21) and using the relations between the cylindrical functions, we obtain     r2 r2 σ rr ðrÞ ¼ σ ∞ 1  02 + σ rr,nonEucl , σ φφ ðrÞ ¼ σ ∞ 1 + 02 + σ φφ,nonEucl r r σ rr,nonEucl ¼  σ φφ, nonEucl ¼

a J1 ðsÞ b Y1 ðsÞ c K1 ðsÞ   γ s γ s γ s

(23)

a J1 ðsÞ b Y1 ðsÞ c K1 ðsÞ a b c + +  J0 ðsÞ  Y0 ðsÞ + K0 ðsÞ γ s γ s γ s γ γ γ

where J1(s), Y1(s), and K1(s) are the first-order Bessel, Neumann and MacDonald functions. Bessel and Neumann functions have an oscillating character. Then (23) implies that the stresses around the opening are spatially oscillating.

Nonclassical boundary conditions To minimize the set of experimental data necessary to verify the constants a,b,c, we note that at the boundary r ¼ r0, the rock mass undergoes significant destruction. Then, the regions corresponding to the maximum value of this parameter modulus should be considered as zones of violation of the medium continuity—zones of the massif pffiffiffiffiffi  destruction. Consequently, the incompatibility function R |γ|r reaches its critical value, which corresponds to the vanishing of its derivative at the boundary: pffiffiffiffiffi  dR |γ|r ¼0 (24) dr r¼r0

Description of rock mass damage at great depths

71

Assuming that the destructive nature of all zones of the rock is the same, it is natural to require the derivative of the incompatibility function to vanish for some of these zones of destruction: pffiffiffiffiffi  dR |γ|r ¼0 (25) dr r¼r1

where r1 is the distance from the initial circuit to the midpoint of the first failure zone, which was obtained from the experimental data. Solutions for the stress field under conditions (24), (25) were constructed in Makarov, Guzev, Odintsev, and Ksendzenko (2016). However, the incompatibility function is a kinematic characteristic, and the destruction around the mine workings occurs at sufficiently great depths, that is, when the stress in the massif reaches a certain critical value. Therefore, when describing the phenomenon of zonal disintegration, the stress arising in the massif should be related to the incompatibility function. From a physical viewpoint, it is clear that the areas of the greatest violation of the medium continuity—the areas of the maximum modulus of the incompatibility function—must coincide with the areas in which the stress becomes critical, or some force failure criterion is fulfilled.

Criteria of failure When choosing the criterion for the rock medium destruction, it is necessary to take into account the physical conditions of this process. Moreover, it is necessary to take into account the scale levels for examination of rocks that have developed in mining practice, with which laboratory and full-scale mine experiments are usually associated. These scale levels include the level of the rock microstructure, studied in the laboratory using electron microscopes, the scale of the rock sample, the scale of the mining openings, and the scale of the deposit. Macrocrack development models were considered in Odintsev (1994). It was found that the development of tensile macrocracks under compression is determined by the collective action of microcracks, and, depending on the nature of the microcracks, the regularities of the macrocracks development differ. Fracture zones appear in areas where crack formation conditions are satisfied:

 KI ¼ ðπlÞ1=2 γ 1 σ 01  γ 3 σ 03  KIC Quantity (πl)1/2(γ 1σ 01  γ 3σ 03) is the expression of the stress intensity factor KI, where l is the half-length of the crack; σ 01, σ 03 are the maximum and minimum principal stresses, respectively; γ 1, γ 3 are empirical coefficients; KIC is the fracture strength of the rock material. As a criterion function,

72

3. Non-Euclidean model of rock masses

K(r) ¼ KI/KIC is introduced. When KI/KIC < 1, no destruction occurs around the mine; when KI/KIC >, destruction takes place. The criterion function, as well as the stress components (23) and the incompatibility function (22), is oscillatory in nature.

Some results To perform calculations on the zonal destruction of the rock mass, algorithms and programs were developed pthat ffiffiffiffiffi included formulas for calculating the incompatibility function R |γ|r , stress, and criterial function K(r). Based on the developed programs, an experiment was conducted with three groups of entered parameters. The first group includes the parameters γ and c in relation (23), the second group characterizes the mechanical properties of the rock mass: parameter of the Lam
e μ; Poisson’s ratio ν; uniaxial compression strength σ c, and the magnitude of gravitational stresses in a rock mass σ ∞. The third group includes parameters characterizing the structure of the rock mass: the crack half-length, the fracture toughness of the rock material KIC. A model experiment showed that the parameters of the zonal structure are little dependent on the values of μ and ν. This conclusion is consistent with laboratory research data: when μ changes by a factor of 10, the critical stress values at which zones are formed change by 2%–5% on average. The parameter γ can be determined based on a statistical analysis of the zonal destruction process for deposits in Russia (the Far Eastern part, Siberia, Donbass) and China. The relationship between the parameters γ and r/r0 is selected linear and is presented in Table 1 (Makarov et al., 2016). The results of the application of the developed method in the study of hard rocks (uniaxial compression strength 150 MPa) to the problem of zonal fracture at the Nikolaevsky mine (Dal’negorsk, Russia) are shown in Table 2 (Makarov et al., 2016). A comparison of the analytical and experimental results for soft rock (Table 3, Makarov et al., 2016) also shows good coincidence of the results. TABLE 1 Values of the model parameter g. γ(m2)

Middle part of the first zone, 1 2 r/r0

r0 5 1.75 m

r0 5 2 m

r0 5 2.5 m

r0 5 3 m

0.7

26.49

20.30

12.98

9.01

0.8

20.31

15.55

9.95

6.91

0.9

16.08

12.31

7.88

5.47

1.0

13.05

9.99

6.39

4.44

73

Acknowledgments

TABLE 2 Predicted depth of zonal failure appearance in the Nikolaevskij ore mine. Number of failure zones

Relative critical stress of zone formation, σ/σ c

Depth of zone appearance, m

1

1.3

520

2

2.3

920

3

2.9

1160

4

3.3

1320

It follows that the basic factor that influences the parameters of zonal failure structure is the value of stress that acts within the rock mass. With the increase of the stress, the number of failure zones and their radial length increase. A detailed description of the algorithm for selecting other parameters, as well as the analytical and experimental results for the phenomenon of zonal disintegration, is presented in (Makarov et al., 2016).

Conclusions We demonstrated that the main idea of the transition from the classical elastic model of the medium to the non-Euclidean model is associated with the rejection of Saint-Venant compatibility condition. The non-Euclidean model has various advantages, but its practical application is limited by the lack of methods for determining the model parameters, so these methods must be developed. Nevertheless, a comparison of theoretical and experimental results shows their satisfactory coincidence. Further application of this model is linked with a description of the materials at different hierarchical levels. For rocks, the general idea of using the non-Euclidean model at different scale levels is determined by the principle of non-Euclidean hierarchy (Guzev, Odintsev, & Makarov, 2018). In this case, each scale level is investigated in the frame of the corresponding non-Euclidean model, each of which covers a certain range of hierarchy.

Acknowledgments The author is much grateful to V.V. Makarov, V.N. Oparin and, V.N. Odintsev for their useful discussions and consultations on rock mechanics. The study was carried out in the frame of tasks № 075-01095-20-00, № 075-01095-19-01.

Experiment 1.03 2.23 3.4 4.54

Elements of zonal failure structure

1

2

3

4

3.97

3.09

2.17

1.28

Theory

12.6

9.1

2.7

24.3

Deviation (%)

Location of the zone boundary, r/r0

TABLE 3 Comparison of the analytical and experimental results of the soft rock.

3.1 –

–

2.1

0.95

Theory

2.7

2.2

1.19

Experiment

–

14.8

4.5

13.9

Deviation (%)

Relative critical zone stresses, σ/σ c

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Majcherczyk, T., Malkowski, P., & Niedbalski, Z. (2005). Describing quality of rocks around underground headings: Endoscopic observation of the fractures. In: Proceedings of the 2005 ISRM international symposium EUROCK 2005, Brno, Czech Republic, May 18–20, pp. 125–137. Makarov, V. V., Guzev, M. A., Odintsev, V. N., & Ksendzenko, L. S. (2016). Periodical zonal character of damage near the openings in highly-stressed rock mass conditions. Journal of Rock Mechanics and Geotechnical Engineering, 8(2), 164–169. https://doi.org/10.1016/j. jrmge.2015.09.010. Metlov, L. S., Morozov, A. F., & Zborshchik, M. P. (2002). Physical foundations of mechanism of zonal rock failure in the vicinity of mine working. Journal of Mining Science, 38(2), 150–155. https://doi.org/10.1023/A:1021111521279. Myasnikov, V. P., & Guzev, M. A. (1999). A geometrical model of the defect structure of an elastoplastic continuous medium. Journal of Applied Mechanics and Technical Physics, 40(2), 331–340. https://doi.org/10.1007/BF02468531. Myasnikov, V. P., Guzev, M. A., & Ushakov, A. A. (2004). Self-equilibrated stress fields in a continuous medium. Journal of Applied Mechanics and Technical Physics, 45(4), 558–564. https://doi.org/10.1023/B:JAMT.0000030334.32046.e6. Odintsev, V. N. (1994). Mechanism of the zonal disintegration of a rock mass in the vicinity of deep-level workings. Journal of Mining Science, 30(4), 334–343. https://doi.org/10.1007/ BF02048178. Oparin, V. N., & Tapsiev, A. P. (1979). On some patterns of crack formation around mine workings. In Rock bumps, methods for assessing and controlling the shock hazard of a rock mass (pp. 342–349). Ilim, Frunze. (in Russian). Pan, Y. S., Li, Y. J., Tang, X., & Zhang, Z. H. (2007). Study on zonal disintegration of rock. Chinese Journal of Rock Mechanics and Engineering, 26(s1), s3335–s3341. (in Chinese). Qian, Q., & Zhou, X. (2011). Non-euclidean continuum model of the zonal disintegration of surrounding rocks around a deep circular tunnel in a non-hydrostatic pressure state. Journal of Mining Science, 47(1), 37–46. https://doi.org/10.1134/S1062739147010059. Shemyakin, E. I., Fisenko, G. L., Kurlenya, M. V., Oparin, V. N., Reva, V. N., Glushikhin, F. P., et al. (1986). Zonal disintegration of rocks around underground workings. Part I: Data of in situ observations. Journal of Mining Science, 22(3), 157–168. Shemyakin, E. I., Fisenko, G. L., Kurlenya, M. V., Oparin, V. N., Reva, V. N., Glushikhin, F. P., et al. (1987). Zonal disintegration of rocks around underground workings. Part III: Theoretical concepts. Journal of Mining Science, 23(1), 1–6. https://doi.org/10.1007/ BF02534034. Sih, G. C., & Tang, X. S. (2005). Screw dislocations generated from crack tip of selfconsistent and self-equilibrated systems of residual stresses: Atomic, meso and micro. Theoretical and Applied Fracture Mechanics, 43(3), 261–307. https://doi.org/10.1016/j. tafmec.2005.03.001. Tan, Y. L., Ning, J. G., & Li, H. T. (2012). In situ explorations on zonal disintegration of roof strata in deep coal mines. International Journal of Rock Mechanics and Mining Sciences, 49(1), 113–124. https://doi.org/10.1007/s00603-011-0146-5. Tropp, E. A., Rozenbaum, M. A., Reva, V. N., & Glushikhin, F. P. (1985). Zonal disintegration of rock around workings at large depths. Preprint of the A. F. Ioffe Physical-Technical Institute Academy of Sciences of the USSR Leningrad, No. 976 (in Russian). Wu, H., Fang, Q., Zhang, Y. D., & Gong, Z. M. (2009a). Zonal disintegration phenomenon in enclosing rock mass surrounding deep tunnels-Elasto-plastic analysis of stress field of enclosing rock mass. Mining Science and Technology, 19(1), 84–90. https://doi.org/ 10.1016/S1674-5264(09)60016-8. Wu, H., Fang, Q., Zhang, Y. D., & Gong, Z. M. (2009b). Zonal disintegration phenomenon in enclosing rock mass surrounding deep tunnels-mechanism and discussion of characteristic parameters. Mining Science and Technology, 19(3), 306–311. https://doi.org/10.1016/ S1674-5264(09)60057-0.

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C H A P T E R

4 A guide to modeling the geotechnical behavior of soils using the discrete element method Daniel Barreto and James Leak Edinburgh Napier University, Edinburgh, United Kingdom

Introduction The discrete element method (DEM) originally developed by Cundall and Strack (1979) for the analysis of rock mechanics problems is significantly popular. Its use is widespread across multiple disciplines (e.g., Fleissner, Gaugele, & Eberhard, 2007; Horabik & Molenda, 2016; Ketterhagen, am Ende, & Hancock, 2009; O’Sullivan, 2011; Richards, Bithell, Dove, & Hodge, 2004; Sarhosis, Bagi, Lemos, & Milani, 2016), the number of publications has also exponentially increased (O’Sullivan, 2011), and open-source and commercial DEM codes are widely available. There is a large body of existing research using DEM, as well as a number of detailed publications on the use of DEM (e.g., O’Sullivan, 2011; Rapaport, 2004; Thornton, 2015). The usefulness of DEM is recognized; however, it is not easy for the novice user to understand in detail its capabilities, limitations, and potential pitfalls. This chapter is aimed to provide a concise source of information that may be of use for anyone interested in using DEM for the simulation of geotechnical problems, but who has limited or no experience or knowledge of the topic. This article/chapter starts by describing the DEM algorithm. Consequently, the steps required are used as a template for the following sections. Boundary types, particle types, specimen generation, as well as some details on contact detection, force calculation, and numerical

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integration procedures are therefore described. Subsequently, an introduction to the common data analysis required for DEM simulations is presented in terms of macro- and micro-variables. While most of the information on the initial sections refers to specimens of spheres, this information provides a theoretical background to then discuss how more realistic granular material simulations can be performed. Particle shape, crushing, fluid coupling, and clay behavior are briefly discussed. The chapter is then concluded with some practical considerations required for running good DEM simulations.

DEM algorithm The essence of DEM is that individual particle movements (i.e., translation and rotation) and their interaction with other particles can be quantified in detail. The first step of any DEM simulation is to define its geometry. This typically involves providing the locations of all boundaries and particles as well as defining the physical and geometrical properties to adequately represent their behavior. Then, the following steps are followed sequentially: 1. Identify interparticle and boundary-particle contacts. 2. Calculate contact forces using an appropriate contact model (forcedisplacement law). 3. Calculate the acceleration of particles using Newton’s second law of motion. 4. Integrate the particle accelerations twice to obtain particle velocities and displacements, respectively. 5. Update particle and boundary positions. 6. Advance simulation time by a time-step increment and repeat steps 1–6 until the simulation is complete. The following sections follow this order to describe important issues of the implementation and the appropriate use of DEM algorithms. Rather than focusing on the detail, the emphasis is on the principles while providing references for readers to deepen their knowledge.

Boundary types In most DEM simulations, three different boundary types may be used, namely, rigid, flexible, and periodic (Fig. 1). Although DEM is a strain/ displacement-based algorithm, all boundary types can be stress-controlled by servo-control algorithms and are generally modeled with no mass/inertia.

Boundary types

FIG. 1

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(A) Rigid, (B) flexible, and (C) periodic boundaries.

Rigid boundaries can take any shape, from a cylindrical/rectangular plate typically seen in geotechnical element tests (i.e., direct shear box, triaxial), and such as those illustrated in Fig. 1A, for a two-dimensional biaxial compression simulation. Rigid boundaries can also represent more complex geometries such as funnels, rotating drums, comminutors, etc. These boundaries are generally specified as a collection of planes and/ or triangular facets or sets of glued/bonded particles. Flexible boundaries (see Fig. 1B) may be used to represent stresscontrolled membranes such as the latex membranes surrounding soil specimens in a triaxial device. In this case, implementations are varied and include movable rigid/plate boundaries (e.g., Kuhn, 1995), elasticity-based models (e.g., Qu, Feng, Wang, & Wang, 2019), bonded particles (e.g., Wang & Leung, 2008), or Voronoi/Delaunay approaches (e.g., Cheung & O’Sullivan, 2008). The approach by Kuhn (1995) uses triangular plates connecting the centroids of particles in contact with the membrane, and to which an equivalent force corresponding to the stress required is applied. On the other hand, Voronoi/Delaunay approaches do not directly model the flexible boundary (see Fig. 1B); they simply use Delaunay triangulation and/or Voronoi tessellation approaches to calculate areas with centroids coinciding with the particle centroids. In this way, a force can be applied to each particle according to the required stress level. Each of these approaches has its own advantages and disadvantages. While the implementation of these approaches is not complex, particular attention must be given to provide the regular updates of the geometry of the boundary which increases the computational cost of the DEM simulation. A challenge when using rigid boundaries (e.g., Fig. 1A), which is also present in physical laboratory experiments and boundary-value problems, is that stress and strain nonuniformities may occur. In other words, the stresses/strains measured in the neighborhood of a rigid boundary may not be the same as those measured far from them. To reduce the effect of these nonuniformities, the geometry of the simulation should be

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carefully considered, and as a result, a significantly large number of particles may be required. An alternative method commonly adopted is to use periodic boundaries. They provide a means to avoid boundary effects and are the equivalent of single element simulations in finite element analyses. As such, it is then possible to simulate a uniform and infinite strain field using a significantly smaller number of particles than that required to simulate the same soil behavior using rigid boundaries. Implementation details for the use of periodic boundaries are clearly described by O’Sullivan (2011).

Particle types In its simplest form, DEM involves the simulation of individual spheres (3D) and discs (2D). These particles provide an efficient means to quantify micro-mechanics, but they are an over-simplification of reality. However, any shape can be simulated using various approaches. One of the most common methods employed to replicate more realistic particle shapes is the use of clumps of individual particles (see Fig. 2A). Any particle shape can be represented for a DEM simulation using this method. Care is, however, needed if there is an overlap between the individual clump particles because the inertia of the clump needs to be adjusted. An approach to avoid this is described by Ferellec and McDowell (2010). Also, the more spheres are used to model each particle, the more accurate its representation, but this also increases the computational effort required for the simulation. To a certain extent, even particle surface roughness could be modeled in this way, albeit at a significant computational expense. Sphero-simplices (Pournin, 2005) are other convenient means to simulate differing particle shapes while benefiting of the computational efficiency related to the simulation of spheres. Their modeling consists of lines and/or polygons across which a sphere can be moved to recreate a particle shape with rounded corners (see Fig. 2B). Superquadric equations are an extension from ellipses and spheres to represent more complex particle shapes using a limited number of mathematical parameters (Fig. 2C) that have been used widely (e.g., Cleary, 2004; Podlozhnyuk, Pirker, & Kloss, 2018; Soltanbeigi et al., 2018). Polyhedra and other simple geometries, such as cylinders and ellipses, are also commonly used in DEM simulations (Gan & Yu, 2020; Lin & Ng, 2004; Lu, Third, & M€ uller, 2015; Xie, Song, & Zhao, 2020; Zhao, Kruyt, & Millet, 2020) as those illustrated in Fig. 2D. In reality, any curve (or surface) in space can be described by analytical/mathematical expressions. As such, potential surfaces and other ‘avatar’ approaches are often used (e.g., Harkness, 2009; Kawamoto, Ando`, Viggiani, & Andrade, 2018), which can accurately replicate more realistic particle shape.

Specimen generation

FIG. 2

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(A) Spheres and clumps, (B) sphero-simplices, (C) superquadrics, (D) polyhedra.

Specimen generation Having defined the geometry of individual particles as well as the boundary types required, an assembly of particles enclosed by the boundaries that adequately represent reality needs to be generated. To do this, various approaches may be used including random generation, radial expansion, gravitational deposition, or other approaches. In many DEM simulations, random generation is usually the first stage. Here, a number of particles are generated within the boundaries while ensuring that there are no interparticle or boundary-particle contacts. There are several approaches to achieve this, but it is common to first (randomly) locate the largest particles and then try several times to fill the spaces between them with the smaller particles. At this point, the particles may have

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the same shape, but not necessarily the same mean diameter as the real soil. If that is the case, the radial expansion method is used to incrementally increase the particle size while the DEM simulation is run. This approach provides some control over the final density of the specimen, but not its resulting confining stress. Furthermore, the resulting particle size distribution may just be a scaled version of the real one. A real particle size distribution’s shape and size can be defined from the outset using the random method, and then servo-control algorithms can be used to achieve the desired stress state. As an alternative, geometrical/sequential specimen generation methods can also be used, but these are generally only applicable to simple particle shapes (e.g., Cui & O’Sullivan, 2003). Since the aim of most DEM simulations is that of replicating reality, the gravity deposition method is worthy of consideration. After random generation, body forces can be switched on during a simulation. This provides weight to individual particles, and therefore, they move in the gravitational direction as cycling progresses. Typically, a significant number of cycles are required for all particles to settle and attain a state of equilibrium. Whatever the method used, control of the initial density of DEM specimens is relatively easy. When interparticle friction is small, particles can slide against each other and accommodate with ease and thus achieving very dense particle assemblies. On the other hand, high interparticle friction restricts interparticle movement and hence results in DEM specimens with low density. Note that it is possible to change the value of the coefficient of interparticle friction during the specimen generation stage to further control the density. However, care must be taken to achieve a stable structure after friction reductions as they may produce excessive interparticle sliding and consequently density increases. However, the effect of interparticle friction on specimen density is nonlinear, and it reduces as the value of interparticle friction increases. Furthermore, it is not commonly recognized that superfluous increases in the shear stiffness of specimens may occur when using high friction values. Another important aspect of specimen generation approaches is their effect on fabric (i.e., the geometric configuration of the particles). Radial and random generation approaches provide specimens with isotropic fabric (i.e., contacts orientations distributed equally in all directions). On the other hand, gravitational deposition methods generate preferred contact orientations in the direction of the gravitational force and therefore, an anisotropic fabric. The use of particle shape beyond spheres or discs also affects fabric anisotropy, particularly when gravity is used because the major particle axis tends to align perpendicularly to the direction of gravity. It is important to be aware of these effects because it is well recognized that fabric anisotropy and specimen preparation affect the mechanical response of granular materials (e.g., Vaid, Sivathayalan, & Stedman, 1999)

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Contact detection To calculate interparticle forces, contacts between individual particles need to be identified. A key assumption of DEM simulations is that although particles are assumed to be rigid, they are allowed to overlap to account for some degree of deformation that would occur in physical experiments. Consequently, contact between two spheres simply occurs when the distance between particle centroids is equal or smaller than the sum of their radii. The process of contact detection is one of the most time-consuming stages in DEM simulations. As a result, there are several implementations of contact detection algorithms aiming to increase computational efficiency. The simplest and most inefficient way is of course to test the existence of contacts between each particle and every other particle in the simulation. There are, however, more efficient ways to detect interparticle contacts that include: (i) neighboring cell approaches, (ii) nearest neighbor approaches, and (iii) bounding box (sweep and prune) approaches, among others. To understand the differences between these approaches is convenient to consider the case of interparticle contacts between discs (Fig. 3). When using neighboring cell approaches (see Fig. 3A), the spatial domain is divided into smaller individual “cells” (boxes in 3D). Normally, a list of particles for each cell is compiled which can be then updated periodically. The gain in efficiency is made because for each particle contact, checks are only made with particles within the same cell and the neighboring cells. Cell size of course then needs to be related to particle size. Nearest neighbor approaches (see Fig. 3B) define a neighborhood for each particle (discontinuous line) and then a list of near-neighbors (i.e., lighter particles) is maintained and periodically updated for each particle. Contacts are then only checked between neighbors. Care must also be taken when considering the size of the neighborhood in relation to the particle size distribution.

FIG. 3 Contact detection algorithms using (A) neighboring cell, (B) nearest neighbor, and (C) bounding box approaches.

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The principle behind bounding box approaches is that each particle has a bounding box (with discontinuous lines) with edges aligned with the global axes (see Fig. 3C). Contacts between particles can only occur if the boxes overlap in the two (or three, for 3D) coordinate directions. Hence first, a sweep is first done to find overlapping bounding boxes and a second round of finer contact detection (i.e., pruning) between overlapping boxes. If particle shapes other than spheres are considered, the approaches described above can still be used. They are, however, used first as “coarse” contact detection approaches using bounding spheres (or boxes) as illustrated in Fig. 3C. A fine detection is then required because several contact types may be found that add to the complexity of the calculations (e.g., point-surface, point-edge, or surface-surface contact types when using polyhedral particles). Accuracy in contact detection is important because the nature of the contacts also determines the magnitude of contact forces as described in the next section. The computational cost reductions that can be achieved by implementing efficient contact detection algorithms are significant. This is a vast research field of its own. It is of the utmost importance to always consider if a specific algorithm is suitable for the specific particle size distribution and particle shape being replicated in the DEM simulation.

Force calculation Once contacts between particles and boundaries have been identified, the next step is the calculation of contact forces based on the chosen contact model (i.e., force-displacement law). Usually, this involves the estimation of both normal and tangential forces at each contact. Considering contact between two spheres and the simplest contact model (linear elastic), the magnitude of the normal contact force (Fn) is given as a function of the particle overlap, Δn as: Fn ¼ kn Δn where kn is the normal contact stiffness. For the case of contact between spheres, the contact overlap (Δn) is the difference between the sum of the particle radii and the centroidal distance between the two particles. Similarly, the tangential force, Ft, can be calculated as: Ft ¼ kt Δs where kt is the tangential/shear stiffness at the contact, and Δs is the cumulative tangential/shear displacement at the contacts. The appropriate implementation Δs is a key aspect of any DEM software. Additionally, slippage at the contacts needs to be considered. This may be done by

Numerical integration procedures

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comparing the magnitude of Ft against the product of the coefficient of interparticle friction (μ) and the magnitude of the normal contact force (Fn). If Ft μFn, then the contact slips and the magnitude of the tangential force is set to Ft ¼ μFn. Contact slippage in DEM simulation is the main form of energy dissipation, and it is aimed to replicate also other forms of dissipation such as heat generation, plastic deformation of contact asperities, etc. The selection of the right value of μ is therefore very important to accurately model geotechnical behavior. Despite their simplicity, linear elastic models are able to replicate many of the complexities of real soil behavior such as stiffness degradation, stress and strain anisotropy, among others. Having said that, more realistic/complex contact may be used. Nonlinear elastic (i.e., Hertzian) contact models, in which the normal contact stiffness is dependent on the contact area, are also very common. Using rheological principles, hysteretic behavior, viscous effects, and long-range forces can also be modeled. Of particular interest for geotechnical applications, long-range contact models may consider electrostatic interactions (as required for the simulation of clay behavior), tensile strength (needed when simulating bonded/cemented soils), and capillary forces (for unsaturated soil behavior), among others.

Numerical integration procedures Having calculated the magnitude and direction of forces at each contact, a summation of all contact forces on individual particles should be made. The resulting force (which is normally characterized by its three components on each of the coordinate directions) on each particle will produce an acceleration (a) in the same direction of the force, and it is calculated using Newton’s second law a ¼ F=m where F is the resultant force acting at the particle and m is the mass of the particle. Particle rotations are considered in a similar manner using the moment of inertia. An important aspect of any DEM formulation is that any force type may be included in the calculation of F above. Therefore, body forces (i.e., gravity) and external forces in addition to those resulting from the contact model can also be considered. The final step in the DEM calculation is to use the defined time-step and using a finite-difference integration scheme, to integrate the accelerations, and to obtain particle velocities and updated positions for particles and boundaries. The new geometry is then used to repeat all calculation steps described.

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As a result of these calculation steps, particle positions, velocities, accelerations, as well as the details of the interparticle forces for each particle at every-step can be obtained. Getting this information in other ways (i.e., experimentally) can be challenging and/or expensive. Nevertheless, the amount of information that can be gathered from a DEM simulation is significant, and its analysis and understanding are not necessarily simple. In the case of three-dimensional simulations, the interpretation of results is not helped by the fact that a clear visualization of results is a challenging task.

Analysis of DEM data Fig. 4A illustrates the geometrical configuration of a typical threedimensional assembly of spheres at a given time-step. Such a plot may be easily modified to visualize (for example) the amount of particle rotation or any other quantity of interest. Visualization of contact force networks is very commonly made. In the case of spherical particles (Fig. 4B), contact vectors connecting particle centroids are illustrated with their thickness proportional to the magnitude of the force. While these visualizations may be attractive and may be used to illustrate specific mechanisms (e.g., arching or shear banding), they are generally just useful in a qualitative manner. Fig. 4B also shows that stress transmission in granular materials is not homogeneous. Interparticle forces are commonly divided into strong and weak force chains/networks according to the average magnitude of the contact forces at a given time-step. Strong and weak force chains have differing and important roles to understand

FIG. 4 3D representation of (A) an assembly of spheres and (B) its contact force network.

Macro-scale variables

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the geotechnical behavior of soils (e.g., Barreto & O’Sullivan, 2012). The illustrations in Fig. 4 also highlight that 3D visualization is challenging and highlight that quantitative approaches to analyze DEM data may be of more use and practical interest.

Macro-scale variables Due to the widespread use of continuum approaches in geotechnical modeling, as well as the obvious use of physical experiments to understand soil behavior, the analysis of DEM results should be presented in a similar manner. In other words, particle-scale variables such as contact forces need to be converted into a stress tensor and particle positions, rotations, and velocities may be converted into strains. The calculation of stresses from interparticle forces is simply a mathematical task, although with exact physical meaning as discussed by Bagi (1996). Homogenization approaches may however differ. A common feature of these approaches is that interparticle forces are averaged over a volume of interest. Due to the heterogeneous nature of stress transmission in granular materials (see Fig. 4B), the choice of such measurement volume is an important decision that depends on the type of simulation performed and what is sought from it. When periodic boundaries are used, the solution is simple because the entire volume can be used for stress calculation, and provided that there is a sufficient number of particles it provides statistically representative data. In contrast, when the problem requires the use of rigid boundaries, the size, shape, and position of the measurement volume need to be considered with care. As discussed before, the stresses near rigid boundaries may be significantly different far from them due to the presence of nonuniformities. Most commercial software uses measurement spheres for quantification of micro- and macroscale variables form DEM simulations. In reality, however, the shape of the volume is not relevant as long as its volume can be calculated accurately. The size of the measurement volume is of significant importance, particularly when a statistically representative sample is required for the calculation of variables. Barreto and O’Sullivan (2012) discussed that for a relatively uniformly graded DEM specimen, they required at least 4000 particles in their simulations using periodic boundaries. A good rule of thumb that may be used when selecting a number of particles in a simulation (or the size) of measurement volume is that at least 5000 particles are required for each significant size fraction (i.e., the fraction that would be retained on a single sieve for a real granulometric analysis). The calculation of strains is relatively straightforward, particularly for simulations considering laboratory element tests. The reason for this is that strains can be calculated in the same way as in physical experiments

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as a function of sample size and boundary displacements. DEM, however, enables to calculate local strains, and these may include variables such as individual particle rotations (e.g., Bagi, 2006; O’Sullivan, Bray, & Li, 2003). Despite this, the most common use of stress/strain calculations in DEM analysis is simply to enable comparison with physical experiments and continuum approaches in which the common stress and strain tensors are used.

Micro-scale variables The biggest benefit of DEM analysis with respect to other experimental and numerical approaches is that it enables to quantify particle-scale interactions. In that regard, the use of coordination numbers is widespread. In its easiest form, the coordination number (Z) is defined as:
Z ¼ 2Nc =Np where Nc is the number of contacts and Np is the number of particles. It represents the average number of interparticle contacts per particle. As such, it is used as a frame of reference characterizing the initial state of particulate assemblies. While its value is dependent on stress level, particle shape, and size distribution, it is normally expected for it to be larger in dense specimens than in loose specimens. Note that alternative definitions of Z may exist. For example, Thornton (2000) defined the mechanical coordination number that considers only the particles with more than one interparticle contact. The rationale behind this definition is that particles with less than two contacts would not contribute significantly to stress transmission within a granular assembly. Of particular interest in geomechanics is that researchers consistently report that coordination numbers are constant at the critical state. A significant contribution of DEM analyses is the fact that the mechanical behavior of granular materials is intimately linked to their fabric evolution (i.e., the geometry and magnitude of interparticle forces). In fact, various fabric-stress-strain relationships have been proposed (e.g., Kruyt & Rothenburgh, 2019; Li & Yu, 2013). Fabric analysis is complex and varied. Possibly, there are as many definitions of the fabric tensor as there are researchers discussing and using it. However, the shared aim of all the existing approaches is that of quantifying the evolution of interparticle forces to aid the analysis of DEM simulations and to gain further insight into the particle-scale interactions that underlie the observed macro-scale response. When considering assemblies of spheres, the fabric tensor is normally quantified by separating the individual (Cartesian) components of the branch vectors (i.e., vectors that join the centroids of particles in contact.

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Once again, definitions may vary, but commonly a second-order tensor is formulated from which three principal values and their orientation can be calculated. The advantage of this approach is that fabric parameters can be made analogous to stress parameters. In other words, in the same way, that mean effective stress, deviatoric stress, principal stresses, and their orientation can be defined from the stress tensor, then mean fabric, deviatoric fabric, principal values of fabric, and their orientation can also be defined. This approach is particularly helpful to understand the effects of fabric anisotropy (e.g., Barreto & O’Sullivan, 2012) and the response of granular materials under generalized stress conditions. Another approach that is commonly used is that of fitting Fourier coefficients to the statistical distribution of contact forces in perpendicular planes as proposed by Rothenburgh and Bathurst (1989) and described in detail by Barreto, O’Sullivan, and Zdravkovic (2007). This approach takes advantage of the 2D nature of the corresponding projections of the contact forces on each of the perpendicular planes, and it is particularly useful when analyzing axi-symmetric DEM simulations. Apart from different definitions for the fabric tensor, different data may be used during fabric analysis. For example, the fabric tensor may be calculated considering all interparticle forces or only those with a magnitude equal or greater to the average contact force magnitude (i.e., strong fabric). It is important to highlight that is also widely accepted that fabric tends toward a critical state, and this has enhanced our current understanding of soil mechanics.

Simulation of realistic soil behavior It was discussed that using DEM it is possible to replicate particle shapes in an accurate manner. This is a significant step toward realistic numerical simulations. The consideration of particle roughness, particle crushing, mineral dissolution, fluid coupling, and the modeling of clay particle interactions and contact laws are other significant aspects where significant progress has been made in recent years. Some of these advances are briefly discussed in this section. With the advance of imaging and experimental techniques, it has been possible to characterize the surface characteristics of individual soil particles in a better way. The key parameters considered here are the interparticle friction and surface roughness. Existing experimental research demonstrates that the coefficient of interparticle friction in most common soils varies in a limited range (0.1–0.3 approximately, but this varies among information sources). This is, however, in contrast to the most common value coefficient of interparticle friction used in DEM simulations which is 0.5. In this regard, it is important to highlight that the choice of interparticle friction coefficient has a significant influence on the

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response of granular materials. Not just on the initial density, as discussed before and the angle of shearing resistance as demonstrated by multiple researchers. The mechanism of interparticle interaction is also highly dependent on this value. Huang, Hanley, O’Sullivan, and Kwok (2014) highlighted that there is a transition from sliding to rolling behavior at the contact points as friction increases. Surface roughness is a highly debated subject. Here, it is sufficient to state that Otsubo, O’Sullivan, Hanley, and Sim (2016) implemented a DEM contact model that captures the influence of surface roughness while demonstrating that it is fundamental to adequately simulate geotechnical behavior at small strains. The crushing of individual particles of silica sand at high stresses and/ or carbonate sand at moderate stresses is a distinctive feature of the behavior of coarse-grained soils. Several relatively simple approaches have been proposed in the past. Cheng, Bolton, and Nakata (2004) used a statistically flawed sphere clump to accurately replicate crushing effects. Similar approaches have been used recently (e.g., Ciantia, Arroyo, Calvetti, & Gens, 2015; de Bono & McDowell, 2018) with success. A particular consideration is, however, that mass may not be conserved in these simulations even though it does not necessarily affect the observed response. These approaches are specifically used for DEM simulations of spheres and/ or sphere clumps. For consideration of both particle shape and crushing effects, the use of more advanced (and computationally expensive) approaches is required. Examples where this is possible include the use of combined DEM-FEM approaches (i.e., Munjiza, 2004). The work by Zhu and Zhao (2019) demonstrates an approach with a significant potential that considers a particulate approach, but not strictly the DEM approach discussed here. The possibility of dynamically changing the particle size of individual sizes is also a simple way to model complex geotechnical behavior. Using this approach, Bym, Marketos, Burland, and O’Sullivan (2015) successfully modeled ground response due to tunneling using 2D DEM simulations. Such approaches can also be used to simulate the effects of mineral dissolution (e.g., McDougall, Kelly, & Barreto, 2013). Note, however, that these studies are limited to the simulation of circular or spherical particles and do not consider any form of particle-fluid interaction. Another aspect that is necessary to replicate real soil behavior is solidfluid interactions. Most commonly, DEM research has focused on saturated and/or dry granular materials. Since stress transmission is via interparticle contacts and governed by the principle of effective stress, simulation of fluid is not strictly necessary for many DEM simulations. In the case of some undrained stress paths, the undrained response can be interpreted by performing simulations of (dry) spheres under constant volume conditions. Similarly, the behavior of unsaturated soil can be modeled with success, particularly if behavior between particles

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is governed by the presence of liquid bridges (i.e., the pendular regime). This is possible by introducing long-range particle interactions that depend on the liquid volume or suction value at the contact as discussed in detail by Zhao et al. (2020). Despite these simplified approaches, modeling of solidfluid interactions in DEM is possible. However, approaches vary and are still very computationally expensive. Some possibilities include the use of CFD-DEM coupling (e.g., Kloss, Goniva, Hager, Amberger, & Pirker, 2012), LBM-DEM coupling approaches (e.g., Han & Cundall, 2012), and SPH-DEM approaches (e.g., Wu, Yang, & Wright, 2016), among others. An important aspect where some recent advances have been made is on the ability of DEM to simulate clay behavior. Progress in this area has been the result of advances in experimental techniques, the accurate modeling of particle shape, and the wider ability of high-performance computing clusters worldwide. It is however the area that may be considered to still be on its infancy compared to progress in other areas. It is however the focus in many research groups around the world, and the work by Pagano, Magnanimo, Weinhart, and Tarantino (2020) is a good starting point for readers interested in this area. For the sake of simplicity, most of the discussion above has been limited to DEM simulations of geotechnical laboratory element tests. It is, however, important to highlight that DEM can be used for any boundary-value problem. However, if that is the case, then simplifications are still unavoidable. In fact, even considering element tests, research examples that demonstrate realistic DEM simulation including particle shape, roughness, crushing, and fluid coupling are very limited. There are inherent difficulties that are not only related to computational cost. For example, drag coefficients used in CFD-DEM simulations normally refer to spherical particles. Progress is of course ongoing, but as geotechnical practitioners/researchers, it is always important to be able to focus on the DEM capabilities that are required to analyze/ understand the problem at hand.

Simulation advice The previous sections have attempted to describe the algorithm and the main features and possibilities for DEM simulations, while for simplicity trying to avoid detailed explanations, equations, etc. Such an approach aims to provide a starting point for researchers and/or practitioners who may have the tools and/or background to use this fantastic tool, but that have not done it in the past. As such it is thought appropriate to list some general advice for those readers with limited or no prior DEM experience. The list includes comments from software selection to practical issues, which are important and rarely discussed in the existing literature.

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Software selection and its validation There is a wide selection of software and/or codes available to perform DEM simulations. Listing and describing the capabilities, limitations, and costs of each possibility, is out of scope here. Instead, discussion is focused on important aspects that may influence your selection. Cost is of course the first consideration. Currently, anyone interested in DEM simulation is spoilt for choice in terms of commercial and/or open-source software. Therefore, considerations refer to the programming language used for simulations, documentation quality and support, ease of software modification/update/control, among others. Existing DEM software may be written using several computer languages that include FORTRAN, C ++, MATLAB, among others. Some of them provide control interfaces using other interpreted languages such as Python. Knowledge of the corresponding programming language is of course beneficial if you are likely to require modification for the implementation of new contact laws, boundary conditions, etc. If that is the case, having a detailed documentation is fundamental to have a firm understanding of how the specific implementation works and even to be able to compile and maintain your versions as they are changed and/or updated. To that extent, the support of the software developers might be key. But perhaps the most important consideration when selecting a DEM program is your own ability to validate the results it produces. It will not only test your own ability to operate the program. You need the reliability of knowing that calculations are made appropriately. Some existing codes provide validation examples, whereas others do not. A good starting point is being able to reproduce the behavior of a single particle resting on a horizontal plane under gravity because its behavior can be contrasted against existing analytical solutions for a single degree of freedom system. Similarly, the rolling of a cylinder along an inclined plane enables to test the implementation of the shear/tangential force calculation (e.g., Ke & Bray, 1995). The replication of the stress ratios at failure for Face-CentredCubic assemblies of spheres developed by Thornton (1979) can be used to validate simulation results using both rigid and periodic boundaries and are a good opportunity to gain experience developing servo-controls as required for simulations of triaxial compression.

DEM specimen preparation and input parameters This stage is one of the most time-consuming parts of a DEM study. It takes significant time, and it increases not just with the number of particles included, but also with the contact detection algorithms required, the complexity of the contact models, etc. The first concern is probably an adequate number of particles. As discussed before, this depends on the

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particle size distribution and the boundary type. But some simplifications may be used. For example, many researchers ignore the largest and smallest particle sizes of real PSD because, in doing so, a significant reduction in the number of particles can be achieved. Particle size and density scaling are both commonly used approaches to accelerate simulations. Always consider, however, if these simplifications will affect your results. Sensitivity analyses may be required. Considering particle crushing with sphere assemblies increases the number of particles required, and it may further increase as particles crush. A comminution limit may also be required. While you may vary the coefficient of interparticle friction to achieve different initial densities, the adequate value that is required for the simulation is significantly important. Experimental validation and calibration of other input parameters may be required, particularly with more complex contact laws, for example, those for cemented soils. Engineering judgment and perhaps (more) sensitivity analyses are warranted. Particle shape, rolling friction? This is a common issue. It is well known that particle shape affects particle rotation. Consequently, some existing research incorporates rolling friction in the DEM contact laws or uses this fact to justify the use of higher values of the coefficient of interparticle friction. This is generally an arbitrary decision that may not adequately replicate soil behavior. Recent research does not recommend the use of rolling friction or high interparticle friction because of the transition between sliding and rolling discussed above. The most commonly used contact models are linear elastic or Hertzianbased. When using linear elastic models, a ratio of tangential to normal contact stiffness between 0.66 and 1.0 needs to be guaranteed to ensure realistic soil behavior. When using Hertzian models, particle characteristics can be easily measured experimentally (e.g., stiffness modulus and Poisson’s ratio). If more advanced contact models are used, calibration may be necessary. In such a case, it is always useful to consider the range of values that are possible for each of the input parameters. This is important because ultimately the same macro-scale response can be obtained with multiple combinations of input parameters. A good approach is to calibrate the response of the material using more than a single stress path (e.g., use triaxial compression and extension tests to calibrate the input parameters). Finally, it has been discussed that 2D simulations are easy to visualize, unlike 3D simulations. It is, however, well recognized that the material of 2D specimens is not realistic due to the lack of interparticle contacts in the out-of-plane direction. While 2D simulations may be useful to demonstrate implementation details for a new contact model or simulation procedure, their use to analyze real granular material response is definitely not recommended by any experienced DEM researcher.

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Simulation control During specimen preparation, it is necessary to guarantee that the particles have reached a state of equilibrium. Approaches to verify these are generally subjective and include the magnitude of the kinetic energy, the amount of variation of a certain micro-scale parameter (i.e., coordination number), and unbalanced force ratios, among others. Whatever the approach used, it is often convenient to perform (yet another) a parametric study to guarantee that the simulation results are not affected. In most geotechnical simulations, the quasi-staticity needs to be guaranteed. This is done by performing the simulation using a strain rate, which is suitably low. While empirical criteria such as the inertial numbers (e.g., Lopera Perez, Kwok, O0 Sullivan, Huang, & Hanley, 2016) may be used to select the adequate strain rate, these are based (generally) on results of DEM simulations of relatively uniformly sized specimens. Therefore, subsequent parametric studies may be required. The time-step of every DEM simulation needs to be significantly small to ensure the numerical stability of the finite difference integration scheme. Most codes use an approach based either on the response of a single degree of freedom system or on the speed of transmission of Rayleigh waves through elastic media. Both of these approaches are valid, but it is always recommended to further reduce the value of time-step, particularly when wide particle size distributions are simulated. For simulations in which a servo-control algorithm is used, the selection of gain parameters may be required, and a trial-and-error process is often required. It is therefore required to always monitor/inspect that the value of stress/required is achieved as expected. This inspection may also require varying the output interval for data to something more frequent before the final choice of gain parameters is made. Note that stress control with servo-control is easier as the simulation progresses, and hence, adequate control needs to be ensured particularly at small strains. When simulating granular materials using DEM, one of the main assumptions is that particles are allowed to overlap to account for (some) particle deformability. This, however, should be strictly limited and monitored during the simulation. A maximum overlap of between 1% and 5% of the particle radius is often cited. Damping should be carefully considered. While using a high value of damping might help to reduce the time required for specimen preparation, small values of damping should be used. The reason for this is that the majority of the energy dissipation in granular materials should occur via friction, heat generation, etc. All of these damping mechanisms are generally incorporated by the contact model when using the correct value of interparticle friction. Furthermore, identifying the response of an overdamped DEM specimen may not be straightforward.

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Discussion and conclusions This chapter has briefly discussed the DEM for the simulation of geotechnical behavior. While the coverage of topics is extensive, the depth is limited. The demonstration of equations and specific simulation results and/or analysis examples are purposely avoided with the aim of increasing an understanding of the method’s capabilities. Extensive references that do include implementation details and analysis are provided. It is expected that any interested readers have the necessary background to refer to those information sources if so required. The reason for doing so is that in the existing literature, there is extensive detail. Unsurprisingly, DEM is one of the most cited numerical methods in recent scientific literature. There are also several textbook type sources. However, this is the first one to briefly describe DEM for somebody with no experience and perhaps not necessarily interested on coding his/her own code or modifying an existing one and maybe just aiming to understand the basics and go ahead with their attempt to perform a DEM simulation and being aware of the possible pitfalls that are only gained with years of experience and which may not be specifically discussed in the existing literature. The initial sections of the article were dedicated to describe the DEM algorithm and the most common analyses that are performed. With this basic background, the steps required to simulate realistic soil behavior were discussed. Finally, in the later sections, some practical advice to consider when performing DEM simulations for somebody with limited experience was listed.

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C H A P T E R

5 Numerical modeling of biocemented soil behavior Neelima Satyam Discipline of Civil Engineering, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India

List of symbols MICP SEM OD R r0 Curea Km Y Z T and T0 C D v R D0, T0, and Υ 0 U U0 pHUL pHLL EC u0 V S

microbially induced calcite precipitation scanning electron microscopy optical density rate of hydrolysis maximum hydrolysis rate concentration of urea at any time half-saturation constant bacterial concentration suspended in growth media OD at 600 nm wavelength temperatures at particular urease activities calcite concentration mass diffusivity fluid velocity source of C fitting parameters urease activity maximum urease activity upper limit of pH lower limit of pH electrical conductivity specific urease activity volume of solution (ml) conductivity (mS/cm)

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00015-0

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Introduction The various challenges associated with the soil are low bearing capacity, high compressibility, liquefaction during earthquake, erosion, landslides, etc. Failure of the soil also results in the complete collapse of the existing superstructure, leading to huge economic and ecological losses. The conventional techniques used to deal with such challenges are compaction, chemical grouting, use of cement, lime or fibers, installation of nails, piles or sheets, etc. Though all these ground improvement methods are associated with limitations and disadvantages. The compaction affects the stability of nearby existing structures and involves energy-consuming heavy equipment. Moreover, it is effective up to a few meters only (Indraratna, Chu, & Rujikiatkamjorn, 2015; Wang et al., 2017). The use of chemicals and cement causes either air pollution or groundwater pollution during manufacturing or after application (Benhelal, Zahedi, Shamsaei, & Bahadori, 2013). Therefore, traditional methods cannot be used for the treatment of large volumes of soil mass, which indicates that there is a need for an economical and eco-friendly ground improvement method. A recently developed technique like microbially induced calcite precipitation (MICP) is a state of art approach, which uses the microbial phase of soil for ground improvement (Ferris, Stehmeier, Kantzas, & Mourits, 1996). MICP is a biogeochemical process that leads to precipitate calcite in soil matrices by urease producing microorganisms. The produced calcite induces cohesion in the form of calcite bonds among sand particles. MICP relies on the hydrolysis of urea for calcite formation, which is a very slow decomposition reaction. However, the rate of hydrolysis of urea can be accelerated up to 1014 times in the presence of a catalyst known as urease, which is commonly found in many plants and bacteria (Whiffin, 2004). The bacterium provides a nucleation site for the precipitation of calcium carbonate as the cell wall of the bacterium is negatively charged, which attracts positive divalent ions of calcium and form cell-Ca bond. Many bacteria have been found to be effective for urease secretion but all of them can’t be used for treatment because either they are pathogenic or not able to survive below the soil in the limited supply of air, space, and light (Mitchell & Santamarina, 2005). Therefore, most of the researchers have preferred Sporosarcina pasteurii (Bacillus pasteurii) as standard bacteria for MICP treatment. Ureolysis involves a series of chemical reactions that generate carbon dioxide (CO2) and ammonia (NH3). Ammonia produced further hydrolyze to give NH4 + and OH
ions, which increase the pH of the solution. The high solubility of CO2 leads to the formation of carbonate and bicarbonate ions to precipitate calcium carbonate in the presence of calcium ion. The overall reactions can be summarized as follow:

Mechanism of MICP

Urease

103

COðNH2 Þ2 + H2 O ƒƒƒ! 2NH3 + CO2

(1)

2NH3 + 2H2 O $ 2NH4+ + 2OH


(2)

2

CO2 + 2OH
! HCO
3 + 2OH $ CO3 + H2 O

(3)

Ca2 + + CO2
3 ! CaCO3

(4)

The biogeochemical mechanism of MICP can be applied in a broad way such as the reinforcement of the historical buildings, bio-clogging, selfhealing of concrete bio-degradation and soil stabilization (Anbu, Kang, Shin, & So, 2016; Bu, Wen, Liu, Ogbonnaya, & Li, 2018; De Muynck, Verbeken, De Belie, & Verstraete, 2010; Ivanov & Chu, 2008). The most common procedure adopted for soil stabilization through MICP involves the cultivation of urease producing bacteria in aerobic and axenic conditions. The bacterial suspension is then bioaugmented to the ground and supplied along with cementation media solution containing urea and calcium chloride. An adequate concentration of calcite is required to gain a noticeable increase in soil strength (Whiffin, van Paassen, & Harkes, 2007). Previous researches have shown sufficient improvement in soil strength after MICP treatment (Choi, Chu, Brown, Wang, & Wen, 2017; Qabany & Soga, 2013). Even few researchers claim higher strength enhancement of bio treated samples over samples treated with traditional methods (Li et al., 2015). The present study includes the usage of master bacterium for urease production, i.e., Sporosarcina pasteurii for the biocementation of poorly graded sand. The laboratory investigation was conducted in plastic tubes filled with sand and treated with two different cementation media concentrations for 10 and 20 days. The field implementation of MICP is not possible until the effects of controlling factors such as cementation media concentration, duration of treatment, temperature, OD of the bacterial solution are predictable. However, a limited study has been carried out on numerical modeling considering the acid-base equilibrium of the chemical reactions involved. Therefore, a numerical code was designed in MATLAB, which considers all the aspects of the biogeochemical mechanisms including acid-base equilibrium. The overall objective of the research was to optimize the condition of MICP treatment to obtain the maximum possible efficiency at minimum cost.

Mechanism of MICP Bio-mediated ground improvement using the MICP technique is one of the potentially developed methods recently, which is sustainable and costeffective. MICP can be achieved through various pathways including

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5. Numerical modeling of biocemented soil behavior

hydrolysis of urea, denitrification, biofilm formation, sulfate and iron reduction. Urea hydrolysis is the well-known and most used calcite precipitation technique (Dhami, Reddy, & Mukherjee, 2013). MICP depends on bacterial growth, urease activity, and temperature, which affects the calcite precipitation kinetics.

Urea hydrolysis Urea hydrolysis depends on the concentration of urea as well as the concentration of calcium ions in the cementation media. Rate of hydrolysis increase with an increase in the concentration of urea can be defined by Michaelis-Menten kinetics equation as follows: r ¼ r0

Curea Curea + Km

(5)

where r0 is the maximum hydrolysis rate, Curea is the concentration of urea at any time and Km is half-saturation constant, i.e., the concentration of urea at which the rate of hydrolysis reduced to 50% of its initial value. The range of Km is 26 to 200 mM (Van Paassen, 2009). However, some researchers have obtained better results with Km as 305 mM (Lauchnor, Topp, Parker, & Gerlach, 2015).

Bacterial growth and urease activity The concentration of bacteria majorly depends on the concentration of inoculum, incubating temperature, and the metabolic state of inoculated strain used. The bacterial concentration suspended in growth media can be calculated by Eq. (6). Y ¼ 8:59  107  Z1:3627

(6)

where Z is known as the optical density of bacteria measured by spectrophotometer at 600 nm wavelength (Ramachandran, Ramakrishnan, & Bang, 2001). The bacterial concentration is one of the main factors, which affects the MICP process (Lauchnor et al., 2015; Zhao et al., 2014). The urease activity is defined as the rate of urea hydrolysis in the aqueous solution, and specific urease activity is defined as urease activity per unit dried biomass. The specific urease activity can range from 6 to 1200 mol-urea L
1 min
1 g DW
1 (Van Paassen, 2009). Different researchers have used different units to measure urease activity. The urease activity and bacterial cell concentration in suspension can be expressed in terms of electrical conductivity (mS cm
1 min
1) and OD respectively resulting in the unit of specific urease activity as mS cm
1 min
1 OD
1 (Whiffin, 2004). The urease activity increases with increasing OD showing a linear relationship between OD and urease activity.

Kinetics of calcite precipitation

105

Temperature Temperature alters the rate of hydrolysis of urea. The upper and lower limit of temperature for urea hydrolysis by urease is 5°C and 70°C, respectively. No urease activity is detected below 5°C and above 70°C. The urease activity increases exponentially with temperature. The increase in urease activity by the factor of 3.4 is observed for per 10° rise in the temperature, in the normal room temperature range of 5°C to 35°C. The variation in the urease activity by temperature in this range can be approximated by the following relation:   ðT
T0 Þ ln 3:4 10 r¼r e (7) 0

where, r0 and r represent urease activity at temperature T0 and T, respectively.

Kinetics of calcite precipitation The calcite precipitation is extensively studied for its applicability in biomineralization process, but its prediction is still difficult due to many influencing factors. The prophecy of the amount of calcite crystals, type of crystals, size of crystals, and surface area is challenging because these properties diversify with distance and time. The formation of calcite consists of many stages, nucleation, growth of crystal, and changes in crystal lattice. Every process has a rate difference, which depends on the condition of precipitation, and these processes subsequently overlap each other (Sohnel & Garside, 1992). CaCO3 is a polymorph which can form in different minerals, under different ambient temperature and pressure. However, calcite is most comply found stable mineral (Stumm & Morgan, 1996). The calcite precipitation varies with time and space which enhances the soil strength and also reduces the porosity of sand. The convectiondiffusion mass transportation equation (E) is used in this study to simulate the calcite concentration with time and distance from the injection point of cementation media. dC ¼ rðDrCÞ
rðvCÞ + R dt Where C ¼ calcite concentration as a function of time and distance, D ¼ mass diffusivity of the grain motion,

(8)

106

5. Numerical modeling of biocemented soil behavior

v ¼ fluid velocity, R ¼ source of C. D can be calculated using the power-law equation as shown below, where D0 (5.4468  10
9 m2s
1), T0 (210.2646 K), and Υ 0 (2.1929) are fitting parameters (Zeebe, 2011).  Υ0 T D ¼ D0
1 (9) T0

Analysis of different parameters The ionic concentration of the solution changes as the decomposition reaction of urea proceeds to generate ammonia and carbon dioxide. The products further react with water to generate ammonium and carbonate ions. The change in ionic concentration and production of ammonia can change the electrical conductivity and pH of the solution respectively. Moreover, the engineering properties of the soil modify due to the formation of calcite precipitation as a result of the biogeochemical reaction. The precipitation occupies the void spaces of soil matrix, which ultimately alter the void ratio, porosity, permeability, and density of the soil. The precipitation also forms bonds among sand particles, which increases the shear strength and unconfined compressive strength of the bio-treated soil. The effects of bio-cementation on various biogeochemical properties of the sand are summarized below:

pH pH is defined as the negative logarithm of hydrogen ion concentration in the solution Eq. (10). The acidic nature of the solution increases as the pH of the solution decreases and vice versa. The decomposition of ammonia generates ammonium ion, which is basic in nature. Therefore, the pH of the solution rises with time, which can affect the equilibrium of the side reversible reactions. The urease activity also depends upon pH. Urease activity is higher at optimum pH, which further decreases as the pH value deviates from optimum pH. Hence the pH of the solution changes during the reaction, the urease activity also fluctuates throughout the biogeochemical reactions. The variation of urease activity due to pH can be approximated by a bell-shaped curve given by Eq. (11) where U and U0 stand for urease activity and maximum urease activity respectively, pHUL and pHLL are the upper and lower value limit of pH for urease activity to reduce by 50% of its optimum value.

Analysis of different parameters

107

pH ¼
log 10 ðH + Þ

(10)

U 1 + 2:10:5ðpHLL
pHUL Þ ¼ U0 1 + 10ðpH
pHUL Þ + 10ðpHLL
pHÞ

(11)

Conductivity Electrical conductivity (EC) is the measure of the ionic concentration of the solution. An increase in conductivity indicates an increase of ions in the solution. The ionic concentration of the solution changes during the MICP process due to urea decomposition and additional reactions. The decomposition of urea to ammonium and carbonate ions increases the ionic concentration of the cementation media. Conversely, the combination reaction between calcium and carbonate ions to form non-ionic precipitation of calcium carbonate, decreases the ionic concentration of cementation solution. The net change in ionic concentration of the solution depends upon the relative rate of these reactions. The rate of change in electrical conductivity can be directly related to the rate of urea hydrolysis. Urease activity and specific urease activity can be obtained by multiplying rate of change of conductivity with 11 and 19 respectively, as shown in Eqs. (12), (13). dCurea 11 dS ¼ V dt dt   Curea d 19 dS OD u0 ¼ ¼ OD dt dt U0 ¼

(12)

(13)

where, U0, u0, V, S represents urease activity (Mol L
1 min
1), specific urease activity (Mol L
1 min
1 g WD
1), volume of solution (mL) and conductivity (mS/cm), respectively in mentioned unit at any time (Whiffin, 2004). Since the analysis and measurement of electrical conductivity is an easier task moreover it has a simple linear relation with urease activity and specific urease activity, therefore, it can be considered as a key factor to ensure the progress of the reaction. Once the concentrations of different reactants and products are known, the rate of ureolysis can be calculated by which the values of conductivity can be predicted using Eq. (12). These electrical conductivity values can be compared with the experimental values for validation of the numerical model.

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5. Numerical modeling of biocemented soil behavior

Experimental study Materials Sand Narmada river (India) sand was used in this study which is classified as poorly graded sand (SP). The grain size distribution curve is shown in Fig. 1. The coefficient of uniformity and coefficient of curvature was 2.60 and 0.72, respectively. The mean grain size (D50) of sand was 0.32 mm.

Bacterial growth and cementation solution The most commonly used urease producing, Gram-positive S. pasteurii bacteria were used in this study, which was stored at
20°C. Himedia nutrient broth was used to prepare bacterial culture solution and the solution was autoclaved at for 20 min at 121°C at 15 psi pressure. The inoculation of cultured bacteria strain was carried out in a laminar airflow chamber followed by aerobic incubation of inoculated solution in an orbital shaking incubator at 200 rpm at 30°C for 24 h. The bacterial cell density or optical density (OD) of the cultured solution was determined using spectrophotometer at 600 nm frequency. The OD was 1.36 which adjusted to 1 by diluting the bacterial solution with distilled water. The bacterial pellets were prepared to centrifuging the solution and removing the supernatant. Fresh nutrient broth was mixed with bacteria pellet before providing it in sand.

FIG. 1

Grain size distribution curve of sand.

109

Experimental study

Cementation media solution was prepared using two equimolar concentrations of urea and calcium chloride dihydrate, i.e., 0.25 and 0.50 M. The cementation media include sodium bicarbonate (2.12 g/L) and ammonium chloride (10 g/L), which acts as a buffer. Nutrient broth (3 g/L) was also added in the cementation media solution, which was autoclaved prior to preventing the chance of bacterial contamination (Mortensen, Haber, Dejong, Caslake, & Nelson, 2011). Experiment details and treatment procedure The biotreatment of sand was carried out in plastic tubes using 40 g of sand and 15 mL of S. pasteurii cultured solution. A similar procedure was adopted to fill the sand and bacteria solution in three layers as used in the previous study (Sharma, Satyam, & Reddy, 2019). The bacteria solution was allowed to hold in tubes for 24 h for bacterial attachment to sand. Further, the bacteria solution was drained out and a cementation media solution was provided. The bacterial augmentation was not carried out in control samples to find out the possibilities of biosimulation in sand. The interval between two consecutive treatment cycles was taken as 12 h. The samples were treated until 10 and 20 days. All samples were placed in three replicas. The samples details are mentioned in Table 1.

Methods The pH and electrical conductivity were measured by Hanna pH and electrical conductivity meters for 12 h of the treatment cycle. The electrodes were dipped in cementation solution filled at the top of sand surface in tube. TABLE 1

Experimental details for biotreatment carried out in this study.

Sample designation

Duration of treatment (days)

Bacteria details

Equimolar concentration of urea and calcium chloride dihydrate in per liter solution

10 SP 0.25

10

S. pasteurii

0.25

10 SP 0.50

10

S. pasteurii

0.50

20 SP 0.25

20

S. pasteurii

0.25

20 SP 0.50

20

S. pasteurii

0.50

10C 0.25

10

Control

0.25

10C 0.50

10

Control

0.50

20C 0.25

20

Control

0.25

20C 0.50

20

Control

0.50

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5. Numerical modeling of biocemented soil behavior

Fann calcimeter pressure gauge model was used to determine the calcite content of biocemented sand after 10 and 20 days of treatment. The highly precipitated layer of 5 mm from top was removed from the treated samples as it was injection point and precipitation was high at the surface. The sub-samples were collected from top and bottom layers to analyze the uniformity of precipitation. The sub-samples were oven-dried over night at 105°C. The sub-samples for SEM analysis were taken from the center of the mid-layer of the biotreated specimen and grounded in powder form. The gold sputter coating was applied to sub-samples to provide conductivity for SEM imaging. The images were taken at 15 kV at a similar magnification of 10,000 X to analyze the presence of microbe beds and calcite crystals.

Validation of numerical model with experimental data Electrical conductivity analysis The electrical conductivity of the solution shows an increasing trend with time (Fig. 2). The conductivity of the solution increases with an increase in the ionic concentration of the solution (Eqs. 1–3). The reason for the increasing trend is the decomposition of a non-ionic reactant + (CO(NH2)2) to ionic products carbonate (CO2
3 ) and ammonium (NH4 ) during hydrolysis. After some time, the conductivity becomes constant, which indicates that all the urea has been hydrolyzed. It can be interpreted from Fig. 2A and B that the increase in conductivity for 0.5 M treated samples was higher than 0.25 M treated samples because of the higher amount of carbonate and ammonium produced in 0.5 M treated samples. The higher concentration of urea yielded higher values of electrical conductivity. A similar positive correlation between electrical conductivity has been reported in previous studies (Wen, Li, Liu, Bu, & Li, 2019a; Whiffin, 2004). The variation in electrical conductivity was almost negligible in the control sample due to the absence of urease producing bacteria or enzyme.

pH analysis The pH of the solution also increases with time. But initially, it increases very rapidly and attains a constant value. The increase in pH is because of the production of ammonium ions, which are basic in nature. Since ammonium ion remains in equilibrium with NH3, a large fraction of total ammonia produced is lost to the atmosphere as NH3 (g). The constant value of pH represents that the dynamic equilibrium has been attained between aqueous and gaseous ammonia. The increase in pH for 0.5 M treated

Validation of numerical model with experimental data

111

FIG. 2 Conductivity vs time plot (A) 0.25 M and (B) 0.5 M cementation media treated sand.

samples was found to be slightly higher than 0.25 M treated samples because of the higher production of ammonia in 0.5 M cementation media, as shown in Fig. 3. Though, the pH variation in control samples was not considerable due to the absence of urease.

Variation of calcite precipitation with time and distance Fig. 4 shows the variation of calcite precipitation with different treatment times and depth of the sample. Fig. 4A and C show that 0.50 M treated samples precipitated 1.6–1.7 times more calcite than 0.25 M treated

112

5. Numerical modeling of biocemented soil behavior

FIG. 3 Variation in pH with time (A) 0.25 M cementation media treated and (B) 0.5 M cementation media treated.

samples in 10 days. While comparing the experimental results of Fig. 4A–D, it can be interpreted that the precipitation of calcite in 10 days almost doubled in 20 days of treatment with 0.25 and 0.50 M cementation media, respectively. Fig. 4B and C show that 0.50 M biotreated samples for 10 days produces approximately 3% lesser amount of precipitation than 0.25 M biotreated samples for 20 days. The control samples didn’t show any calcite bonds and amount of calcite precipitation was also negligible. Hence, it is beneficial to use 0.50 M cementation media in combination with S. pasteurii as it provides a significant amount of calcite precipitation

Variation of calcite with time and distance: (A) and (B) show 10 and 20 days biotreated samples with 0.25 M cementation media; (C) and (D) show 10 and 20 days biotreated samples with 0.5 M cementation media.

FIG. 4

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5. Numerical modeling of biocemented soil behavior

and also reduces the number of treatments, which saves time and efforts. The amount of calcite in the top, middle, and bottom layers was compared to check the uniformity of precipitation. Approximately the same amount of precipitation was measured in all three layers of all biotreated samples. Though, the uniform distribution of calcite precipitation throughout the depth of treatment is one of the biggest challenges of the MICP process in large scale applications (Dejong et al., 2013). Cementation media concentration significantly affects the biocementation process. The rate of urea hydrolysis depends on the amount of urea and calcium chloride used. If the molarity of urea is increased in cementation solutionn a greater number of ammonium ions will be generated, which increase the pH and precipitates the calcite rapidly. The higher concentration of cementation media leads to bioclogging near the injection point and lower concentration of cementation media requires more treatment cycles, which will be time- and energy-consuming (Sharma et al., 2019; Wen, Li, Liu, Bu, & Li, 2019b). Therefore, the cementation media concentration should be optimized to avoid these challenges. Fig. 4 shows the comparison of experimental results with predicted results using the biogeochemical numerical model. The predicted data is nearly similar to experimental results, which validates the model for its applicability to assess the large-scale field applications.

SEM analysis The SEM images of biocemented samples are shown in Fig. 5. The presence of bacteria beds can be encountered in Fig. 5A and D, which denotes cell-Ca bond. The presence of calcite crystals can be seen in all images of Fig. 5, which shows the precipitation of calcite on and between the sand grains. The precipitated calcite increases the strength of cohesionless soils.

Summary In this research, a biogeochemical numerical model was developed to predict the changes in pH, EC, and amount of calcite precipitation in the MICP process which was further validated by experimental data. Smallscale tube experiments were carried out to investigate the factors affecting biogeochemical reactions and mass of calcite precipitation. The urease activity of S. pasteurii during urea hydrolysis was examined through pH and EC changes. The following conclusion can be drawn from this research: 1. The biotreatment of sand with S. pasteurii shows a significant amount of calcite precipitation, which shows the applicability of MICP to Narmada sand.

Summary

115

FIG. 5

SEM analysis of biotreated samples: (A) and (B) show 0.25 M cementation media treated samples up to 10 and 20 days, respectively; (C) and (D) show 0.5 M cementation media treated samples up to 10 and 20 days, respectively.

2. The electrical conductivity of the treatment solution increased with time. For higher cementation media concentration, the rapid increase in EC was measured. 3. The pH value of the treatment solution also increases with both time and higher cementation media concentration. The pH stabilized around 8–8.5 after 5–6 h, which indicated that the dynamic equilibrium has been achieved. 4. The rate of biogeochemical reactions of hydrolysis of urea should not be very high or very low. The higher rate of ureolysis promotes calcite precipitation near the injection point, whereas a lower rate increases the number of treatment cycles to achieve a sufficient amount of precipitation. For cementation media concentrations of 0.25 M and 0.50 M, biotreated samples for 20 days produced almost twice the amount of calcite precipitation as compared to the corresponding samples treated for 10 days. The maximum amount of calcite precipitation (18.1%) is found in 0.50 M cementation media treated samples for 20 days. 5. The proposed biogeochemical numerical model incorporates the bacteria cell concentration (OD) and kinetics of urea hydrolysis into the biocementation process, which is useful in simulation of the

116

5. Numerical modeling of biocemented soil behavior

experimental results. The predicted data was in close agreement with experimental data, which validates the code for large scale applications. Prediction of results will help to reduce the cost and time of treatment. 6. SEM analysis proves the formation of calcite and cell-Ca bonds between the sand grains. The research confirms the influence of pH and electrical conductivity on urea hydrolysis and calcite precipitation. The biogeochemical numerical model was validated by experimental results, which can be used to predict the ureolysis rate and biocementation for large-scale applications.

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C H A P T E R

6 Stability analysis of rock slope against planar failure with irregular discontinuity Subhadeep Metya and Neeraj Chaudhary Department of Civil Engineering, NIT Jamshedpur, Jharkhand, India

List of symbols A b c Fi Fr FS H i T U V W z zw αh αv Ψf Ψp Ψs θ γr γw φu ϭn τ

Base area of sliding rock mass (m2/m) Top width from crest to tension crack (m) Cohesion of rock mass along failure plane (kN/m2) Forces which induces sliding (kN/m) Forces which endure sliding (kN/m) Factor of safety of rock slope Height of rock slope (m) Angle of inclination (degree) Rock anchor stabilizing force (kN/m) Water pressure along the failure plane (kN/m) Pressure exerted by water in the tension crack (kN/m) Weight of sliding rock mass (kN/m) Depth of tension crack (m) Depth of water filled up in tension crack (m) Horizontal seismic acceleration Vertical seismic acceleration Angle of slope face with horizontal (degree) Angle of failure plane with horizontal (degree) Angle of upper slope face with horizontal (degree) Inclination angle of rock anchor stabilizing force with the normal to failure plane (degree) Unit weight of rock (kN/m3) Unit weight of water (kN/m3) Peak friction angle (degree) Normal stresses along sliding plane (kN/m2) Shear stresses along sliding plane (kN/m2)

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00006-X

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6. Stability analysis of rock slope against planar failure

Introduction Rock slope stability analysis has always been a topic of immense interest for researchers and geotechnical practitioners because of its safety and economical damages. Rock slopes fail generally into these five modes namely planar, wedge, circular, toppling, and buckling (Hocking, 1976; Hoek & Bray, 1981; Kliche, 1999; Lee & Wang, 2011; Ramamurthy, 2014; Tang, Yong, & Ez Eldin, 2017). The type of rock failure is governed by the pattern of discontinuity. Large landslides in mountains also found to have complex failure mechanisms. There are so many methods available in the literature for assessing the stability of a rock slope, such as the kinematic method, the empirical method, the limit equilibrium method (Goodman, 1988; Hoek & Bray, 1981; Karaman, Ercikdi, & Kesimal, 2013), the upper and low bound method (Chen, 2004; Zhou & Wang, 2017), the finite element method (Baba, Bahi, Ouadif, & Akhssas, 2012; Eberhardt, 2003; Jiang, Qi, Wei, & Zhou, 2015; Wang, Yin, Chen, & Lee, 2004) and the discrete element method (Son & Adedokun, 2015; Stead, Eberhardt, & Coggan, 2006). Despite all these techniques, the use of the factor of safety is still the most commonly adopted method to analyze the stability analysis. Limit equilibrium method and strength reduction method are generally used to determine the factor of safety. Also, these two methods are quite convenient to analyze 2D problem of failures. Limit equilibrium analysis with the plane failure mechanism is the focus of attention in this study. On the other hand, many researchers have also devoted a great number of contributions in developing a probabilistic method for analyzing the stability of a rock slope (Li, Zhou, Lu, & Jiang, 2009; Ma et al., 2019; Metya & Bhattacharya, 2013, 2016, 2020; Park, Um, Woo, & Kim, 2012; Ya-Fen, Yun-Yao, Hsein, & Der-Her, 2012). Theoretically, these methods have some shortcomings embedded in it. In general, the available methods on probabilistic stability analysis of a rock slope require necessarily a complicated and inconvenient formula for determining the factor of safety, and thus these methods are sometimes not preferred in assessing the stability of a rock wedge (Park et al., 2012). Under the purview of stability analysis of a rock slope, the strength values are, in recent years, been regarded as a variable in the range from the peak strength to the residual strength (Bhattacharya, Chowdhury, & Metya, 2019; Metya, Bhattacharya, & Chowdhury, 2016a, 2016b), as because the sliding failure of a rock mass is generally a progressive failure process (Metya, Bhattacharya, & Chowdhury, 2016c; Metya, Dey, Bhattacharya, & Chowdhury, 2019; Pan, Sun, Wu, & L€ u, 2017; Scholte`s & Donze, 2012) and the softening behavior of a rock mass is also been demonstrated through a series of laboratory tests (Casagrande, Buzzi, Giacomini, Lambert, & Fenton, 2018; Sow et al., 2017; Tiwari & Latha,

121

Introduction

2019). Although accounting for strain-softening behavior in stability analysis of rock slopes is outside the scope of this study, in this direction, studies can be made considering the uncertainties associated with residual factor in the reliability analysis (Bhattacharya et al., 2019). Plane failure of rock mass occurs when the discontinuities are parallel to rock slope and dips toward the slope face. Fig. 1A shows the failure of rock mass as plane failure. Following are the conditions of plane failure: (a) discontinuity plane must strike slope face within 20 degree and (b) angle of failure plane must be more than angle of internal friction (Fig. 1B) (Bell, 2007; Hocking, 1976; Hoek & Bray, 1981; Raghuvanshi, Negassa, & Kala, 2015; Sharma, Raghuvanshi, & Anbalagan, 1995). Hoek and Bray (Hoek & Bray, 1981) gave an analytical expression for the factor of safety of rock mass against the planar mode of failure using a limit equilibrium approach. They assumed the horizontal top surface of rock slope and analysis was based on Mohr-Coulomb shear strength criteria. Sharma et al. (Sharma et al., 1995) modified the approach for inclined top and inclined tension crack. Later on, many researchers proposed the limit equilibrium approach and modified the Hoek’s model for various practical possible situations (Ahmadi & Eslami, 2011; Hoek, 2007; Ling & Cheng, 1997; Price, 2009; Sharma, Raghuvanshi, & Sahai, 1999; Shukla, Khandelwal, Verma, & Sivakugan, 2009; Tang et al., 2017; Zheng, Liu, & Li, 2005). Among these, Ling and Cheng (Ling & Cheng, 1997) also gave the formulation using coulomb criteria for large width using plane strain conditions. Mohr-Coulomb assumed the smooth failure surface similar to soil. However, in hard rock, failure plane is never smooth (Fig. 2). Patton’s demonstrated the influence of roughness on shear strength by conducting a direct shear test on a saw-tooth specimen (Fig. 3A) and gave the theory as. τ ¼ σ n tan ðφb + iÞ

Yf (A)

(1)

Yp j

(B)

FIG. 1 Plane failure of rock slope: (A) sliding of rock mass along planar failure and (B) 2D representation of planar failure.

6. Stability analysis of rock slope against planar failure Peak strength

Shear displacement 6

Shear stress t

Shear stress t

122

Residual strength

Normal stress sn Displacement d

(B)

Shear stress t

(A)

Peak strength

ju

Residual strength

jr

C

(C)

Normal stress sn

FIG. 2 Shear testing of discontinuities (A) rock sample sliding plane in direct shear test assumed by Coulomb (B) Shear strength vs. Shear displacement curve of rock direct shear strength test (C) Mohr’s strength envelope for rock.

Shear strength t

Normal stress sn

i

Failure of intact rock mass

Shear stress

Shearing on saw-tooth surfaces

(A) (yn+i)

(B)

Normal stress sn

FIG. 3 Shear testing of Saw-tooth specimen (A) Shearing on a saw-tooth specimen and (B) Patton’s shear strength graph.

where τ is shear strength along discontinuity, ϭ n is the normal stress, φb is the basic friction angle, and i is the inclination angle of discontinuity. In this chapter, Hoek’s model is modified for the inclined free surface of rock slope and considered other factors such as partially water filled up

Limit equilibrium analysis

123

tension crack, rock bolt stabilizing force, horizontal and vertical seismic forces. Two cases were considered for the analysis, Case (1) Tension crack is on the inclined free surface of rock slope and Case (2) tension crack is on the slope face. A factor of safety model is developed using Patton’s theory for the factors and cases as mentioned above. To validate the developed models and to make a comparison between the developed models, a benchmark example has been reanalyzed. Sensitivity of rock slope has also been studied with the variation of the upper slope face angle, water level in tension crack, and seismic coefficients.

Limit equilibrium analysis Geometry of the rock slope considered for the present analysis is shown in Fig. 4. PRS shows the sliding rock mass of weight W inclined at an angle Ψ f with horizontal with failure plane at an angle Ψ p from horizontal and upper slope face is inclined at an angle Ψ s with horizontal, anchored with single rock bolt (for stabilizing rock bolt) inclined at an angle θ with the normal to the failure plane. Rock mass is subjected to gravitational acceleration induced by earthquake separated by tension crack (tc) at the upper slope surface in Fig. 4A and at slope face in Fig. 4B of depth z partially filled with water up to depth zw. PR, PQ, RQ shows the slope face, failure plane, and upper slope face, respectively. U represents the pressure exerted by the pore water along the failure plane and V represents the pressure exerted by water-filled in tension crack. αh and αv are the horizontal and vertical seismic coefficients respectively. +/ sign shows the vertical acceleration toward/opposite to the direction of gravity. Factor of safety (FS) of rock slope, FS (Hoek & Bray, 1981), FS ¼

Resisting force ðFr Þ Driving force ðFi Þ

(2)

where Fr is the summation of forces which resists sliding and Fi is the summation of forces which induces sliding. Resisting force Fr ¼ τA

(3)

Using Mohr-Coulomb shear strength criteria. Resisting force Fr ¼ ½c + σ n tan ðφu ÞA

(4)

From Fig. 1, the total resisting force, Fr ¼ cA h n

o
 i + W ð1  αv Þ cos ψ p  αh sin ψ p  V sin ψ p  U + T cosθ tan ðφu Þ (5) where A is the base area of sliding rock mass.

124

6. Stability analysis of rock slope against planar failure

b Q

Ys tc’ R T

Z

ahW

ZW

V

q

tc”

U Yp

H

Yf

±avW

P

(A)

Q b

T

ahW

Ys

R

Yp

tc’

q V

H

ZW Z

tc”

U Yf

±avW

P

(B) FIG. 4 Slope Geometry showing inclined free slope surface. (A) Slope geometry with Tension crack in upper slope surface. (B) Slope geometry with Tension crack in slope face.

Total forces that induce sliding, n

o
 Fi ¼ W ð1  αv Þsin ψ p  αh cos ψ p + V cos ψ p  T sin θ

(6)

Combining Eqs. (2), (5), and (6). h n

o
 i cA + W ð1  αv Þ cos ψ p  αh sin ψ p  V sin ψ p  U + T cosθ tan ðφu Þ n

o
 FS ¼ W ð1  αv Þ sin ψ p  αh cos ψ p + V cos ψ p  T sin θ (7)

Using Patton’s shear strength criteria.

Limit equilibrium analysis

125

h n

o
 i W ð1  αv Þcos ψ p  αh sin ψ p  V sin ψ p  U + T cos θ tan ðφu + iÞ n

o
 FS ¼ W ð1  αv Þsin ψ p  αh cos ψ p + V cos ψ p  T sin θ (8)

Eqs. (7), (8) present the analytical expressions for the factor of safety developed by modification of Hoek’s model and using Patton’s shear strength criteria respectively, for considered slope geometry and forces. Modified equations are also being developed for the two cases namely, Case (1) and Case (2). Case (1): Tension crack in upper slope face where,








 o 1 n + bH 1  tan ψ p cot ψ f + bz W ¼ γ r H2 cot ψ f 1  tan ψ p cot ψ f 2 (9)




 z ¼ H 1  cot ψ f tan ψ p + b tan ðψ s Þ  tan ψ p

(10)

Generally, a small movement of rock at the free surface defines the location of tension crack. So that rock slope geometry can be ascertained for stability analysis. However, this small movement may not be visible due to surficial soil and vegetation over slope. In such cases, critical tension crack location can be calculated by using Eq. (11) given by Hoek and Bray (Hoek & Bray, 1981). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


 b ¼ cot ψ f cot ψ p  cot ψ f (11) H The derivation of this equation assumes a dry slope and horizontal upper slope surface, but it is probably adequate for most cases. However, probabilistic analyses need to be carried out when parameters are not well defined. Area of sliding plane.

 
 A ¼ H cot ψ f + b sec ψ p (12) Case (2): Tension crack at slope face. Depth of tension crack.

 


 z ¼ H cot ψ f  b tan ψ f  tan ψ p Weight of the block, 




  1 z 2 2 1 cot ψ p cot ψ p tan ψ f  1 W ¼ γrH 2 H Area of sliding plane,

(13)

(14)

126

6. Stability analysis of rock slope against planar failure



 
 A ¼ H cot ψ f  b sec ψ p

(15)

For both cases (a) and (b). Pressure exerted by pore water along failure plane. 1 U ¼ γ w zw A 2

(16)

Pressure exerted by water filled in tension crack. 1 V ¼ γ w z2w 2

(17)

Illustrative example of planar slope failure Geotechnical investigation on this rock slope reveals that the failure plane is well developed at 35 degree. The rock slope is 35 m high with an orientation of 70 and 11 degrees upper slope face indicates the presence of tension crack 15 m behind the slope crest with water filled up depth 9 m. The shear strength parameters include cohesion as 98 KN/m2, friction angle as 28 degree, and the roughness value was estimated to be 15 to 25 degrees. The rock mass is found to have unit weight 26 KN/m3. The slope designer is required to perform analyses to demonstrate the existing factor of safety. Consider horizontal and vertical seismic coefficients as 0.45 and 0.15 (Fig. 5). It may be noted that the roughness value was estimated to be 15 to 25 degrees. Because of this crude estimation, a parametric study needs to perform taking (φu + i) as 43, 48, 53 degrees.

FIG. 5

Slope geometry considered in the illustrative example.

127

Results and discussions

Results and discussions In the following sections, comparative results obtained by using developed models for the illustrative example is presented (in Table 2), the sensitivity analysis of rock slope with upper slope face angle (Fig. 6), depth of water in tension crack (Fig. 7), seismic acceleration (Fig. 8) is also discussed. For these analyses, the parameters calculated are given in Table 1, and analyses are performed using Eq. (4).

1.73 1.72

FS

1.71 1.7 1.69 1.68 1.67 5

10

15

25

20

30

35

40

Inclination of upper slope face (degree) FIG. 6

Effect of upper slope face angle on the factor of safety.

1.6 1.5

FS

1.4 1.3 1.2 1.1 1 0

0.2

0.4

0.6

0.8

Zw/Z

FIG. 7

Effect of depth of water in tension crack on the factor of safety.

1

1.2

128

FIG. 8

6. Stability analysis of rock slope against planar failure

Effect of the vertical seismic coefficients on the factor of safety.

TABLE 1 Various parameters calculated for the reanalysis of the example problem. Sl. No.

Parameters

Value

Unit

1

Depth of tension crack (z)

18.49

m

2

Area of sliding block (A)

33.86

m2/m

3

Pressure exerted by water in the tension crack (V)

397.3

kN/m

4

Wt. of sliding mass (W)

13,008.25

kN

5

Water pressure along the failure plane (U)

1494

kN/m

Table 2 shows the results obtained from the analysis for the example problem considered in this study and shows the comparison between the modified Hoek-model and developed model based on Patton’s criteria. It is observed that the results are in agreement with the developed models which in turn validates the future use of the analytical expression for the factor of safety using Patton’s criteria as it accounts for the irregular failure surface. TABLE 2 Comparative values of the factor of safety using a modified Hoek model and developed Patton’s model. Models used

Values of the factor of safety obtained

Modified Hoek model

1.825

Patton’s criteria

φu + i ¼ 43 degree

1.627

φu + i ¼ 48 degree

1.932

φu + i ¼ 53 degree

2.315

Summary and conclusion

129

Fig. 6 shows the sensitivity analysis of the stability of rock slope to an inclination of the upper slope face with horizontal. For the considered example, parametric studies have been made to know the effect of variation of Ψ s. It is observed that values of the factor of safety decrease with the inclination of the upper slope face linearly. This is because when the inclination of the upper slope face increases, mass over the sliding plane also increases causing instability in the rock mass. Fig. 7 presents the sensitivity of rock slope to the depth of water filled in tension crack. This study aims to know the quantitative measurement in the changes of factor of safety due to the change of water level in tension crack. It is observed that the factor of safety reduces as the depth of water increases as the presence of water in tension crack causes the pressure to slide the rock mass away from the failure plane. When tension crack fully filled up with water, the factor of safety reduces approximately 28% while comparing it from the dry state. Fig. 8 shows the effect of horizontal and vertical seismic coefficients on the factor of safety. The horizontal seismic forces are considered to act outward of the slope while vertical seismic forces are considered to act toward and opposite to the direction of gravity. In addition to the direction, the magnitude of vertical seismic forces is also varied while horizontal seismic forces kept constant by taking horizontal seismic coefficient as 0.45. For this analysis, the vertical seismic coefficient is taken as half and less than half of the horizontal seismic coefficients following IS 1893-2016 (IS 1893 Part 1, 2016) where it is mentioned that the value of vertical seismic coefficients should be taken as half or two-thirds of the horizontal seismic coefficients. It is observed from Fig. 8 that the factor of safety decreases (in a nearlinear fashion) with an increase in the value of vertical seismic coefficient toward the direction of gravity and increases nonlinearly with increases in the value of vertical seismic coefficient against the direction of gravity. Effect of vertical seismic coefficient against the direction of gravity on the factor of safety of rock slope has been found to be predominant as the rate of increase/decrease of the factor of safety with vertical seismic coefficient is more against the direction of gravity.

Summary and conclusion An analytical formulation for the factor of safety of rock slope using limit equilibrium approach has developed using Patton’s theory for the inclined upper slope face considering partially water filled up tension crack, horizontal and vertical seismic forces, and rock bolt stabilizing forces. Hoek’s model of factor of safety for rock slope against plane failure

130

6. Stability analysis of rock slope against planar failure

is also modified for considered slope geometry and forces. A benchmark example problem is reanalyzed for a comparative study between these two developed models. Results show a very good agreement between these two models. The model based on Patton’s theory is found to be better for the analysis of the stability of rock slope as it accounts for the irregular discontinuity pattern along the failure plane. Sensitivity of rock slope with the inclination of upper slope face, depth of water in tension crack and seismic coefficients have been studied and it is observed that factor of safety reduces as the inclination of upper slope face and depth of water in tension crack increase. When tension crack fully filled up with water, the factor of safety reduces approximately 28% while comparing it from the dry state. The factor of safety decreases with an increase in the value of vertical seismic coefficient toward the direction of gravity and increases nonlinearly with increases in the value of vertical seismic coefficient against the direction of gravity. Effect of vertical seismic coefficient against the direction of gravity on the values of factor of safety is very severe, thus the effect of vertical seismic forces should not be ignored while analyzing stability analysis of rock slopes.

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C H A P T E R

7 Modeling of unsaturated soils slopes considering the residual shear strength behavior Xiuhan Yang and Sai Vanapalli Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada

List of symbols a, n, and m c0 cp0 cr0 FOS FOSP FOSR S (ua 2 uw) θ θi θr θs κr (σ n 2 ua) τ τr Ψb Ψi Ψr

SWCC parameters effective cohesion of saturated soils peak effective cohesion of saturated soils residual effective cohesion of saturated soils factor of safety factor of safety calculated based on peak shear strength factor of safety calculated based on residual shear strength degree of saturation matric suction volumetric water content volumetric water content corresponding to the inflection point of SWCC residual volumetric water content saturated volumetric water content fitting parameter for residual shear strength net normal stress shear strength of soils residual shear strength of soils suction corresponding to the air entry value suction corresponding to the inflection point of SWCC suction corresponding to the residual volumetric water content

ϕb ϕ0 ϕp0

friction angle with respect to matric suction effective friction angle of saturated soils peak effective friction angle of saturated soils

ϕr0

residual effective friction angle of saturated soils

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00001-0

133

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7. Modeling of unsaturated soils slopes

Introduction Fine-grained soil slopes that are in a state of the unsaturated condition typically undergo a large deformation prior to reaching failure conditions (Ng, Zhan, Bao, Fredlund, & Gong, 2003; Widger & Fredlund, 1979). In such scenarios, the soil shear strength in the sliding zone will drop from the peak shear strength (PSS) to the residual shear strength (RSS) due to the large shear strain. The phenomenon is widely referred to in the literature as strain-softening behavior (Skempton, 1964). Many researchers have highlighted the role of RSS in the long-term slope stability analysis of saturated fine-grained soil slopes (Locat, Jostad, & Leroueil, 2013; Potts, Kovacevic, & Vaughan, 1997). These studies suggest that the distribution of shear strength is not uniform within a deforming slope. In the sliding zones that underwent large deformation, the shear strength of the soils reduces due to strain-softening behavior. The factor of safety (FOS) would be overrated if only PSS was used for slope stability analysis because the RSS that governs the slope failure is typically lower than the PSS. More recently, the importance of RSS was highlighted in the slope stability analysis of unsaturated expansive soils (Qi & Vanapalli, 2016). Therefore, the strain-softening behavior based on RSS must be considered for the reliable modeling, analysis and design of the unsaturated slopes that can undergo large shear deformation prior to reaching the failure condition. The RSS behavior of saturated soils has been widely investigated in the literature (Lupini, Skinner, & Vaughan, 1981; Skempton, 1964). During the last decade, the RSS behavior of unsaturated soils was investigated by a few researchers from experimental studies (Hoyos, Velosa, & Puppala, 2014; Infante Sedano & Vanapalli, 2011; Romero, Vaunat, & Mercha´n, 2014). The experimental results have highlighted the significant influence of matric suction on the RSS and strain-softening behaviors of unsaturated soils. However, experimental data presently available in the literature on the RSS of unsaturated soils are rather limited. More importantly, studies related to simple yet reliable approaches for predicting the RSS of unsaturated soils are lacking. A model for predicting the RSS of unsaturated soils (Infante Sedano & Vanapalli, 2011) is summarized below: τr ¼ c0r + ðσ n  ua Þ tan ϕ0r + ðua  uw Þ ðSκr Þ tan ϕ0r

(1)

where (σ n  ua) is the net normal stress; (ua  uw) is the matric suction; cr0 and ϕr0 are the RSS parameters of saturated soils; S is the degree of saturation; κ r is the fitting parameter. A reasonable performance has been validated using the above equation based on the experimental results of suction-controlled ring shear tests conducted on unsaturated soils. However, S in this model should be estimated from an apparent soil-water characteristic curve (SWCC) obtained

Introduction

135

using a conventional pressure plate test from a specimen which has been sheared to the residual state. It will be time-consuming to measure such an SWCC. To simplify the required input parameters, a new model was developed to predict the RSS of unsaturated soils (Yang & Vanapalli, 2018), which is based on the conventional SWCC. In this model, the residual shear strength of unsaturated soils can be expressed as:   θ  θi κr τr ¼ c0r + ðσ n  ua Þ tan ϕ0r + ðua  uw Þ tan ϕ0r (2) θs  θ i where θs is the saturated volumetric water content; θi is the volumetric water content corresponding to the inflection point (i.e., the peak point of the derived function curve of SWCC equation, as shown in Fig. 1); κ r is the fitting parameter for RSS, which is suggested to be 0.4 for a glacial till (Indian Head till) from Saskatchewan, Canada. This RSS prediction model requires only the information of a conventional SWCC, which can be obtained relatively easily from the laboratory. However, this model can only be applied within a relatively low suction range (i.e., 0 < Ψ < Ψ i). Equation 2 shows when θ is less than θi, the RSS due to matric suction is zero. In other words, the suction cannot contribute to the RSS of unsaturated soils when it exceeds Ψ i. This is inconsistent with the experimental observations reported by Romero et al. (2014) that suction can still contribute to the RSS of unsaturated soils even when suction values are as high as 300 MPa. Therefore, Eq. (2) can only be used when 0 < Ψ < Ψ i. However, this is typically the range in which suction values are associated with the changes in the liquid phase flow and are important in conventional geotechnical engineering practice.

FIG. 1

The salient features of the soil-water characteristic curve (SWCC).

136

7. Modeling of unsaturated soils slopes

Geotechnical engineers understand the importance of RSS in the longterm stability analysis of both saturated and unsaturated slopes. However, there are limited case studies in the literature where the RSS concept was considered to interpret the behavior of unsaturated slopes in comparison with saturated soil slopes. This chapter introduces an old landslide that was reactivated recently due to the combined influence of rainfall infiltration and the Yangtze River water level variation at the slope toe. A series of direct shear test results were reported to study the RSS behavior of the saturated and unsaturated specimens. Based on the RSS parameters, a series of slope stability analyses were conducted using commercial software, which is discussed in later sections. The results of the study are used to rationally interpret the landslide behavior at residual state conditions.

Site investigation studies Study area The investigated Outang landslide is in Anping, Fengjie, China, which is about 177 km away from the Three Gorges Dam. The annual mean temperature and mean precipitation are 16.3°C and 1147.9 mm, respectively. 70% of total annual precipitation occurs during the summer season (i.e., from May to September). The studied landslide is on the south bank of the Yangtze River, inclined from the South to the North. The Yangtze River flows in front of the slope toe, as shown in Fig. 2. The water level of the Yangtze River varies between 145 and 175 m periodically every year.

Description of landslide The studied slope has a length of about 1800 m and a height of 600 m from the crest to the toe with an average gradient of 1:25. The average thickness of the sliding mass is about 50.8 m. The electron spin resonance (ESR) tests were used to determine the age and sequence of landslides. The results indicated that the old landslide has occurred tens of thousands of years ago in three stages as shown in Fig. 2. Three Gorges Dam that was constructed to the east of the Outang landslide and began to store water since 2003, has significantly influenced the behavior of Outang landslide. The field observations indicated that the old landslide has started deforming again (Dai, 2016). First, large displacements of sliding mass and obvious sliding zones could be found at some monitoring points. A great number of cracks could be observed on the ground of the first-stage landslide. Similarly, settlements of roads and

Site investigation studies

137

FIG. 2 Typical cross section of the Outang landslide. Modified after Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

cracks on the ground could be observed in the third-stage landslide every year since 2003, especially during the rainy season. However, the secondstage landslide was still relatively stable. All these pieces of evidence support local large deformations which was increasing and extending progressively in the slope with time.

Rainfall and river water level data Fig. 3 summarizes the rainfall data along with the variation of the Yangtze River water level from December 1, 2010 to December 1, 2014. Two typical variation curves of the ground surface displacement of the third-stage landslide from May 2012 to December 2014 are also presented in Fig. 3. This information is valuable for analyzing the relationship between the deformation behavior of the landslide and the variation of hydraulic/ climatic conditions, which is useful in the Outang landslide investigation. From Fig. 3, it can be found the precipitation varied periodically during the measurement period (Dec 1st, 2010–Dec 1st, 2014). Accordingly, the water level was also changed periodically based on the precipitation by the Three Gorges Reservoir. Every year the water level was decreased to the lowest value (145 m) during the rainy season to alleviate possible floods; then, it was increased to the highest value (175 m) during the dry season. The ground surface displacement kept increasing for the measurement period, i.e., from May 2012 to December 2014. However, there is a significant increase in the ground surface displacement from May to September in 2012 and 2014, which is much greater than the period in 2013. In addition, the precipitation rate is significantly higher from May to September

138

7. Modeling of unsaturated soils slopes

FIG. 3

Variation of the Yangtze River water level, precipitation and accumulated displacement of the ground surface. Modified after Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

in 2012 and 2014 than that in the same period during 2013 under the same condition of water level. These comparison studies suggest that the displacement variation was consistent with the variation in the precipitation rate. In addition, during the dry season (e.g., from January to May in 2013 and 2014), even though the precipitation is relatively low, there is still a slight increase in the ground surface displacement. This may be attributed to the decrease in the water level during this period. Therefore, it is reasonable to conclude that the displacement of the studied landslide can be attributed to the combined effect of the precipitation and water level variation.

Material properties The site investigation studies have shown that the sliding mass of the Outang landslide has two layers (Fig. 2): (i) a 3–20 m silty clay layer with gravels; (ii) a 10–85 m broken rock layer. Three weak zones (R1, R2, and R3) are sandwiched between the sliding mass and bedrock. The site investigation results show that the sliding mass moved along the R1 (10–35 cm claystone layer) and R3 (40–70 cm clay layer) weak zone. Several undisturbed soil specimens were collected from the silty clay layer, R1 and R3 weak zone. A series of laboratory tests were conducted to determine the physical properties of the soils, which were reported in

139

Site investigation studies

TABLE 1

Physical properties of soils. Natural specimen (unsaturated)

Saturated specimen

Plasticity index

Liquid index

Specific gravity

Water content (%)

Density (kg/m3)

Water content (%)

Density (kg/m3)

Silty clay

12.3

0.02

–

0.05

2040

21.63

2070

R3

11.1

0.17

2.70

19.48

2070

20.51

2080

R1

10.7

0.15

2.71

18.00

2007

19.80

2100

Summarized from Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

(Dai, 2016) and summarized in Table 1. A series of consolidated undrained direct shear tests were conducted on both the natural and saturated soil specimens of R1 and R3. In addition, a series of quick undrained direct shear tests were conducted on the natural and saturated soil specimens of silty clay. The shear strength envelopes of these soils are presented in Fig. 4. It can be found the shear strength of natural soil specimens is greater than that of saturated soil specimens at both peak and residual state. This is attributed to the reduction in matric suction associated with an increase

FIG. 4 Shear strength envelopes of soils. Modified after Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

140

7. Modeling of unsaturated soils slopes

in the degree of saturation. The degree of saturation of the natural specimens can be calculated from their water content and density relationships (shown in Table 1), which was equal to 89.53% and 94.18% for R1 and R3. It can also be found that the residual shear strength is lower than the peak shear strength for both unsaturated and saturated soil specimens. In addition, the reduction in shear strength from peak to the residual state for unsaturated soil is greater than that of the saturated soil. That is because, for unsaturated soil, the contribution of matric suction to shear strength is also reduced (i.e., ϕb reduces) in addition to the reduction in saturated soil shear strength parameters (i.e., c’ and ϕ’). Patil, Puppala, Hoyos, and Pedarla (2017) indicated that the water menisci along which suction acts are destroyed beyond peak state when the soil undergoes a large deformation, which contributes to the reduction in ϕb of unsaturated soils.

Slope stability analyses Schematic of slope used for numerical modeling A series of numerical modeling studies were conducted based on the PSS and RSS parameters using the commercial software Geostudio (GeoSlope International Ltd, 2012). Fig. 5 presents the numerical model of the landslide. In this numerical model, several assumptions were made to simplify the analyses. First, the surface of the slope was simplified by neglecting some minor changes in the gradient of the real slope surface. Second, only R1 and R3 weak zone were considered in the numerical model. The R2 weak zone was neglected in the numerical model because the slip surface did not go through it. The thickness of the R1 and R3 were both assumed equal to 0.5 m. Lastly, the three landslides were considered to have occurred at the same time. In other words, one slip surface was used in the numerical model to replace the slip surfaces in three stages.

FIG. 5

Schematic of the slope used in the numerical modeling.

Slope stability analyses

141

Calculation procedures In this numerical model, the combined influence of rainfall infiltration and variation of the water level was simulated. The variation of the FOS of the slope with time under the combined influence of these two factors was studied. For this purpose, three steps were involved in the numerical analysis. They can be described as: (1) Steady-state seepage analysis: The boundary conditions on the left boundary and the slope surface below the water level (elevation ¼ 175 m) were set as the constant total head boundary condition. Then, a steady-state seepage analysis can be conducted using Seep/W to simulate the pore water pressures in the slope under no precipitation and variation of the water level. (2) Transient seepage analysis: Based on the numerical model in Step (1), a specific boundary condition was applied on the slope surface to model the precipitation and variation of the water level. Then, a transient seepage analysis was conducted to calculate the pore water pressures in the slope at different times. (3) Slope stability analysis: Based on the numerical model in Step (2), the values of FOS of the slope at different times can be calculated using the Slope/W and RSS (or PSS) parameters.

Boundary conditions The unit flux on the bottom and right boundary was set as 0 m/s; i.e., no flow occurred on those two boundaries. The total head on the left boundary was constant at 525 m as suggested by Dai (2016). In the steady-state seepage analysis, the total head on the slope surface below the elevation of 175 m (i.e., the highest water level) was set as 175 m. In the transient seepage analysis, the actual hydraulic/climatic condition was simulated by using the boundary conditions on the slope surface. From Fig. 3, it can be found that both the precipitation and water level varied periodically with time. In the present study, only a representative period of 1 year (Feb 2012–Feb 2013) was selected for investigation. During the wet season of this year, the shear strength decreased due to the rainfall infiltration and the seepage force increased due to the decrease in the water level. The composite influence of water level variation and precipitation can accelerate the slope failure. This means the wet season should be considered as vulnerable period. Therefore, only the hydraulic/climatic condition for 7 months (i.e., Feb 2012–Sep 2012) was studied in this research. Fig. 6 presented the actual and assumed values of precipitation and water level variation. The unit flux on the slope surface above the elevation of 175 m was defined as a function of the elapsed time (continuous

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7. Modeling of unsaturated soils slopes

FIG. 6 Variation of precipitation and Yangtze River water level used in the numerical model (Feb 2012–Sep 2012).

solid line with solid circles in Fig. 6). In addition, the total head on the slope surface below the elevation of 175 m was also defined as a function of the elapsed time (continuous solid line with solid squares in Fig. 6).

Material properties In this numerical model, five types of materials are used, including silty clay, R1, R3, broken rock, and bedrock (Fig. 5). Mohr-Coulomb model was used for all those materials. The hydraulic and shear strength parameters of those materials are described below. The PSS and RSS parameters shown in Fig. 4 were used for the silty clay, R1, and R3. The effective shear strength parameters were not available (Dai, 2016) for these materials. The parameters of R1 and R3 were derived from consolidated undrained direct shear test results and the parameters of silty clay were derived from quick undrained direct shear test results. By using this approach, the magnitudes of FOS will be underestimated. However, the research focus in this chapter is directed on the slope stability considering the RSS behaviors. Although effective shear strength parameters were not used, the trends in variation of FOS of the simulated slope with time will be similar to that calculated using effective shear strength parameters. In other words, the slope behavior can still be analyzed considering the RSS. In addition, experimental results for the shear strength parameters of the broken rock and bedrock and the hydraulic properties of all those five materials were not provided by Dai, 2016. However, a series of shear strength parameters and hydraulic properties have been suggested for

143

Slope stability analyses

TABLE 2

Shear strength parameters in the numerical model. cp0 (kPa)

ϕp0 (o)

cr0 (kPa)

ϕr0 (o)

Broken rock

70

16.2

70

16.5

Bedrock

700

42

700

42

Summarized from Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

TABLE 3

Parameters of SWCC. a

n

m

θs

θr

Ψ i (kPa)

Silty clay

10.65

1.40

0.81

0.32

0.059

20

Broken rock

9.12

1.73

0.76

0.27

0.051

16

R1 and R3

9.07

1.53

0.59

0.30

0.094

20

Bedrock

9.81

1.46

0.70

0.05

0.012

20

Summarized from Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

all the five materials in the numerical models reported by Dai (2016), which are summarized in Table 2, Table 3, and Fig. 7. The SWCC parameters in Table 3 were obtained by fitting the SWCC suggested in Dai (2016) using the Fredlund and Xing equation (Fredlund & Xing, 1994). These properties were also used in this study.

FIG. 7 Coefficient of permeability functions for different materials. Modified after Dai, Z.W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis, Chang’an University, Xi’an, Shanxi, China (in Chinese).

144

7. Modeling of unsaturated soils slopes

The RSS parameters of the broken rock and bedrock were assumed equal to the PSS parameters (shown in Table 2). That is because the slip surface only passed through the R1 and R3 weak zone; for this reason, the shear strength parameters of the broken rock and bedrock did not influence the FOS of the slope. Therefore, the assumption will not influence the FOS. In addition, the same SWCC and permeability function were used for the R1 and R3 weak zone, since they are both clays. The prediction model for PSS of unsaturated soils was proposed by Vanapalli, Fredlund, Pufahl, and Clifton (1996) as Eq. (3). The prediction models for RSS of unsaturated soils (e.g., Eqs. 1 and 2) were described in the earlier sections of the chapter. However, the apparent SWCC of the sheared specimens were not available in Dai (2016). Therefore, Eq. (1) cannot be used. In addition, the seepage analysis in this research showed the suction in the studied slope can reach as high as 800 kPa, which is much greater than Ψ i of these soils. Therefore, Eq. (2) also cannot be used. For this reason, a model was postulated in this research to calculate the RSS in analogy with Eq. (3), which is expressed as Eq. (4).   θ  θr 0 0 τ ¼ c + ðσ n  ua Þ tan ϕ + ðua  uw Þ (3) tan ϕ0 θs  θr   θ  θr 0 0 (4) τr ¼ cr + ðσ n  ua Þ tan ϕr + ðua  uw Þ tan ϕ0r θs  θr where θr is the residual volumetric water content. Eq. (4) was used to approximate the RSS of unsaturated soils in this research, since there is no RSS prediction model. The parameter, (θ  θr)/(θs  θr), is typically used to describe the reduction in suction contribution at peak state with increasing suction. However, after peak state, some water menisci are destroyed, which contributes to a greater reduction in suction contribution at residual state than at peak state. In Eq. (4), (θ  θr)/(θs  θr) was still used to describe the reduction in suction contribution. It tends to underestimate the reduction in suction contribution at residual state, i.e., Eq. (4) tends to overrate the RSS. Finally, based on the PSS and RSS, the FOS of the slope can be calculated by using Morgenstern-Price method. The slip surface as shown in Fig. 5 was specified manually and used in the slope stability analyses.

Analyses of results Fig. 8 presents the values of FOS at peak state (FOSP) and FOS at residual state (FOSR) under the combined influence of precipitation and water level variation. It can be found FOSP was always greater than 1.0

Analyses of results

145

FIG. 8 Variation of FOS at peak and residual state with time under the combined influence of precipitation and water level variation.

throughout the 7 months in spite of decreasing with time. In other words, the slope was stable within the study period based on the PSS. These results however are not consistent with the field observations because during this period significantly local large displacements occurred (Fig. 3). This means the FOS was overrated due to the use of PSS. Therefore, the RSS should be considered for a reliable analysis of this landslide. From Fig. 8, it can be found that FOSR decreased almost linearly by about 6.6% within the first 3 months. However, in the following 4 months, the decrease in FOSR was much slower. From the 3rd to 7th month, FOSR decreased by 0.66%. To study the mechanisms associated with the landslide reactivation, soil suction profiles along the slip surface under the combined influence of the precipitation and water level variation are presented in Fig. 9. A sketch of the slope (shown in Fig. 5) is also presented in Fig. 9 to show the positions where the studied suctions were obtained. Two reasons can be found to interpret the reactivation of the landside. The first reason is associated with the suction decrease caused by rainfall infiltration. The rainfall infiltration can cause an increase in pore water pressure. Conversely, the decrease in water level can cause a decrease in pore water pressure. From Fig. 9, it can be found the pore water pressures increased (i.e., the suction decreased) on the slip surface at the top of R1 and the connection between R1 and R3. This means the rainfall infiltration mainly influenced these two zones. Such behavior may be attributed to the slip surface being shallow at the top of R1 (Fig. 9); thus, the infiltrated rainfall can reach the slip surface within a short time period. At the connection between R1 and R3, the bedrock is almost horizontal (Fig. 9); thus, the infiltrated rainfall tended to accumulate above the bedrock. Once the suction

146

7. Modeling of unsaturated soils slopes

FIG. 9 Pore water pressure profiles along the slip surface under the combined influence of precipitation and water level variation.

decreased on the slip surface, the RSS of the slide zone soils reduced as a result, which results in a decrease in FOSR. The second reason is the seepage force increase caused by a decrease in the water level. From Fig. 9, it can be found that the pore water pressures decreased on the slip surface near the river, but they are still positive values. Therefore, the decrease in the water level did not influence the RSS of the slide zone soils. However, with decreasing water level, the difference between the water levels at the rear and front part of the slope increased. Consequently, the seepage force increased as a driving force in the sliding mass, which contributes to the decrease in FOSR. To further study the influence of precipitation and water level decrease, two more numerical modeling studies were conducted based on RSS, which considered the influence of precipitation and water level decrease, separately. The variations of FOSR with time in those two scenarios were presented in Fig. 10. In the first 3 months, the decrease in FOSR caused by the water level decrease was much greater than that caused by the rainfall infiltration. This is because the slip surface which was influenced by the rainfall infiltration was far above the phreatic line. This means the matric suction on this part of the slip surface was high and the volumetric water content was low. The contribution of the matric suction to RSS is much lower than that of the overburden pressure. Thus, the reduction in matric suction contribution can only cause a relatively small decrease in the RSS.

Analyses of results

147

FIG. 10 Variation of FOS at residual state with time under different hydraulic/climatic conditions.

In addition, the extent of the zones which were influenced by the rainfall infiltration was still limited. On most part of the slip surface, the RSS was not influenced by the rainfall infiltration. Therefore, the decrease in FOSR caused by the rainfall infiltration was relatively small. In other words, it was the decrease in the water level that had a dominant contribution to the reduction in FOSR within the first 3 months. From the 3rd to 7th month, the continuous precipitation caused a further decrease in FOSR. However, FOSR increased slightly for the scenario where only the water level decrease was considered. This means, once the water level attains a relatively constant value, the rainfall infiltration should be the dominant factor that contributes to the reduction in FOSR. In addition, it can also be found that the FOSR was less than unity initially when the precipitation and water level decrease did not start. This result is inconsistent with the field observations which indicated that the slope was stable before the precipitation and water level variation occurred. This can be attributed to an assumption in the numerical model that the RSS parameters were used throughout the slope, which did not fit the actual case. In reality, the magnitude of shear strength should depend on the level of deformation. Initially, when no large deformation occurs, the shear strength of soils should be interpreted based on PSS parameters. After the precipitation increases and water level decreases, large deformations occur in some zones of the slope. As a result, in those zones with large deformations, the shear strength will be reduced to the RSS; however, the shear strength in other zones will be somewhere between the PSS and RSS depending on the magnitudes of deformation. Therefore, different values of shear strengths should be used in different zones of the

148

7. Modeling of unsaturated soils slopes

slope according to the deformation level for reliably determining the FOS. If the RSS is used throughout the slope, the FOS will be underestimated. However, this approach provides a conservative value of FOS for the slope stability analysis.

Summary In this chapter, a reactivated landslide was revisited and analyzed. A series of site investigations and direct shear test results were reported to study the progressive failure behaviors of the landslide. The landslide was reactivated by the combined influence of the precipitation and water level variation. A number of numerical modeling were conducted using Geoslope to study the slope stability of the landslide based on the peak and residual shear strength. The key results are summarized below: (1) The FOS will be overrated if PSS is used for slope stability analysis. Therefore, the RSS should be considered for reliable modeling of the slope behavior. (2) FOSR decreased during the seven-month study period. This can be attributed to the reduction in the RSS of the slide zone soils caused by the rainfall infiltration and the increase in the seepage force caused by the water level decrease. (3) During the period of the water level variation, the water level decrease should be the dominant factor that contributed to a significant reduction in FOSR. Once the water level reached a relatively constant value, precipitation became the dominant factor contributing to the FOSR reduction; however, the reduction rate was relatively smaller. (4) In reality, different values of shear strengths should be used in different zones of the slope according to the deformation level. In other words, the RSS should be only used in the zones with large deformations; however, the values of shear strength between the PSS and RSS should be used in other zones depending on the magnitudes of deformation. If the RSS was used throughout the slope, the FOS would be underestimated. The slope stability analyses in this study are based on RSS; for this reason, they provide conservative results.

References Dai, Z. W. (2016). Study on the deformation and failure mechanism of Outang landslide in the Three Gorges Reservoir Region, China. Ph. D. Thesis Xi’an, Shanxi, China: Chang’an University. (in Chinese). Fredlund, D. G., & Xing, A. (1994). Equations for the soil-water characteristic curve. Canadian Geotechnical Journal, 31, 521–532.

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GeoSlope International Ltd (2012). Stability modeling with SLOPE/W: An engineering methodology. Calgary, Alberta, Canada: GEOSLOPE International Ltd. Hoyos, L. R., Velosa, C. L., & Puppala, A. J. (2014). Residual shear strength of unsaturated soils via suction-controlled ring shear testing. Engineering Geology, 172, 1–11. Infante Sedano, J. A., & Vanapalli, S. K. (2011). Experimental investigation of the relationship between the critical state shear strength of unsaturated soils and the soil-water characteristic curve. International Journal of Geotechnical Engineering, 5, 1): 1–8. Locat, A., Jostad, H. P., & Leroueil, S. (2013). Numerical modeling of progressive failure and its implications for spreads in sensitive clays. Canadian Geotechnical Journal, 50(9), 961–978. Lupini, J. F., Skinner, A. E., & Vaughan, P. R. (1981). The drained residual strength of cohesive soils. Geotechnique, 31(2), 181–213. Ng, C. W. W., Zhan, L. T., Bao, C. G., Fredlund, D. G., & Gong, B. W. (2003). Performance of an unsaturated expansive soil slope subjected to artificial rainfall infiltration. Geotechnique, 53 (2), 143–157. Patil, U. D., Puppala, A. J., Hoyos, L. R., & Pedarla, A. (2017). Modeling critical-state shear strength behavior of compacted silty sand via suction-controlled triaxial testing. Engineering Geology, 231, 21–33. Potts, D. M., Kovacevic, N., & Vaughan, P. R. (1997). Delayed collapse of cut slopes in stiff clay. Geotechnique, 47(5), 953–982. Qi, S., & Vanapalli, S. K. (2016). Influence of swelling behavior on the stability of an infinite unsaturated expansive soil slope. Computers and Geotechnics, 70, 154–169. Romero, E., Vaunat, J., & Mercha´n, V. (2014). Suction effects on the residual shear strength of clays. Journal of Geo-Engineering Sciences, 2(1–2), 17–37. Skempton, A. W. (1964). Long-term stability of clay slopes. Geotechnique, 14(2), 77–102. Vanapalli, S. K., Fredlund, D. G., Pufahl, D. E., & Clifton, A. W. (1996). Model for the prediction of shear strength with respect to soil suction. Canadian Geotechnical Journal, 33 (3), 379–392. Widger, R. A., & Fredlund, D. G. (1979). Stability of swelling clay embankments. Canadian Geotechnical Journal, 16(1), 140–151. Yang, X., & Vanapalli, S. K. (2018). Slope stability analysis of a slope based on the peak and the residual shear strength of unsaturated soils. In Proceedings of 7th international conference on unsaturated soils (UNSAT 2018), Hong Kong, China.

C H A P T E R

8 Multi-objective optimum design of geosynthetic reinforced soil foundation using genetic algorithm Manas Ranjan Dasa, Madhumita Mohantyb, and Sarat Kumar Dasb a

Civil Engineering Department, ITER, SOA University, Bhubaneswar, Odisha, India, bCivil Engineering Department, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, India

List of symbols $ B BCRp BCRr c d/B di fb and fc i M Mc, Mq, and Mᵧ Mp n Nc, Nq, and Nᵧ ɸ pr q Ta Tu γ μ

US Dollar width of the footing bearing capacity ratio with respect to pull out bearing capacity ratio with respect to rupture cohesion of the fill relative depth of reinforcement depth of ith layer bond coefficients and taken as 0.6 number of a particular layer to extract maximum benefit Mc, Mq, Mᵧ are taken equal to M standard bearing capacity coefficients due to reinforcement a coefficient number of layers standard bearing capacity coefficients internal friction angle of the fill bearing capacity of reinforced soil with one layer of reinforcement ultimate bearing capacity allowable tensile strength of the geosynthetic reinforcement ultimate tensile strength of geosynthetic reinforcement unit weight of the fill soil reinforcement friction coefficient

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8. Multi-objective optimum design

Introduction Reinforced soil or mechanically stabilized soil consists of soil that is strengthened by tensile elements such as metal strips, geotextiles, or geogrids. The development of polymeric materials in the form of geosynthetics has brought major changes in geotechnical engineering. The beneficiary effects of soil reinforcement are derived from (a) the increased tensile strength of soil and (b) the shear resistance developed from the friction at the soil reinforcement interfaces. So, there is an increase in applications of geosynthetics in geotechnical structures such as foundation, embankments, retaining wall, etc. The design of shallow foundation considers two criteria: the bearing capacity and settlement. Bearing capacity generally depends on the strength of the soil while settlement generally depends on the compressibility of soil (Das, 2007). In the case of weak soil, the improvement in bearing capacity and decrease in settlement can be achieved by geosynthetic reinforcement (Shukla & Yin, 2006). The analysis of reinforced foundation is considered in terms of pullout and rupture (break out) failure of geosynthetic reinforcement. Various small scale laboratory experiments were performed on various soils like clay (Sakti & Das, 1987) and sand (e.g., Guido, Knuppel, & Sweeny, 1987; Khing, Das, Puri, Cook, & Yen, 1993; Yetimoglu, Wu, & Saglamer, 1994) using single or multilayered geosynthetic reinforcement. It was confirmed by most researchers that there was a significant increase in bearing capacity and a decrease in settlement of soil reinforced with geosynthetics. The increase in bearing capacity of reinforced foundation is defined in terms of bearing capacity ratio (BCR), which is defined as the bearing capacity of reinforced soil to that of unreinforced soil. Regarding the design of shallow foundation using geosynthetics, Das, Shin, and Singh (1996) proposed a procedure regarding strip foundation in geogrid reinforced clay. Wayne, Han, and Akins (1998) discussed some design issues of geosynthetic reinforced foundation. Shin and Das (2000) conducted a small-scale laboratory model test to determine the ultimate bearing capacity of a strip foundation supported on sand reinforced with multiple layers of geogrid and proposed a design procedure based on the model study. However, the rupture strength of the geogrid was not taken into consideration while determining the bearing capacity ratio. Michalowski (2004) presented recommendation for the design of reinforced foundation using a kinematic approach of limit analysis. Both pull out strength and rupture strength were considered, and the improvement was presented in terms of bearing capacity ratio. The design of reinforced soil is a trial-and-error process in which the position, layer, and length of the reinforcement are estimated based on the desired BCR value.

Introduction

153

Optimization is an integral part of engineering design. Wang and Kulhawy (2008) discussed the optimum dimension and design of ordinary reinforced cement concrete shallow foundation. Basudhar, Vashistha, Deb, and Dey (2007) presented an optimization-based design of reinforced earth retaining wall. However, to the best of knowledge of the authors, such a study on reinforced foundation bed is not available in the literature. The optimization algorithm presented in Basudhar et al. (2007) and Wang and Kulhawy (2008) are based on the traditional optimization algorithm. These algorithms have the shortcoming of initial point dependent and the penalty function used for the constrained optimization may distort the true optimum value (Deb, 2001). In the recent past, evolutionary algorithms like genetic algorithm (GA) (Cheng, Li, & Chi, 2007; Das, 2005; Goh, 1999), simulated annealing, particle swarm optimization, simple harmony optimization and Tabu search (Cheng et al., 2007) have been used in geotechnical engineering with success. In the above problems, single objective has been considered in terms of cost (Basudhar et al., 2007; Wang & Kulhawy, 2008) or factor of safety (Cheng et al., 2007; Das, 2005; Goh, 1999). However, in certain cases, there are more than single objectives and these objectives may be conflicting. When objectives are conflicting, it is considered as multi-objective optimization problems. In traditional optimization algorithms, multi-objective problems are considered as single objective considering other objectives as constraints (Deb, 2001). Hence, in such cases Pareto-optimal (trade-off ) solutions are obtained with a number of runs. Fig. 1 shows the variation in traditional and evolutionary multi-objective optimization algorithm. It can be seen that in case of traditional multi-objective optimization, it is converted to single-objective optimization problem with importance attached to each objective or taking other objectives as constraints. But an ideal multiobjective algorithm should find out a set of Pareto-optimal solutions considering all the objectives as equally important. Then one of the solutions is chosen considering higher-level information. Population-based evolutionary multi-objective optimization (EMO) is able to generate the required Pareto front in a single run. A comprehensive review of EMO algorithms can be found in Deb (2001) and Coello, Veldhuizen, and Lamont (2002). But, the application of multi-objective optimization is limited in geotechnical engineering (Deb & Dhar, 2011; Deb, Dhar, & Bhagat, 2012). In the above cases, the multi-objective algorithm has been used for parameter estimation. With above in view, an attempt has been made in this work the optimum design of reinforced foundation bed is considered in terms of achieving an optimum BCR value with a minimum cost of foundation. As the two objectives are conflicting, genetic algorithm-based multi-objective optimization method (NSGA-II) is considered here for analysis. Parametric studies have

FIG. 1

Soultions evaluation

Ranking

Optimal Limits Identification

Optimization problem generation

Population ranking

Optimization

Traditional Optimization method

Initialization

NSGA II implementation

One optimum solution

Multiple trade-off solutions

High-level information

Typical diagram showing variation between traditional and evolutionary multi-objective optimization model.

Optimization contraints

Optimization objectives

Decision variables

Model formulation

Choose one solution

155

Methodology

been made to find out the effect of soil properties and tensile strength (Tu) of geosynthetic reinforcement on the optimum design.

Methodology The methodology consists of the development of the optimization model based on the physical problem and solution of the optimum function using NSGA-II. The methodology for reinforced soil foundation as presented in Michalowski (2004) has been considered. A brief introduction about the above method is presented as follows: The ultimate bearing capacity (q) without reinforcement is given by the Terzaghi equation as given in Eq. (1). 1 q ¼ cNc + qNq + γBN γ 2

(1)

with coefficients N dependent only on internal friction. A typical footing on reinforced earth foundation is shown in Fig. 2. The bearing capacity of reinforced soil with one layer of reinforcement as per Michalowski (2004) is given in Eq. (2). 

 1 1 d pr ¼ c ðNc + nfcMcÞ + q Nq + nμμMq + γB Nγ + μMγ (2) d 2 B 1  μMp B The soil reinforcement friction coefficient, μ is expressed in Eq. (3). μ ¼ fb tan Ø

(3)

fb and fc are bond coefficients and taken as 0.6. Mc, Mq, and Mγ are standard bearing capacity coefficients due to reinforcement, B is the width of the footing and (d/B) is the relative depth of reinforcement.

B Df u d

SOIL

Geogrid layer 1

s 2 s 3

FIG. 2

A general footing on reinforced earth foundation (Das, 2009).

156

8. Multi-objective optimum design

Standard bearing capacity coefficients are taken as functions of ϕ only (Michalowski, 2004). The values of Nc, Nq, and Nᵧ are expressed in Eqs. (4)–(6).   (4) Nc ¼ Nq  1 cot ϕ
Nq ¼ tan 2

 π ϕ π tan ϕ + e 4 2

Nγ ¼ e0:66 + 5:11tan ϕ tan ϕ

(5) (6)

To extract the maximum benefit Mc, Mq, Mᵧ are taken equal to M when the reinforcement intersects the failure mechanism. The coefficient M can be estimated by Eq. (7).   (7) M ¼ 1:6 1 + 8:5tan1:3 ϕ Coefficient Mp can be approximated by Eq. (8). Mp ¼ 1:5  1:25  102 ϕ

(8)

The value of ϕ is in degrees. For two- or three-layer reinforcement, the bearing capacity equation is presented in Eq. (9). "

n X     1 di N pr ¼ + nf M + q N + nμM + γB + μM c N c q γ c n X 2 B di i¼1 1  μMp B i¼1

1

!#

(9) where n is the number of layers and di is the depth of ith layer. Coefficient M is approximated with the following expressions:   M ¼ 1:1 1 + 0:6 tan 1:3 ϕ

(10)

for second layer foundation, and   M ¼ 0:9 1 + 11:9 tan 1:3 ϕ

(11)

for the third layer foundation. Coefficient Mp approximated as. Mp ¼ 0:75  6:25  103 ϕ for second layer reinforcement, and

(12)

Methodology

157

Mp ¼ 0:50  6:25  103 ϕ

(13)

for third layer reinforcement (ϕ in degrees). Bearing capacity ratio with respect to pull out is calculated as given in Eq. (14). BCRp ¼

pr q

(14)

Bearing capacity ratio with respect to rupture is calculated as given in Eq. (15) BCRr ¼

pt q

(15)

Where, 1 nTt Mr pt ¼ cN c + qN q + γBN γ + 2 B

(16)

n is number of layers and


   π ϕ π ϕ  e 4 + 2 tan ϕ Mr ¼ 2 cos 4 2

(17)

The multi-objective optimization technique, NSGA-II used in the present study is not very common in civil engineering in general and geotechnical engineering in particular. Hence, a brief discussion about the NSGA-II algorithm is presented here for completeness and the details can be found in Deb (2001).

Multi-objective algorithms A real coded Nondominated Sorting Algorithm (NSGA-II) is utilized in this study. In a comparison presented in Deb (2001), it is shown that the NSGA-II performs well enough for highly nonlinear test problems, even for the case of a discontinuous Pareto front. The NSGA-II utilizes simulated binary cross-over (SBX) and polynomial mutation for cross over and mutation. Both approaches use a distribution index for the generation of offspring. Generally, the distribution indices for SBX and polynomial mutation vary between 4 and 50 and 5 and 50, respectively. A larger value of the distribution indices gives a higher probability of “near parent” solutions. However, a small value allows distant solutions to be selected as offspring. In the NSGA-II, constraint violations are directly incorporated when classifying the population into different fronts. The algorithm does not have any information about the true Pareto front, since there is no

158

8. Multi-objective optimum design

formal way to verify whether a given front is a Pareto-optimal front or not. So the NSGA-II never really knows when to stop. In the present study, the program is terminated when the nondominated front has not improved over a significant number of generations. However, details about NSGA-II is available in Deb (2001).

Results and discussion To validate the developed code, the problem as considered in Michalowski (2004) was taken with footing width B ¼ 1.2 m and the sand under the footing has an internal friction angle ϕ ¼ 35 degree and unit weight γ ¼ 17 kN/m3. The allowable tensile strength of Geosynthetic reinforcement was taken 16 kN/m. The results obtained in the present study and compared with that obtained as per Michalowski (2004) are presented in Table 1.

Objective function The objective functions considered here are the cost of reinforcing the soil with geosynthetics expressed in US Dollar ($) per meter run and bearing capacity ratio (BCR). The basic rate/unit cost of different activities for the construction of the reinforced foundation is taken as per Basudhar et al. (2007) and presented in Table 2.

Variation of optimum BCR value In the reinforced soil foundation, it is very important to achieve the maximum BCR value with the combination of soil and reinforcement parameters. In this section, the objective function is considered as maximization of the BCR value and minimization of the total cost per meter run. Like any numerical method, the effective application of the NSGA-II depends upon the parameters like crossover and mutation parameters and number of generation. A typical diagram showing the variation of TABLE 1 Validation of the present study with Michalowski (2004). Reinforcement with one layer

Reinforcement with two layers

Reinforcement with three layers

BCRp

BCRt

BCRp

BCRt

BCRp

BCRt

Michalowski (2004)

1.64

1.35

1.86

1.71

–

–

Genetic Algorithm (Present study)

–

1.35

–

1.71

–

2.08

Results obtained

159

Results and discussion

TABLE 2 Unit cost of various items of reinforced soil foundation. Items

Unit price

Cost of earthwork

$2.6/m3

Cost of fill

$3/1000 kg

Cost of geosynthetic

$(Ta*0.03 + 2.6)/m2

Engineering and testing cost

$10/m2

BCR and minimum cost with different generation is shown in Fig. 3 for the crossover probability of 0.9 and mutation probability of 0.07. It can be seen that number of Pareto solutions are very less up to 10th generation and increases with an increase in generation. At 100th generation all the population could reach the Pareto (nondominated) solutions. In the present study, the Pareto solutions could reach after 100th generation. It can be also seen that the total cost per meter run increase with an increase in BCR value. Hence, higher-level information in terms of engineering decisions regarding the cost and BCR value can be made to find out the desired BCR value at the cost overrun. A parametric study is also made to find out the influence of various parameters on the maximization of BCR value and minimization of the total cost per meter run. Various parameters considered for the analysis were angle of internal friction of the fill (ϕ), unit cohesion (c), unit weight of the fill (γ) and ultimate tensile strength of geosynthetic (Tu). 80

Total cost per metre (In USD)

70

5th Generation 10th Generation 50th Generation 100th Generation

60

50

40

30

20 1.5

2.0

2.5

3.0

BCR

FIG. 3

Variation in the total cost and BCR during different generation.

3.5

160

8. Multi-objective optimum design

100 Single layer Double layer Triple layer

90

Total cost per metre (In USD)

80 70 60 50 40 30 20 10

1.0

1.5

2.0

2.5

3.0

3.5

BCR

FIG. 4 Variation of the total cost with BCR for different number of layers at Tu ¼ 100 kN/m.

The variation of BCR value for different reinforcement layers and total cost for a particular ϕ and Tu value (100 kN/m) is plotted in Fig. 4. It can be seen that for single layer BCR value goes up to 2.0, whereas for higher BCR value it needs two or three layers of reinforcement. It can be also seen that for the double layer after the BCR value of 2.75, the rate of increase in cost is more. Similarly different rate of increase in the total cost observed for three-layer reinforcement. Such a diagram helps the designer and policy planner to decide an optimum configuration for the reinforced soil foundation bed. To study the variation of the optimum BCR value and total cost with angle of internal friction the value of cohesion (c) is taken as 0 and γ ¼ 17 kN/m3 and is shown in Fig. 5. It can be seen that there is a steeper increase in total cost with ϕ value of 20 degree in comparison to ϕ value of 35 degree. This suggests the advantages of using coarse sand with higher ϕ value. The effect of cohesion value on the optimum BCR value and the total cost is considered for ϕ ¼ 35 degree and γ ¼ 17 kN/m3. The variation is shown in Fig. 6. The variation of optimum BCR and total cost with different values of γ is shown in Fig. 7, considering cohesion ¼ 0 and ϕ ¼ 35 degree. It can be seen that the variation in the total cost and BCR value is negligible with variation in a unit weight of soil. This observation may help in using lightweight materials like fly ash or fly ash blended with sand having equivalent shear strength parameters. Finally, the variation of optimum BCR value and total cost for different values of Tu is plotted in Fig. 8. It can be seen that to get higher BCR value it

161

Results and discussion

100

f = 25° f = 30°

Total Cost per metre (inUSD)

90

f = 35°

80

70

60

50

40

30 1.4

1.6

1.8

2.0

2.2

2.4

2.8

2.6

3.0

3.2

3.4

3.6

BCR

FIG. 5 Variation of the total cost with BCR for different value of angle of internal friction.

60 c = 20kN c = 25kN c = 30kN c = 35kN c = 40kN c = 45kN

Total Cost per metre (in USD)

55

50

45

40

35 1.5

1.6

1.7

1.8

1.9

BCR

FIG. 6

Variation of the total cost with BCR for different unit cohesion of soil.

2.0

162

8. Multi-objective optimum design

100 g = 15kN/m3

90

g = 16kN/m3

Total Cost per metre (in USD)

g = 17kN/m3 g = 18kN/m3

80

g = 19kN/m3 g = 20kN/m3

70 60 50 40 30 20 1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

BCR

FIG. 7

Variation of the total cost with BCR for different unit weight of soil.

90 Tu = 50kN/m Tu = 75kN/m

Total cost per metre (In USD)

80

Tu = 100kN/m Tu = 125kN/m Tu = 150kN/m Tu = 175kN/m

70

Tu = 200kN/m

60

50

40 1.5

2.0

2.5

3.0

3.5

4.0

BCR

FIG. 8

Variation of the total cost with BCR for different tensile strength of geosynthetic.

References

163

is also essential to have higher Tu value. In these curves, it can be also seen that the optimum BCR value is close to 2.75 beyond which the rate of increase in total cost is more. At this point, it may be mentioned here that the cost of reinforcement is related to the Tu value, hence at the same BCR value the total cost increases with increase in Tu value.

Conclusion In this study, an attempt has been made to model a geosynthetic reinforced soil foundation to have maximum BCR value with minimum cost. An evolutionary multi-objective optimization algorithm, NSGA-II has been used to analyze the above problem. Based on the above study, following conclusions can be made: (i) The number of Pareto-optimal solutions are found to depend upon the number of generations keeping other NSGA-II parameters constant. The NSGA-II is found to be efficient in finding out the Pareto solutions. (ii) Based on the parametric study, it was observed that for single layer BCR value goes up to 2.0, whereas for higher BCR value it needs two or three layers of reinforcement. (iii) It was observed that there is a steeper increase in total cost with ϕ value of 20 degree in comparison to ϕ value of 35 degree. (iv) The variation in the total cost and BCR value was negligible with variation in unit weight of soil suggesting the use of fly ash or sand blended fly ash as a fill material. (v) Higher BCR value was obtained with higher Tu value and optimum BCR value is close to 2.75 beyond which the rate of increase in total cost is more. Such a study will help engineers and policy planners to identify a tradeoff solution of total cost and BCR value.

References Basudhar, P. K., Vashistha, A., Deb, K., & Dey, A. (2007). Cost optimization of reinforced earth walls. Geotechnical and Geological Engineering, 26(10), 1–12. Cheng, Y. M., Li, L., & Chi, S. C. (2007). Performance studies on six heuristic global optimization methods in the location of critical slip surface. Computers and Geotechnics, 34(6), 462–484. Coello, C. A. C., Veldhuizen, D. A. V., & Lamont, G. B. (2002). Evolutionary algorithms for solving multi-objective problems. New York: Kluwer Academic Publishers. Das, S. K. (2005). Slope stability analysis using genetic algorithm. Electronic Journal of Geotechnical Engineering, 1.

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8. Multi-objective optimum design

Das, B. M. (2007). Principles of geotechnical engineering. New York (USA): Cengage Publishing House. Das, B. M. (2009). Shallow foundations: Bearing capacity and settlement (2nd ed.). London, UK: Taylor & Francis Group. Das, B. M., Shin, E. C., & Singh, G. (1996). Strip foundation on Geogrid-Reinrorced clay: A tentative design. In: Proceedinhs of the international offshore and polar engineering conference, Los Angeles, USA, Vol. 1. Deb, K. (2001). Multi-objective optimization using evolutionary algorithms. Chichester; UK: Wiley. Deb, K., & Dhar, A. (2011). Optimum design of stone column-improved soft soil using multiobjective optimization technique. Computers and Geotechnics, 38(1), 50–57. Deb, K., Dhar, A., & Bhagat, P. (2012). Evolutionary approach for optimal stability analysis of geosynthetic-reinforced stone column-supported embankments on clay. KSCE Journal of Civil Engineering, 16(7), 1185–1192. Goh, A. T. C. (1999). Genetic algorithm search for critical slip surface in multiple-wedge stability analysis. Canadian Geotechnical Journal, 36(x), 382–391. Guido, V. A., Knuppel, J. D., & Sweeny, M. A. (1987). Plate loading tests on geogridsreinforced earth slabs. In: Proc. Geosynthetics ‘87, Industrial Fabrics Assoc. Int., St. Paul, Minn, Vol. 1, (pp. 216–225). Khing, K. H., Das, B. M., Puri, V. K., Cook, E. E., & Yen, S. C. (1993). Bearing capacity of strip foundation on geogrid-reinforced sand. Geotextiles and Geomembranes, (12), 351–361. Michalowski, R. L. (2004). Limit loads on reinforced foundation soils. Journal of Geotechnical and Geoenviromental Engineering, 130(4), 381–390. Sakti, J. P., & Das, B. M. (1987). Model tests for strip foundation on clay reinforced with geotextile layers: (pp. 40–45). Transportation research record 1153Washington DC: Transportation Research Board. Shin, E. C., & Das, B. M. (2000). Experimental study of bearing capacity of a strip foundation on geogrid reinforced sand. Geosynthetics International, 7(1), 59–71. Shukla, S. K., & Yin, J. H. (2006). Fundamentals of geosynthetic engineering. London; U.K: Taylor & Francis Group. Wang, Y., & Kulhawy, F. H. (2008). Economic design optimization of foundations. Journal of Geotechnical and Geoenviromental Engineering, 134, 1097–1105. Wayne, M. H., Han, J., & Akins, K. (1998). The design of geosynthetic reinforced foundations, geosynthetics in foundation reinforcement and erosion control systems: (pp. 1–18). Reston, VA: ASCE Geotechnical Special Publication, No. 76. Yetimoglu, T., Wu, J. T. H., & Saglamer, A. (1994). Bearing capacity of rectangular footings on geogrid-reinforced sand. Journal of Geotechnical Engineering, 120(12), 2083.

C H A P T E R

9 Analysis of laterally loaded pile V.A. Sawanta and Sunita Kumarib a

Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India, bDepartment of Civil Engineering, National Institute of Technology Patna, Patna, Bihar, India

List of symbols B L kh nh EI R, T e cu Dr H M Hu Mmax E ν ks Kn u θy IρH IρM

pile diameter length of pile subgrade reaction modulus coefficient of modulus variation flexural rigidity stiffness factors eccentricity undrained shear strength relative density horizontal load at top applied moment at top ultimate lateral pile resistance maximum bending moment modulus of elasticity Poisson’s ratio interface shear stiffness interface normal stiffness horizontal displacement rotation about y-axis influence factor for top displacement due to H influence factor for top displacement due to M

Introduction Pile foundations are used for jacket-type offshore structures, which are subjected to large magnitudes of lateral loads due to waves and winds. The ultimate failure of a pile-soil system is catastrophic and excessive lateral deflection of the pile creates operational difficulties. In view of this, the bending moment in the pile is required to be predicted in a rational

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9. Analysis of laterally loaded pile

manner. The p-y concept developed by Reese and Matlock (1956) is popularly being used to estimate pile top deflection and bending moments. The ultimate internal resistance of a vertical pile and the deflection of the pile are complex matters involving the interaction between a structural element and the soil. Taking the case of a vertical pile unrestrained at the head, the lateral loading on the pile head is initially carried by the soil close to the ground surface. At a low loading, the soil compresses elastically, but the movement is sufficient to transfer some pressure from the pile to the soil at a greater depth. At a further stage of loading, the soil yields plastically, and the pile transfers its load to greater depths. A short rigid pile unrestrained at the top and having a length-to-width ratio of less than 10 to 12 rotates, and passive resistance develops above the toe on the opposite face to add to the resistance of the soil near the ground surface. Eventually, the rigid pile will fail by rotation when the passive resistance of the soil at the head and toe are exceeded. The short rigid pile restrained at the head by a cap or bracing will fail by translation in a similar manner to an anchor block that fails to restrain the movement of a retaining wall transmitted through a horizontal tied rod. The failure mechanism of an infinitely long pile is different. Theoretically, the passive resistance to yielding provided by the soil below the yield point can be considered infinite and rotation of the pile cannot occur, the lower part remaining vertical while the upper part deforms. Failure occurs when the pile yields at the point of maximum bending moment, and for analysis, a plastic hinge capable of transmitting shear is assumed to develop at this point. In the case of a long pile restrained at the head, high bending stresses develop at the point of restraint, for example just beneath the pile cap, and the pile may yield at this point.

Pile behavior The first step is to determine whether the pile will behave as a short rigid pile or as an infinitely long flexible pile based on the relative stiffness of pile and soil. This is achieved by computing the stiffness factors R and T for the particular combination of pile and soil. The stiffness factors are governed by the flexural rigidity (EI) of the pile and the soil modulus kh. The soil modulus kh has been related to Terzaghi’s concept of a modulus of horizontal subgrade reaction. It is related to the compressibility of the soil. It is not constant for any soil type but depends on the diameter of the pile Dand the depth of the particular loaded area of soil being considered. In the case of stiff over-consolidated clay, the soil modulus is generally assumed to be constant depth. For this case, stiffness factor R is ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith computed as R ¼ 4 EI=kh B having units of length.

167

Pile behavior

TABLE 1

Relationship of kh with cu of stiff over-consolidated clay.

Consistency 2

Undrained shear strength cu (kN/m ) 3

Range of kh (MN/m )

Firm to stiff

Stiff to very stiff

Hard

50–100

100–200

>200

15–30

30–60

>60

After Tomlinson, M., Woodward, J. (2008). Pile design and construction practice, 5th ed., Taylor and Francis.

For short rigid piles, it is sufficient to take kh in the above equation as equal to the Terzaghi modulus k1, as obtained from load/deflection measurements on a 305 mm square plate. It is related to the undrained shearing strength of the clay, as shown in Table 1 (Tomlinson & Woodward, 2008). For most normally consolidated clays and for coarse-grained soils, the soil modulus is assumed pffiffiffiffiffiffiffiffiffiffiffiffi to increase linearly with depth, for which Stiffness factor T ¼ e EI=nh in units of length and. Soil modulus kh ¼ nh  z/B, and z is the depth below ground level. Values of the coefficient of modulus variation nh recommended by Reese, Cox, and Koop (1974) and Terzaghi (1955) are given in Fig. 1 (Garassino, Jamiolkowski, & Pasqualini, 1976).

Coefficient of subgrade reaction (MN/m3)

45 40 35 30 25

Reese et al.

20

Terzaghi 15 10 5 0 0

10

20

30

40

50

60

70

80

90

100

Relative density Dr FIG. 1 Relationship between coefficient of subgrade reaction nh and relative density Dr. After Garassino, A., Jamiolkowski, M. and Pasqualini, E. (1976). Soil modulus for laterally-loaded piles in sands and NC clays. Proceedings of the 6th European conference, ISSMFE, Vienna, Vol. I(2), pp. 429–434.

168

9. Analysis of laterally loaded pile

TABLE 2 Criteria for behavior as a short rigid pile or as a long elastic pile. Linearly-increasing soil modulus

Constant soil modulus

Rigid (free head)

L2T

L2R

Elastic (free head)

L 4 T

L 3.5 R

Pile type

Other observed values of nhare as follows: Soft normally consolidated clays: 350–700 kN/m3 Soft organic silts: 150 kN/m3 From stiffness factors R or T using estimates of nh and kh appropriate to ground conditions, the criteria for behavior as a short rigid pile or as a long elastic pile should be checked. It is related to the embedded length L and stiffness factors as shown in Table 2.

Ultimate lateral resistance of short rigid piles For short-term loading in uniform cohesive soils, Broms method is quick and convenient. Broms assumed variation in soil reaction using simplified diagrams as shown in Fig. 2 for pile with free and fixed heads. A zone of zero pressure for (1.5B) depth is assumed to represent the effect of soil shrinkage away from the pile. The depth f of point of zero shear below 1.5D can be calculated from H  9cu B  f ¼ 0

(1)

Maximum bending moment or free head pile Mmax ¼ Hðe + 1:5B + f Þ  9cu B  0:5f ¼ Hðe + 1:5B + f Þ  H  0:5f ¼ Hðe + 1:5B + 0:5f Þ

(2)

The part of pile of length g resists the bending moment Mmax and for equilibrium g 3g g g 9cu B g2  9cu B  ¼ Mmax ¼ 9cu B  2 4 2 4 4

(3)

The short rigid fixed-headed pile behaves as a cantilever carrying a load over part of its length.   9cu B L2  2:25B2 Mmax ¼ (4) 2

Ultimate lateral resistance of short rigid piles

169

FIG. 2 Soil reactions and bending moments for short pile under horizontal load in cohesive soil. After Broms, B. (1964a). The lateral resistance of piles in cohesive soils. Journal of the Soil Mechanics Division 90(SM2), 27–63 and Broms, B. (1964b). The lateral resistance of piles n cohesionless soils. Journal of the Soil Mechanics Division 90(SM3), 123–56.

Moreover, the ultimate lateral resistance of fixed-headed piles can also be found out using Fig. 3 shown below. For cohesionless soil, the distributions of soil reaction and bending moment are represented in Fig. 4.At a particular depth (z) of soil the soil reaction on the pile is given by pz ¼ 3 B p0z Kp

(5)

The ultimate lateral pile resistance of pile can be found out using Fig. 5 for different pile-head conditions and eccentricity ratio e/B. Here, B is the width of the pile transverse to the direction of rotation. poz is the effective overburden pressure at depth z and Kp is Rankine’s coefficient of passive earth pressure. High value of passive resistance at the toe of a pile in cohesionless soil can be replaced by a concentrated horizontal force P. From the established graphical relationships of Broms (1964a, 1964b) the ultimate lateral resistance Hu can be determined as follows: Hu ¼

γ B L3 Kp 2ðe + LÞ

(6)

For short rigid fixed-headed piles in cohesionless soils, failure is due to simple translation and the ultimate lateral resistance Hu is expressed as Hu ¼

3γ B L2 Kp 2

(7)

170

9. Analysis of laterally loaded pile

soil 60

Restrained Ultimate lateral resistance Hu/cuB2

50 e/B

40

=

0

30

20

10

Free head 0

0

4

8

12

16

20

Embedment length L/B

FIG. 3

Ultimate lateral resistance of short pile in cohesive soil related to the embedded

length.

Au

Hu

Mmax

e

L L

Deflection

P Deflection

3 bg L Kp Soil reaction

(A)

3 bg L Kp Soil reaction

Free head

Mmax

Bending moment

Bending moment

(B)

Fixed head

FIG. 4 Soil reactions and bending moments for short pile under horizontal load in cohesionless soil. After Broms, B. (1964a). The lateral resistance of piles in cohesive soils. Journal of the Soil Mechanics Division 90(SM2), 27–63 and Broms, B. (1964b). The lateral resistance of piles n cohesionless soils. Journal of the Soil Mechanics Division 90(SM3), 123–56.

171

Long flexible piles

200

Restrained

Applied lateral load H/kpB3

160

Free head 120

80

40

0

0

4

8

12

16

20

Length L/B

FIG. 5 Ultimate lateral resistance of short pile in cohesionless soil related to the embedded length.

The simplified equation for ultimate lateral resistance is only valid when the maximum negative bending moment at pile head is lesser than the ultimate resistance moment Mu of pile at this point. The bending moment is expressed as follows:; M max ðveÞ ¼ γ B L3 Kp

(8)

Long flexible piles The basic problem of laterally loaded pile (LLP) is three dimensional in nature. However, it is solved using suitable assumptions to reduce computational efforts. It is idealized as one-dimensional beam problem or two-dimensional problem with plane-strain axi-symmetric idealizations. It is presumed that the error involved with this assumption is not beyond 10%–15%. With these assumptions number of simultaneous equations to be solved are an order of 1%–10% (for 1D and 2D) as compared to the actual three-dimensional problem. With a smaller number of equations, it is convenient to introduce nonlinear complexities in the soil behavior. Available methods of analysis can be broadly classified into three groups. • Finite difference method • Structural analysis method like moment area method • Finite element method

172

9. Analysis of laterally loaded pile

The above methods are further divided into two parts by the way soil is modeled in the analysis. The subgrade reaction model of soil behavior, which was originally proposed by Winkler in 1867, characterizes the soil as a series of unconnected linearly elastic springs. The obvious disadvantage of this soil model is the lack of continuity. Real soil is at least to some extent continuous as the displacements at a point are influenced by stresses at other points within the soil. A further disadvantage is that the spring modulus of the model (the modulus of subgrade reaction) is dependent on the size of the foundation. In spite of these drawbacks, the subgrade reaction approach has been widely accepted in practice because it provides a relatively simple means of analysis and enables factors such as nonlinearity, variation of soil stiffness and depth, and layering of the soil profile to be taken into account readily. From a theoretical point of view, the representation of the soil as an elastic continuum is more satisfactory, as the continuous nature of the soil is taken into account. The use of this model for the analysis of the settlement of piles has been found to provide a convenient and relatively reliable means of describing pile behavior under axial loading. While the elastic model is an idealized representation of real soil, it can be modified to make allowance for soil yield, and can also be used to give approximate solutions for varying modulus with depth and for layered systems. In addition, it has an important advantage over the subgrade reaction approach of enabling the analysis to be made of group action of piles under lateral loads; also it provides a means of analyzing the behavior of battered piles subjected to a general system of loading. The major drawback of the application of the elastic method to practical problems is the difficulty of determining the appropriate soil modulus; however, this difficulty exists to a certain extent with the subgrade reaction method. To account for the realistic non-homogeneous nature of the soil one has to go for a complete three-dimensional finite element approach for analysis. But it requires more time and memory. The analysis of laterally loaded pile using the above-mentioned numerical tools for the two approaches of modeling soil behavior is discussed in subsequent sections.

Finite difference method The pile is usually assumed to act as a thin strip with flexural rigidity EI. To employ finite difference formulation, pile is divided in n equal parts having n+1 nodes as shown in Fig. 6. The governing differential equation in terms of displacement u in x-direction along depth z EI

d4 u +q¼0 d z4

(9)

Finite difference method

FIG. 6

173

Discretization of a pile for a finite difference approach.

Derivatives of u can be approximated as given below: d ui ðui1  ui + 1 Þ ¼ 2h dz d2 ui ðu1  2u0 + u1 Þ ¼ h2 d z2 d3 ui ðu2  2u1 + 2u1  u2 Þ ¼ h2 d z3

(10)

d4 ui ðui2  4ui1 + 6ui  4ui + 1 + ui + 2 Þ ¼ h4 d z4 Boundary Conditions at Pile top (z ¼ 0) and tip (z ¼ L): EI

EI

d2 u0 ðu  2u0 + u1 Þ d3 u0 ðu  2u1 + 2u1  u2 Þ ¼ EI 1 ¼ M and EI ¼ EI 2 ¼H d z2 d z3 h2 h2

d2 un ðu  2un + un + 1 Þ d3 un ðu  2un1 + 2un + 1  un + 2 Þ ¼ EI n1 ¼ 0 and EI ¼ EI n2 ¼0 d z2 d z3 h2 h2

174

9. Analysis of laterally loaded pile

Applying boundary conditions, the final system of equations can be expressed in the matrix form as given below. 9 8 2H 2M > > > >  > > > > > h h2 > > > > > > > > > M > > > > > > > > > > 2 > > h > > > > > > 0 > > > > > > > > > > > > > > 0 > > > > > > > > > > > > > > 0 =
u0 > > > > > 7> 6 > 7> 6 2 5 4 1 > > u1 > > 7> 6 > > > > 7> 6 > > 7> 6 1 4 6 4 1 u > > > 2 > 7> 6 > > > 7 6 > > 1 4 6 4 1 u 7> 6 > > 3 > > > > 7> 6 > > > 7> 6 1 4 6 4 1 u > = 7< 4 > 6 EI 6 7 7 6 + dfpg ¼ 1 4 6 4 1 u5 > > > 4 7 6 > > > h 6 > > 7> > > 1 4 6 4 1 > > > 7> 6 > > > > > > 7> 6 > > > > 7> 6 1 4 6 4 1 > > > > > 7> 6 > > > > > 7> 6 > > > > > 6 1 4 6 4 1 7 > > > > > > 7> 6 > > > > > > 7 6 > > 1 4 5 2 5 > u > > > 4 > > n1 > > > > ; > : > > 2 4 2 un > > > > > > : 2

2

4

2

0

0 0

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ;

¼ fBg

(11)

Unknown soil pressure is related to displacement at a given node in the subgrade reaction approach. 9 2 9 8 38 p0 > kh u0 > > > > > > > > > > > > > 7> kh = 6 < p1 > 6 7 < u1 = 6 7 ¼ or fpg ¼ ½kh  fug ⋱ (12) 7> > 6 > > > 5> kh > 4 > > pn1 > > un1 > > > > > ; ; : : pn kh un Then, the final set of equations in terms of unknown displacements {u} can be written in the following form. EI ½D fug + d ½kh  fug ¼ fBg h4

(13)

In the elastic continuum approach, unknown displacements {u} are related to unknown pressure {p} using Mindlin’s equation. 9 2 9 8 38 f11 f12 p0 > f1n > u0 > > > > > > > > > > > > > f2n 7 = 6 < u1 > 7 < p1 = 6 f21 f22 7 6 ¼ or fug ¼ ½ f   fpg (14) 7> > 6 > > > 5> > 4 > > un1 > > pn1 > > > > > ; ; : : fn1 fn2 fnn pn un Here, [ f ] is a matrix of flexibility coefficients. Hence, unknown pressure {p} can be related to unknown displacements {u} using inverse relation. fpg ¼ ½ f 1  fug

Simplified finite element analysis with subgrade reaction approach

175

Then, the final set of equations in terms of unknown displacements {u} can be written in the following form. EI ½D fug + d ½ f 1 fug ¼ fBg h4

(15)

Finite element method Finite element method is a robust numerical tool for handling complex problems. Heterogeneity in the material properties, irregular domain sizes can be easily handled in the analysis. Basic problem of laterally loaded pile (LLP) is three dimensional in nature. However, it is solved using suitable assumptions to reduce computational efforts. One can use a complete three-dimensional analysis. Both pile and soil are to be assumed as a continuum and discretize pile-soil domain using threedimensional elements. In the subgrade reaction approach, soil is considered a series of discrete springs.

Simplified finite element analysis with subgrade reaction approach A more simplified finite element approach is also possible with the spring approach. Beam element, plate element, and spring elements are used to model pile, pile-cap, and soil respectively. In this way, a three-dimensional system of the pile group is replaced by onedimensional beam element, two-dimensional plate element, and spring elements. The memory requirement is nearly one-tenth of the actual three-dimensional formulation. In this manner, more complex problems can be handled with significant accuracy. Interaction between the pile cap and soil is also taken into account. This is the area where not much attention is given in the past. A parametric study is conducted to investigate the behavior of pile groups. Results also show that a lateral response of the pile group depends on spacing between piles. The most salient feature of the analysis is that an interaction between the pile cap and soil has a significant effect of response of the pile group. This can be revealed from comparing the results with one in which interaction is not considered.

Beam element Beam element has six degrees of freedom at each node, which includes lateral displacement u and v, axial displacement w, and rotation about three axes. If rotation about the z-axis is not considered, the degree of

176

9. Analysis of laterally loaded pile

freedom is reduced to 5 at each node. Nodal displacement vector, {δ}e,{δ}Te ¼ {u1v1w1θx1θy1u2v2w2θx2θy2}. Stiffness matrix of the element [k]e, is given by the expression ðL ½ke ¼ ½BT ½D½Bdz

(16)

0

Here, [B] is strain-displacement transformation matrix and [D] is a constitutive relation matrix for beam element. 2 3 12 6h 12 6h 6 7 4h2 6h 2h2 7 EI 6 6 7 ½ke ¼ 3 6 h 4 12 6h 7 5 2 4h

Spring element Soil support at various nodes of beam element is simulated by using a series of equivalent and independent elastic springs in three directions (x, y, z). Soil stiffness can be found out using the principle of virtual work. A virtual displacement {Δδ} is applied to the spring system, and by equating internal work done to external work, soil stiffness can be worked out. Soil reactions at any point {px, py, pz} within the element are given by 8 9 2 3 9 Esx 0 0 8 > = < px >

> : ; : ; w p 0 0 E z

sz

{Esx, Esy, Esz} are soil subgrade reaction modulus at depth z. Soil support element stiffness matrix, [K]s, can be obtained as: 2 3 Esx 0 0 ðL 7 T6 ½ksoil e ¼ ½N  4 0 Esy 0 5½N  dz 0

0

(18)

0 Esz

After integration, 3 3 2 2 156 22h 54 13h 72 14h 54 12h 7 7 6 6 4h2 13h 3h2 7 ðEs2  Es1 Þh 6 3h2 14h 3h2 7 Es1 h 6 soil 7 7 6 6 + ½ke ¼ 6 420 6 840 156 22h 7 240 20h 7 5 5 4 4 2 2 4h 5h

177

Three-dimensional finite element analysis

Three-dimensional finite element analysis In a more realistic and rigorous analysis, both pile and soil are to be assumed as a continuum. The pile, soil, and pile cap are discretized into a number of 20 node isoparametric continuum elements. The interface between pile or pile cap and soil is modeled using 16 node isoparametric surface elements with zero thickness (Ladhane & Sawant, 2016; Sawant & Dewaikar, 1999). These interface elements are useful in simulating the mechanics of stress transfer along the interface.

Continuum element The co-ordinates (x, y, z) of a point within the element are expressed in terms of local co-ordinates (ξ, η, ζ) of the element as, x¼

20 X

Ni xi ; y ¼

i¼1

20 X

Ni yi ; z ¼

i¼1

20 X

N i zi

(19)

i¼1

where (xi, yi, zi) are nodal coordinates of the element and Ni, are the shape functions of the elements. Shape functions describing geometry and displacements of the element are expressed as follows: 1 Ni ¼ ð1 + ξi ξÞð1 + ηi ηÞð1 + ζ i ζ Þðξi ξ + ηi η + ζ i ζ  2Þ for i ¼ 1 to 8 8  1 Ni ¼ 1  η2 ð1 + ξi ξÞð1 + ζ i ζ Þ for i ¼ 9 to 12 4  1 Ni ¼ 1  ζ 2 ð1 + ξi ξÞð1 + ηi ηÞ for i ¼ 13 to 16 4  1 Ni ¼ 1  ξ2 ð1 + ηi ηÞð1 + ζ i ζ Þ for i ¼ 17 to 20 4

(20)

where (ξi, ηi, ζ i) are local coordinates of ith node. The same shape functions as shown in Eq. (20) are used to define displacements {u, v, w}, within the element: u¼

20 X i¼1

Ni ui ; v ¼

20 X i¼1

N i vi ; w ¼

20 X

Ni wi

(21)

i¼1

where u, v, and w are the displacements in the x, y, and z directions, respectively. Other types of continuum elements can be used which can define the same variation in displacements. Sawant and Shukla (2014) developed an 18-node triangular prism element to model the pile-soil domain. Plaxis-3D has 10-node tetrahedron elements to model the soil domain (Chandluri & Sawant, 2020).

178

9. Analysis of laterally loaded pile

The strain-displacement relationship is: f ε g ¼ ½ B f δ g e

(22)

where [B] is strain-displacement transformation matrix (6  24), {ε}, is strain vector and, {δ}, is vector of unknown displacements. A stress-strain relationship is given as: 8 > > > > > > > > > > > > >
> > > > > σy > > > > > > = σz >

6 6 6 6 6 6 E 6 6 ¼ fσ g ¼ > > 6 ð 1  ν Þ ð 1  2ν Þ > > τ > 6 xy > > > > > 6 > > > > 6 > > > > 6 τ > > yz > > 4 > > > > ; : τxz

1ν

ν

ν

0

0

0

ν

1ν

ν

0

0

0

ν

ν

1ν

0

0

0

0

0

0

0:5ð1  2νÞ

0

0

0

0

0

0

0:5ð1  2νÞ

0

0

0

0

0

0

0:5ð1  2νÞ

3

9 8 7> εx > 7> > > > 7> > 7> > > εy > > > > 7> > = 7< ε > z 7 7 > 7> > > γ xy > 7> > > > 7> > > γ yz > 7> > > > 7> 5: γ ; xz (23)

in which E, is the modulus of elasticity and ν is the Poisson’s ratio. Eq. (23) can be written in the abbreviated form as: fσ g ¼ ½D fεg ¼ ½D½Bfδge

(24)

where [D] is the constitutive relation matrix. The stiffness Matrix, [K]e of an element is given as ð1 ð1 ð1 ½BT ½D½BjJ jdξdηdζ

½Ke ¼

(25)

1 1 1

Interface element Shape functions describing the geometry and displacements of the interface element are given as: 1 Ni ¼ ð1 + ξi ξÞð1 + ηi ηÞðξi ξ + ηi η  1Þ for i ¼ 1,3, 5,7 4  1 Ni ¼ 1  ξ2 ð1 + ηi ηÞ for i ¼ 2 and 6 2  1 Ni ¼ 1  η2 ð1 + ξi ξÞ for i ¼ 4 and 8 2

(26)

where (ξi and ηi) are local co-ordinates of the ith node. The same shape functions are used to define the displacements, (u, v, w) within the element. u¼

8 X i¼1

Ni ui ; v ¼

8 X i¼1

Ni vi ; w ¼

8 X i¼1

Ni wi

(27)

Three-dimensional finite element analysis

179

The relative displacements, Δu0 , Δv0 , Δw0 in ξ and η directions are the components of the generalized strains for the element and are given by a relation: fεge ¼ ½Bf fδge

(28)

where {ε}e is the stain vector and {δ}e is a vector of unknown displacements. The element stiffness matrix, [k]e is obtained by usual expression, ð1 ð1 ½Ke ¼

  ½BTf ½D ½Bf V ξ  V η  dξ dη

(29)

1 _1

where [D] is the constitutive relation matrix for the interface given as: 2 3 ks 0 0 (30) ½D  ¼ 4 0 ks 0 5 0 0 kn The stiffness matrices of interface elements are assembled with the stiffness matrices of continuum elements to obtain overall stiffness matrix.

Equivalent nodal force vector The lateral force, H, acting on pile cap, is considered as uniformly distributed shear force over the cross-sectional area A. The intensity, q, of this uniformly distributed shear is, q ¼ H/A. Equivalent nodal force vector, {Q}e, is then expressed as ð1 ð1 q ½N T jJ j dξ dη

fQge ¼

(31)

1 _1

where [N] represents a matrix of shape functions. The stiffness matrices for all elements are evaluated and assembled into the global stiffness matrix, [A], in skyline storage form. Similarly, the load vector is assembled in vector [B]. From assembled global stiffness matrix and known load vector, overall equilibrium equations are formulated. The active column solution technique developed by Felippa (1975) is used for the solution of equilibrium equations of the system.

Validation Prakash (1962) carried out an experimental study for studying the behavior of the long pile embedded in sand and subjected to lateral load. The same data were employed for 3D FEA. The bottom boundary was considered rigid and rough, whereas the lateral boundary was considered rigid and smooth. The pile-soil contact surface was assumed

180

9. Analysis of laterally loaded pile

smooth for which contact shear stiffness was taken zero and contact normal stiffness had a very high value. The section for the pile was hollow circular with a diameter of 1.600 . However, for the sake of convenience, the hollow circular section of the pile is converted into an equivalent square section of 1.600 .An equivalent modulus of elasticity of sand was defined. As detailed information pertaining to stress-strain behavior was not available, this modulus was approximately computed from the relation E ¼ Jγz as given by Terzaghi and Peck (1948) where z is the depth from surface and J is the dimensionless parameter. Lateral load of magnitude 2.75 lb. was applied at the top of the pile. Variation in displacement and moment along depth is compared in Fig. 7. Quite a Displacement (inch) –0.002

0

0.002 0.004

0.006

0.008

0.01

0.012 0.014 0.016

0 2 4 6

FEM Expt

Depth (inch)

8 10 12 14 16 18 20

Validation with Prakash

22 24

(A) –1

0

1

2

Moment (Ib-in) 3

4

5

6

0 2 4 6

Depth (in)

8 10 12 14

FEM Expt

16 18 20 22

Validation with Prakash (1962)

24

(B) FIG. 7 (A) Comparison of displacement profile. (B) Comparison of bending moment profile.

181

Conclusion

fair agreement is seen in the results obtained by FEM and experimentally.

Comparative study of all methods For comparison of different methods, analysis of single pile is carried out by the above approaches. Properties of pile are kept constant with unit diameter, L/d ratio 25 and E ¼ 2.85  107 kN/m2. Soil modulus Es is varied in the parametric study. Top displacement Δ, can be expressed in terms of nondimensional factors (Influence Factors) IρH and IρM as given below. Δ ¼ IρH

H M + IρM Es L Es L2

(32)

Values of IρH and IρM are tabulated in Table 2 for a comparison of all methods along with those reported by Barber (1953) for different pile flexibility factor kR ¼ EI/EsL4. From Table 3, pile top displacement Δ can be calculated for a given pile-soil combination. For intermediate values, pile flexibility factor kR interpolation can be used.

Conclusion Different methods of analysis of laterally loaded piles are discussed in this chapter. Initially, the behavior of the short rigid pile is discussed. For long flexible piles, numerical tools like finite difference method and finite element method are used in the analysis. The subgrade reaction approach and an elastic continuum approach are considered in modeling the surrounding soil. The analysis of laterally loaded pile using the aboveTABLE 3

Influence Factors IrH for different pile flexibility factor kR. SRA FDM

SRA FEM

ECA FDM

Barber (1953)

19.625

43.36

18.550

45.61

17.499

24.96

17.868

25.60

12.496

14.13

14.991

13.69

7.649

7.96

10.374

8.11

4.742

4.82

6.706

5.06

10

4.034

4.09

4.594

4.10

1

3.951

4.01

5.453

4.00

kR 6

10

5

10

4

10

103 2

10

1

182

9. Analysis of laterally loaded pile

mentioned numerical tools for the two approaches of modeling soil behavior is discussed. End results are expressed in the form of influence factors for the convenience in applying the solutions to field problems.

References Barber, E. S. (1953). Discussion to paper by S M Gleser: (pp. 96–99). ASTM. STP 154. Broms, B. (1964a). The lateral resistance of piles in cohesive soils. Journal of the Soil Mechanics Division, 90(SM2), 27–63. Broms, B. (1964b). The lateral resistance of piles n cohesionless soils. Journal of the Soil Mechanics Division, 90(SM3), 123–156. Chandluri, V. K., & Sawant, V. A. (2020). Influence of sloping ground on lateral load capacity of single piles in clayey soil. International Journal of Geotechnical Engineering, 14(4), 353–360. https://doi.org/10.1080/19386362.2017.1419538. Felippa, C. A. (1975). Solution of equations with skyline–stored symmetric coefficient matrix. Computers & Structures, 5, 13–25. Garassino, A., Jamiolkowski, M., & Pasqualini, E. (1976). Soil modulus for laterally-loaded piles in sands and NC clays. (Vol. 1). Institut Fuer Grundbau und Bodenmechanik, TU Wien, pp. 429–434. Ladhane, K. B., & Sawant, V. A. (2016). Effect of pile group configurations on nonlinear dynamic response. International Journal of Geomechanics, 16(1), 04015013. https://doi. org/10.1061/(ASCE)GM.1943-5622.0000476. Prakash, S. (1962). Behaviour of pile groups subjected to lateral loads. Ph. D. ThesisUrbana: University of Illinois. Reese, L. C., Cox, W. R., & Koop, F. D. (1974). Field testing and analysis of laterally loaded piles in sand. In: Proceedings of the VI annual offshore technology conference, Houston, Texas, pp. 473–485. 2(OTC 2080). Reese, L. C., & Matlock, H. (1956). Non-dimensional solutions for laterally-loaded piles with soil modulus assumed proportional to depth. In Proceedings of the 8th Texas conference on soil mechanics and foundation engineering, Austin, Texas (pp. 1–41). Sawant, V. A., & Dewaikar, D. M. (1999). Analysis of pile groups subjected to cyclic lateral loading. Indian Geotechnical Journal, 29(3), 199–220. Sawant, V. A., & Shukla, S. K. (2014). Effect of edge distance from the slope crest on the response of a laterally loaded pile in sloping ground. Geotechnical and Geological Engineering, 32(1), 197–204. https://doi.org/10.1007/s10706-013-9694-7. Terzaghi, K. (1955). Evaluation of coefficients of subgrade reaction. Geotechnique, 5(4), 297–326. Terzaghi, K., & Peck, R. B. (1948). Soil mechanics in engineering practice. New York: Willey. Tomlinson, M., & Woodward, J. (2008). Pile design and construction practice (5th ed.). Taylor and Francis.

Further reading Gleser, S. M. (1953). Lateral load tests on vertical fixed head and free head piles: (pp. 75–93). ASTM. STP 154.

C H A P T E R

10 Role of localized elevated pore pressures and strain localization mechanisms in slope stability problems Rennie B. Kaunda Department of Mining Engineering, Colorado School of Mines, Golden, CO, United States

Introduction In certain geological environments and slopes subjected to external forces, soil or rock does not completely fail, but deformation zones are created due to intense strain localization, resulting in morphological and geotechnical changes. Strain localization is a feature of elastoplastic materials where shear bands are formed as a result of inhomogeneous material deformation leading to permanent expressions of intense strain zones. Catastrophic failure can be a result of the strain-softening behavior of the materials involved, combined or not with mechanisms such as excess pore water pressure leading to the sudden acceleration of the landslide. Instantaneous and localized strain localization due to localized excessive pore pressures have been key issues in many previous slope failures. The rapid infiltration of precipitation (i.e., snowmelt or rainfall), causing soil saturation and a temporary rise in pore pressure, is generally believed to be the mechanism by which most landslides in soils and weathered rock are generated in steep slopes (Chase, Kehew, & Montgomery, 2001; Kaunda, Chase, Kehew, Kaugars, & Selegean, 2008, 2009; Wieczorek, 1996). Pore water pressure build-up can reduce the effective strength of saturated slope materials, and thus triggering slope failure. Transiently elevated pore pressures have also been recorded in

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00010-1

183

© 2021 Elsevier Inc. All rights reserved.

184

10. Role of localized elevated pore pressures

hillside slopes and shallow bedrock during rainstorms associated with abundant shallow land sliding (Sidle, 1984; Wilson & Dietrich, 1987; Reid, Nielsen, & Dreiss, 1988; Wilson, 1989; Johnson & Sitar, 1990; Simon, Larsen, & Hupp, 1990). Pore pressure build-up can also occur due to increased groundwater levels within slopes following periods of prolonged above-normal precipitation or during the raising of water levels in lakes and reservoirs adjacent to the slope, creating unstable conditions (Wieczorek, 1996). The role of localized elevated pore pressures can be demonstrated using the example of coastal slope stability problems. For instance, it was held generally that wave activity was the primary culprit for slope instability problems leading to coastal erosion during high lake levels near Lake Michigan (Edil & Vallejo, 1977; Mickelson, Edil, Bosscher, & Kendziorski, 1991). However, site-specific field measurements documented significant slope movements during periods of low lake levels when there was less wave activity (Chase et al., 2001). Detailed monitoring of the region has shown that the movements are at their maximum during the winter-to-spring cycle of the year. Montgomery (1998) showed that slope zones of mixed sand and lacustrine clay with high hydraulic head experienced the greatest slope failures. Elevated water pore pressures due to water trapped by the frozen slope surface are the primary culprit. Laboratory tests have shown that freezing can seal pores and microcracks in the soil (Othman & Benson, 1993), which can prevent sufficient drainage to alleviate pre-pressure build-up. Elevated pore pressures or reduced matric suction also result in decreased shear strength of a potential failure surface (Ng & Shi, 1998). Freezing action disrupts the soil skeleton such that when the slope surface thaws, groundwater flows at great pressure toward seepage zones through cracks and pores expanded by multiple freeze–thaw cycles (Chase et al., 2001). Most slope stability evaluations use methods such as limit equilibrium to calculate factors of safety (Duncan, 1996) or finite element modeling to investigate slope deformations (Chen, Yin, & Lee, 2004; Griffiths & Lane, 1999; Kim & Lee, 1997; Yang & Yin, 2004). However, it is important to bear in mind that software are only decision support tools and should never substitute for engineering judgment. Further, most slope failures often occur in a progressive manner with factors of safety close to 1.0 (Cooper, Bromhead, Petley, & Grant, 1998; Kamai, 1998; Nieuwenhuis, 1991), and as such, an accurate understanding of the stress–strain behavior of the slope materials is needed. For instance, studies have shown that a slower rate of post-peak strength decrease in brittle soil could cause a large difference in the factors of safety calculated for brittle and ideally plastic soils (Srbulov, 1997). It is important, therefore, to have an accurate understanding of sitespecific ground truth, geotechnical field conditions, slope failure

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mechanisms, and slope displacement kinetics to anticipate the future of a slope. This chapter discusses the roles of localized elevated pore pressures and strain localization mechanisms in slope stability problems. The chapter is organized as follows. “Mechanisms of strain localization and their influence on the behavior of geomaterials” section provides a detailed overview of strain localization in geomaterials. Next, the role of localized pore pressures and strain localization during slope instability is discussed. Finally, a case example is discussed using the Oso landslide, which occurred in Washington State, the United States in 2014, to highlight likely implications of strain localization during a major landslide, including lessons that could be drawn for other slope instability incidents with similar geotechnical or geological conditions.

Mechanisms of strain localization and their influence on the behavior of geomaterials Mechanisms of strain localization in rock material Strain localization typically occurs over long periods of geologic timescales in rock material. In porous geomaterials such as sandstone, deformation bands are the most common strain localization feature (Fig. 1) (Aydin, Borja, & Eichhubl, 2006). The localized strain bands can occur in shear or compaction form (Wong, Baud, & Klein, 2001) (Fig. 2). Particle grains within deformation bands tend to be smaller, more compact, possess stronger preferred orientations, and have more elongate shapes than particles outside the band (Aydin, 1978; Cashman & Cashman, 2000). Strain localization can be considered an instability in material constitutive behavior (Bernabe & Brace, 1990; Borja & Aydin, 1994; Rudnicki & Rice, 1975). The material within a deformation band is thought to strain harden as a result of the deforming mechanism (Aydin, 1978; Aydin & Johnson, 1983; Bernabe & Brace, 1990; Borja, 2004; Wibberley, Petit, & Rives, 2000), although some studies have also described a type of cataclastic deformation indicating strain softening (Torabi, 2007). One possible explanation for the strain hardening in rock is grain crushing and pore collapse during cataclastic flow leading to embrittlement (Wong, Szeto, & Zhang, 1992). Grain crushing and pore collapse seem to be concentrated in most weakly cemented grain clusters (Menendez, Zhu, & Wong, 1996). During strain localization, a complete spectrum of combined shear and volumetric deformation modes have been observed ranging from compaction bands (shortening), shear bands (shear) to dilation bands (extension) (Aydin et al., 2006; Besuelle, 2001; Cilona et al., 2012; Du Bernard, Eichhubl, & Aydin, 2002; Issen & Rudnicki, 2001; Olsson, 1999). The differences in deformation modes can be attributed to rheology, loading

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FIG. 1 Examples of natural strain localization bands in rock: (A) disaggregation band from Gullfaks, North Sea (B) Phyllosilicate band from Huldro field, North Sea, (C) Cataclastic band, Sinai, and (D) Dissolution band, San Rafael Dessert, Utah. Phyllosilicate bands and cataclastic deformation bands show the largest reduction in pore space, while disaggregation bands, where non-cataclastic granular flow is dominant, show little influence. Modified from Torabi, A. (2007). Deformation bands in porous sandstones. PhD thesis, University of Bergen, Norway.

conditions and frictional strength (Aydin et al., 2006, Borja & Aydin, 2004, Sample, Woods, Bender, & Loveall, 2006). For instance, the frictional strength of a granular deformation zone is sensitive to grain shape, particle size distribution, and their evolution (Mair, Frye, & Marone, 2002). Studies by Menendez et al. (1996) showed that in a brittle regime, shear localization does not develop under compression until post-peak strength and is characterized by very little prior intragranular cracking. In addition, before reaching peak stress, dilatancy during deformation is due to shear rupture of cemented grain boundaries (Menendez et al., 1996). The very high density of intragranular microcracking (i.e., Hertzian fracture) and pronounced stress-induced microscopic post-failure are consequences of shear localization and compactive processes operative inside the shear band (Menendez et al., 1996). In contrast, throughout a cataclastic flow regime, Hertzian fracturing is a primary cause for shear enhanced compaction and strain hardening (Menendez et al., 1996). In other words, the propagation of shear bands under pre-peak conditions is not impeded by their geometry, unlike brittle fractures (Schultz & Balasko, 2003).

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Extension

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FIG. 2 Schematic shows distinction between deformation bands and slip planes. Deformation can occur under compaction, shear, or extension due to strain localization. Particles within the deformation bands tend to be smaller, more compact, with stronger preferred orientations, and more elongate shapes than particles outside the band. Modified from Aydin, A., Borja, R. I., & Eichhubl, P. (2006). Geological and mathematical framework for failure modes in granular rock. Journal of Structural Geology, 28, 83–98.

It is possible to detect the onset of strain localization in the laboratory microscopically prior to peak stress during axial compression (Besuelle, 2001; Regnet et al., 2015). Laboratory compaction on sandstone samples displayed discrete compaction bands similar to deformation bands observed in the field (Baud, Klein, & Wong, 2004). The discrete bands were associated with episodic surges in acoustic emission activity corresponding to strain hardening trends (Baud et al., 2004). Shear bands, such as cataclastic deformation bands, typically show shear displacements less than about 10 cm, which is much smaller than seismic resolution and therefore cannot be identified by seismic imaging on a large scale (Fowles & Burley, 1994). Olsson and Holcomb (2000) measured acoustic emissions from crack initiation and propagation during axial compression to track compaction strain localization zones. By correlating strain measurements and acoustic emissions during triaxial tests on Castlegate sandstone, Olsson and Holcomb (2000) also observed that compaction localization progressed as a propagation front 2 cm thick. Therefore, laboratory-scale

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studies of microstructure and their influence on porosities and permeabilities can be extremely invaluable (Cilona et al., 2012, Regnet et al., 2015, Wong & Baud, 1999, Ogilvie & Glover, 2001). A major challenge, however, remains in translating laboratory-scale investigations and interpretations to field-scale realities and observations. Baud, Vajdova, and Wong (2006) conducted comprehensive sets of experiments on saturated Bentheim sandstone samples to investigate how cataclastic flow and compaction localization develop in watersaturated sandstones. Cylindrical sandstone samples were subjected to triaxial compression at an axial strain rate of 1.3  10 5 s 1 under a range of confining stress of 40–350 MPa, while monitoring strain localization development and porosity loss using acoustic emission signatures. Their findings confirmed previous studies that the critical stress for the onset of shearing enhanced compaction is inversely proportional to porosity and grain size. Baud, Wong, and Zhu (2014) further examined the evolution of shear-enhanced compactive yield behavior using constitutive modeling. Their results showed that as the porous sandstone transitioned from the brittle to the ductile regime, the failure mode changed from shear band to compaction band to homogeneous cataclastic flow. Unlike observations in classical critical state soil mechanics, there was no separation of the yield behavior into a regime associated with compactancy at high mean stress, and a second one associated with dilatancy at low mean stress. Also unlike during inelastic compaction of soils, the porosity of the sandstone did not reduce to a “critical state” serving as a transition between compactancy and dilatancy, due to non-associative flow behavior. Therefore, the shear-enhanced compaction observed was appreciably much more than predicted by theory as a consequence of inelastic behavior over a broad range of effective pressures.

Mechanisms of strain localization in rock-soil mixtures Strain localization can also occur under mixed conditions, such as weathered or highly altered rock slopes at relatively more rapid timescales. When granular materials are loaded, deformation progresses systematically from localized to more pervasive geometry with increasing confining stress, reflecting gradual transitions between brittle and semi brittle behavior (Mair et al., 2002). Mandl, DeJong, and Maltha (1977) used ring shear tests on granular material to investigate changes in texture and stress states of shear zones. Mandl et al. (1977) observed Coulomb-type slip during the deformation of softened shear zone material. Crosta, Imposimato, and Roddeman (2003) applied numerical modeling using associated and non-associated flow rules from plasticity theory to calculate the runout of large landslides. The landslide materials comprised

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highly fractured and tectonized diorite, gabbro, ortho-quartzite, amphibolite and fine-grained gneiss, alluvial and lacustrine sediments, phyllites, glacial till silty-sandy cataclastic zone. Heavy rainfall and seismic loading were used independently as slope instability triggering mechanisms. The authors noted the challenge presented by the broad range of grain sizes— from massive blocks to very fine particles (clay size). Numerical modeling could thus help alleviate some of the morphologic and physicalmechanical constraints by selection of suitable boundary conditions, constitutive laws, and material laws. Their results showed a progressive continuous fracturing and subdivision during flow and degradation of mechanical properties during mass flow. Further, they showed that depth-averaging techniques to simulate run-out can be inconsistent with morphologic features such as sharp changes in flow direction; and thus an approach that considers and analyses internal material strains is more meaningful than an approach that does not. Crosta et al. (2003) concluded that accumulating debris cover during flow can increase mobility through mechanisms such as undrained loading, strain localization, and sediment and water entrainment. Darve and Laouafa (2000) showed numerically that it is possible for a globally stable slope to experience catastrophic failure due to localized perturbations. Such perturbations might include a reduction in effective stress due to elevated pore pressures, experiencing cohesion loss after heavy rainfall, or a bifurcation of strain from quasi-homogeneous for the entire slope to strictly localized and shear banding. Darve and Laouafa (2000) explained that in cases where slope angles are low (i.e., less than about 14 degrees), the behavior of landslides cannot be explained by accounting for plastic strain localization alone. In those instances, unstable landslide behavior should be evaluated mechanically also, by considering the axisymmetric stress state conditions inside the plastic limit condition. Iverson, Baker, and Hooyer (1997) studied the deformation of subglacial till with contrasting clay content using a specialized-ring shear device capable of shearing large annular specimens to high strains at variable rates. The distribution of shear strain and measurements of local stresses normal to the direction of shear could be monitored continuously. Experimental results showed that the percentage of clay present had important effects on mechanical behavior during shear. Subglacial tills (or comparable buried sediments) with high clay contents under similar effective stress conditions would be expected to be weaker than coarser-grained tills. Further, the shear strength of tills with high clay content would be expected to be much less sensitive to in-situ water-pressure fluctuations. Although strain localization was observed in both low (4%) and high (32%) clay content tills, the residual strength of the latter was less than the former by a factor of 2. The amount of clay present had no effect on

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whether strain localization occurred or not, and neither type of till behaved like a linear or Bingham-viscous fluid. Rudnicki (1984) explored the effects of dilatant hardening on the development of shear bands in fissured rock masses. Using an analytical technique, the shear of an inelastically deforming rock mass with an initially weakened layer was examined. The formulation consisted of combined pure shear and uniaxial compression and was built on previous work by Rudnicki and Rice (1975) demonstrating that homogeneous dilatantly hardened deformation grows unstable when the conditions for localization of deformation are suitable. In other words, infinitesimal spatial perturbations led to exponential growth of shear strain and pore pressure at peak-drained shear stress. Rudnicki’s (1984) work showed that dilatant hardening effects are significant in delaying the development of concentrated shear deformation within an embedded weak layer. Instability in the embedded layer may not occur at the peak of its drained response curve, but when the weak layer reaches the peak of its undrained response curve instead as a result. The results suggest a rapid acceleration of strain within the weakened layer and diminish with distance from the zone of weakness. As the relatively stronger material experiences unloading due to a prescribed material velocity at the boundary of the rock mass, the strain imposed on it is decreased. The decreased strain in the stronger material results in accelerating the strain in the weak layer even further. Rudnicki (1984), however, noted that his assumption of combined pure shear and uniaxial compression was particularly favorable for localization, given that the development of localized deformation is dependent on the deformation state. Gylland, Jostad, and Nordal (2014) investigated the failed state material behavior of Norwegian soft clay in slopes and the development of strain localization induced by strain softening. The authors used a modified triaxial cell which allowed for strain localization and shear band formation under undrained conditions. Their results indicated that during shear band initiation and development, there was a non-uniform material response attributed to both softening and hardening regimes within individual samples, leading to localized zones. Shear resistance was found to increase with strain rate for nonlocalized samples, while the opposite was reported for localized samples. The contrast was explained as a result of pore pressure gradients which were built up due to the shear bands in the localized samples and validated by numerical simulations. Also observed were attenuation of X-ray scans within shear bands indicating increased densities and reduced porosities. The diminished porosities were interpreted as an indication of a contractant material response at failure, and hence the migration of water. The authors concluded that the timing and rate-dependent nature of soft clays is significant in the initiation and evolution of landslides in these materials.

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Lade (1992) conducted soil triaxial tests and slope stability modeling under drained and undrained conditions to study possible stress paths inside the failure surface envelope. The size of local instability would be expected to increase with slope height. Relatively smaller (less than about 90 ft. high) slopes would be unlikely to acquire sufficient stresses to critical stress regions. For larger slopes, perturbations such as extra rainwater could increase the stresses. Lade (1992) further determined that upon the formation of a local zone of instability, the resulting porepressure build-up continued propagating leading to an even larger region of instability within the slope and likely to failure by static liquefaction. Lade (1992) concluded that for loose, fine sands and silts, unstable behavior under undrained conditions could occur as the soil compresses during undrained shear. This can occur as a result of hardening inside the failure surface, resulting in large plastic strain, under decreasing stresses.

Strain localization and pore pressure Deformation bands may be local or regional, ranging in size from less than a millimeter to hundreds of meters long, with lower porosities than original parent material (Aydin, 1978; Fowles & Burley, 1994; Pittman, 1981). Consequentially, deformation bands and slip planes at a large scale can significantly modify pore space and the corresponding fluid flow properties (up to two or more orders of magnitude) enabling zones to act as seals or conduits (Antonellini & Aydin, 1994; Gibson, 1994; Issen & Rudnicki, 2000; Main, Mair, Kwon, Elphick, & Ngwenya, 2001). Phyllosilicate bands and cataclastic deformation bands (Fig. 3) (Torabi, 2007) show the largest reduction in pore space, while disaggregation bands, where non-cataclatisc granular flow is dominant, show little influence (Fossen, Schultz, Shipton, & Mair, 2007).

FIG. 3 Graph superimposed on a cross-section image of a petrographic sample showing porosity variations along individual deformation band lengths due to variations in microstructure. Modified from Torabi, A., Braathen, A., Cuisiat, F., & Fossen, H. (2007). Shear zones in porous sand: Insights from ring-shear experiments and naturally deformed sandstones. Tectonophysics, 437, 37–50.

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Reductions in porosities under compression have been related to rock grain size, grain shape, grain mineralogy and sorting (Chauhan, Kjeldstad, Bjørlykke, & Høeg, 2002; Mair, Main, & Elphick, 2000), and elevated fracture strengths (Baud et al., 2014; Dunn, LaFountain, & Jackson, 1973). Crawford (1998) used a laser particle sizing technique to measure reductions in shear-band permeabilities of laboratory sandstone samples under triaxial compressive stress states, with changes ranging from 2.5 to 3.5 orders of magnitude. Flodin, Prasad, and Aydin (2003) used field and laboratory studies to show that deformation style was controlled by porosity and that the nature of intergranular bonds had an impact on compressional (P-) and shear (S-) wave velocities. Torabi (2007) used low order, spatial correlation functions to analyze high resolution, high magnification, back scatter images to detect porosity variations along individual deformation band lengths due to variations in microstructure (Torabi, 2007). Strain localization in deformation bands is known to be sequential (Friedman & Logan, 1973; Mair et al., 2000; Main et al., 2001; Olsson, Lorenz, & Cooper, 2004). For instance, Friedman and Logan (1973) captured different states of deformation banding in Solenhofen Limestone based on strain, strain rate and normal stress. It was thus possible to back analyze the normal stress based on the condition of the deformation band (Friedman & Logan, 1973). Sternlof, Rudnicki, and Pollard (2005) utilized displacement discontinuity boundary element modeling of compaction bands to back-analyze induced stress fields and porosity loss. Lothe, Gabrielsen, Bjørnevoll-Hagen, and Larsen (2002) monitored the progressive development of progression bands in triaxial compression tests on sandstone using P- and S-wave velocities and identified five main stages: (1) closure of micro cracks and pores, (2) tighter grain packing perpendicular to maximum stress direction, (3) reduction in P-and S-wave velocities during progression of micro cracking and tighter grain packing, (4) secondary fracturing, and (5) development of a slip plane. Strain energy relationships have been applied to predict deformation band nucleation, where drastic changes in porosities are encountered (Okubo & Schultz, 2005, 2006). Therefore, by analyzing the condition and character of strain localization at each different sequences and scales of deformation, it is possible in principle to evaluate the pore size at that stage, and the corresponding hydrological and geotechnical conditions. Pore space and volumetric reduction can have significant implications for pore pressures. For instance, Viesca and Rice (2012) showed that for submarine landslides, localized high pore pressure distributions can lead to accelerated growth of a slip surface resulting in run-away effects of downward mass movement, under conditions where friction weakens with slip. Continued acceleration of such slopes is dependent on whether there are further increases in pore pressure or on whether there are other

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mechanisms available to arrest the rapid slip growth. The pore pressure increases have been found to be proportional to the length of the induced rupture surface and landslide dimensions.

Example case study: The Oso landslide Description of the Oso landslide On Saturday, March 22, 2014, the Oso Landslide struck Snohomish County, Washington, resulting in 43 fatalities and several injuries. During a follow-up field reconnaissance investigation by a team of interdisciplinary experts (sponsored by the National Science Foundation’s Geotechnical Extreme Events Reconnaissance (GEER) program), multiple field information was collected including geologic conditions, climatic data, eye witness accounts, lidar, aerial photographs, and seismologic data. The details of the report can be found in (Keaton et al., 2014) and include discussions on likely triggering mechanisms, landslide initiation, mobilization, and landslide transition behavior into a rapidly moving debris flow. The clues are indicative of likely rapid strain localization occurrence. Keaton et al. (2014) and Wartman et al. (2016) describe the Osos landslide as “…the remobilization of an existing landslide, engaging the ancient slide surface (phase 1), and retrogressive rotational slides (phase 2)…” Although the nucleation of the failure surface can explain the development of the failure mechanism, it is not a prerequisite for the sudden acceleration and development of the catastrophic failure which occurred and has attracted the interest of the research community (e.g. Iverson et al., 1997; Iverson & George, 2016; Wartman et al., 2016). In light of observed sand boils (signs of confined elevated water pressure reaching or exceeding total overburden pressure leading to liquefaction at depth) at the site in Zones E and F (Fig. 4), localized pore pressure jumps may have led to strain localization. Previous studies, such as Viesca and Rice (2012), have confirmed that it is possible for the growth of a landslide slip (failure) surface to be driven by locally elevated pore pressure under submerged conditions, in a manner which is similar to earthquake nucleation. Keaton et al. (2014) further reported that: “…no significant seismic activity in the days preceding the landslide and therefore it is unlikely that it had a siesmogenic origin. Instead, it is highly probable that the intense 3-week rainfall that immediately preceded the event played a major role in triggering the landslide… This corroborates with our data found during the reconnaissance, which provides evidence of multiple stages of failure.” The field reconnaissance conducted by the GEER team after the Oso landslide (Keaton et al., 2014) observed several additional phenomena,

FIG. 4 The main features of the Oso landslide as mapped by the field investigation team showing the six major zones (Zones A–F). Modified from Keaton, J.R., Wartman, J., Anderson, S.A., Benoıˆt, J., de la Chapelle, J., Gilbert, R., Montgomery, D.R. (2014.) The 22 March 2014 Oso landslide. Washington, Geotechnical Extreme Events Reconnaissance Association Report GEER-036, http://www.geerassociation.org/GEER_Post%20EQ%20Reports/Oso_WA_2014/.

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described in “Effects of strain localization on Oso landslide activity” section, which are indicative of strain localization occurrence during Stages 1 and 2 of the landslide, including features and characteristics which have been reported during submarine landslides by others, related to elevated pore pressure responses and strain localization (e.g., Lade, 1992; Viesca & Rice, 2012).

Effects of strain localization on Oso landslide activity The GEER team made detailed observations and records after the Oso landslide in each of the six zones (Fig. 4). Zone A consisted of recessional outwash and glacial till. The observations made in Zone A included seepage at sharp contacts between oxidized recessional outwash and unoxidized gray glacial till, constant raveling of noncohesive material from unstable head scarp face, evidence of down-drop, back rotated block from orientation of trees, and blocky debris field of intact blocks of till up to 10 ft. (>3 m) high. Zone B consisted of extensional field of back-rotated blocks marked by transverse ridges and depressions. Observations made in Zone B included slicken slides on rotational blocks, standing water infilling several depressions, near-surface gray glacioluctustrine deposits, varved clays and silts, and side ejecta emanating from cracks near high ridges. Zone C consisted of debris flows along margins and exposed flat-lying lacustrine deposits. Observations made in Zone C included seepage and internal erosion (or piping). Zone D consisted of sheared dark lacustrine sediments beneath approximately 5 ft. of glacial outwash. Zone E consisted of high standing blocks, oxidized sand, and gravel colluvium, original forest floor with rooted in-place ferns, and lacustrine deposits. Observations made in Zone E included splashed veneer on both sides of the contact between Zones D and E, sand ejecta, sand boils and evidence of liquefaction. Zone F consisted of gray silty debris flow with disintegrated blocks of exotic material. Observations made in Zone F included soft deposits, sand boils and liquefaction, and a ripped asphalt roadway surface in two locations. Iverson et al. (1997) interpreted the Oso landslide as a high mobility, debris avalanche-flow (or flow slide in geotechnical engineering terminology), based on an (length/maximum height) L/H value of 9.5. The high mobility was unusual for a landslide only 180 m high, with a slope of less than 20 degrees. The researchers attributed the unusual high mobility to liquefaction of wet basal sediments. Rapid compressional loading of wet basal sediment from collapsing material upslope combined with the initial porosity of basal sediments is believed to have triggered the liquefaction. Iverson and George (2016) simulated landslide liquefaction and mobility bifurcation at Oso based on a computational evolution model of coupled

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solid volume fraction and basal pore pressures. They found that the ultimate mobility and behavior of the landslide was highly dependent on the initial soil conditions. Slight changes in initial soil conditions including basal slip surface geometry, basal friction angle, critical state solid volume fraction, and pore pressure had significant impacts, resulting in landslide mobility bifurcation dynamics. Wartman et al. (2016) concluded that landslide failure at Oso occurred in two stages triggered by 3 weeks of rainfall, alteration of previous hydrogeological regime from previous landslides, weakening of landslide mass due to previous landsliding, and changes in stress redistribution due to removal and re-deposition of material. The first stage was interpreted as a remobilization of a 2006 landslide which mobilized as a debris avalanche/flow. The second stage occurred 2 min later in response to the unloading and involved headward extension into previously unfailed material. Similarly, the observations recorded after the landslide by the field reconnaissance teams can be examined for possible clues for strain localization mechanisms. The conditions required for strain localization are highly prescriptive, as discussed in “Mechanisms of strain localization and their influence on the behavior of geomaterials” section. Correlations can be drawn between select, key observations at Oso, and strain localization mechanisms discussed in “Mechanisms of strain localization and their influence on the behavior of geomaterials” section. The combined steep slope geometry at Oso, elevated pore pressures, and the evidence of internal erosion provide strong clues for possible strain localization effects. For landslide slopes greater than about 14 degrees, localized perturbations might include reduction in effective stress due to elevated pore pressures, experiencing cohesion loss after heavy rainfall or a bifurcation of strain from quasi-homogeneous for the entire slope to strictly localized and shear banding. Relatively smaller (i.e., less than about 25 m high) slopes would be unlikely to acquire sufficient stresses to critical stress regions. For large slopes, perturbations such as extra rainwater may increase stresses. The presence of unconsolidated till fragments in Zone A and glacial outwash above glaciolucustrine sediments in Zone B provides a recipe for a strain localization mechanism, where elastoplastic material behavior is exhibited (i.e. beyond elastic limit) under high confining stress. The observed down-drop, back rotation in Zones A and B, could have created compaction and the presence of thinning and extensional field could have led to shear in Zone B. Evidence of shear in the gray-to-blackish lacustrine material exposed in Zone D supports the notion that a shear or compaction strain localization mechanism was possible in Zone D. A weak lacustrine sediment layer is present in Zone D, and it is well known that soft soil behavior is strain rate-dependent. Strain localization can be induced by strain softening in soft clays, which can affect the behavior of landslides. Previous studies (“Mechanisms of

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strain localization and their influence on the behavior of geomaterials” section) indicate that during shear band initiation and development, a non-uniform material response attributed to both softening and hardening regimes within a loaded region can lead to localized zones. Since the critical stress for the onset of shearing enhanced compaction is inversely proportional to porosity and grain size, the wide range of grain sizes of the glacial deposits and outwash present in all the zones suggest that small changes in stress state would be sufficient to create shearing-enhanced compaction. Lacustrine clay sediments are present in Zones B, C, and D, and the effects of the overlying load of coarser till fragments and colluvial materials provide an additional recipe for strain localization. Subglacial tills (or comparable buried sediments) with high clay contents under similar effective stress conditions would be expected to be weaker than coarser grained tills. In previous studies discussed in “Mechanisms of strain localization and their influence on the behavior of geomaterials” section, strain localization occurred whether the amount of clay was high or not and was observed in both low (4%) and high (32%) clay content tills. However, the shear strength of tills with high clay content would be expected to be much less sensitive to in-situ water-pressure fluctuations. A follow-up question would be whether strain localization occurred in the lower regions of the landslide (e.g. Zone F). Mass flow was observed in Zones C and F, which were characterized by debris flows, and there is evidence of localized perturbations, elevated pore pressures, and rapid shear strain rate. It is generally accepted that changes in material properties and strain localization are possible during debris flows. As discussed in “Mechanisms of strain localization and their influence on the behavior of geomaterials” section, previous studies have shown a progressive continuous fracturing and subdivision during flow as well as degradation of mechanical properties during mass flow (including possible strain localization). Further, the sand boils observed in Zones E and F, and the sand deposit ejecta observed in Zones B and E were probably expressions of the same phenomena, i.e., static liquefaction. According to Lade (1992), localized, pore pressure build-up-induced instability will propagate adversely leading to slope failure by static liquefaction. The correlations amount to evidence supporting the likelihood of a mechanism responsible for strain localization during Stage 1 or Stage 2 of the landslide. Stage 1 was interpreted by the response team of field investigators as an initial unloading and remobilization of previous landslide deposits, while Stage 2 was interpreted as a retrogression of the slip surface toward a preexisting Stage 1 slip surface due to readjustments of stress states and groundwater seepage forces (Fig. 5) (Keaton et al., 2014.) However, additional studies are needed to further support the correlated field observations with strain localization occurrence during the various reported stages.

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FIG. 5 Interpreted cross-section profile of the Oso landslide showing Stages 1 and 2 of the failure. Stage 1 was interpreted as the remobilization of old existing landslide material, engaging the ancient slide surface, and Stage 2 consisted of retrogressive rotational slides. The positions of the river in 2013 and 2014 are also shown. Modified from Keaton, J.R., Wartman, J., Anderson, S.A., Benoıˆt, J., de la Chapelle, J., Gilbert, R., Montgomery, D.R. (2014.) The 22 March 2014 Oso landslide. Washington, Geotechnical Extreme Events Reconnaissance Association Report GEER-036, http://www.geerassociation.org/GEER_Post%20EQ%20Reports/Oso_ WA_2014/; Wartman, J., Montgomery, D.R., Anderson, S.A., Keaton, J.R., Benoıˆt, J., de la Chapelle, J., & Gilbert, R. (2016). The 22 March 2014 Oso landslide, Washington, USA. Geomorphology, 253, 275–288.

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Summary and conclusions This chapter provides a detailed review of strain localization mechanisms in different types of geomaterials, including their roles in slope instability problems such as landslides. Triggering mechanisms and necessary conditions for strain localization to occur are discussed. In summary, strain localization in geomaterials can occur either gradually or very rapidly based on prevailing conditions and may lead to slope instabilities. Irrespective of timescale for development, strain localization can avail a three-dimensional interface (i.e., preexisting or new failure surface) for slip in slopes and landslides and is possible under a variety of loading conditions in soil, rock, or mixed rock-soil conditions. The broad range of possibilities can be explained due to conditions such as differential stress states imposed upon heterogeneous materials. During the Oso landslide which occurred on Saturday, March 22, 2010, resulting in 43 fatalities and several injuries, the GEER investigation team observed sand boils (signs of confined elevated water pressure reaching or exceeding total overburden pressure leading to liquefaction at depth) in Zones E and F of the landslide, raising the question as to whether strain localization occurred during Stage 2 of the failure triggering the 300 m length shearing. The question is triggered by similar conditions reported in previous studies (e.g., Viesca & Rice, 2012), given that locally elevated pore pressures can drive landslip surface growth in a manner not unlike that observed during earthquake nucleation. If the observations at Oso are examined in light of the requirements and broad range of conditions required for strain localization to occur, some similarities can be noted. The presence of unconsolidated till fragments in Zone A and glacial outwash above glaciolucustrine sediments in Zone B implies available poorly lithified sediments likely to exhibit elasto-plastic material behavior and strain localization. Evidence of shear was observed in the lacustrine material exposed in Zone D. It is also possible that the backdrop rotation in Zones A and B could have created compaction, as compaction strain localization mechanisms have been reported in previous studies. Further, the wide range of grain sizes of the glacial deposits suggests that minor changes in stress states would be sufficient to create shearing-enhanced strain localization. The debris flows observed in Zones C and F could also have led to changes in material properties and possible strain localization. Other observations and possible clues include the high landslide slope angles, the weak embedded clays, the localized elevated pore pressures, the variable shear strain rates, material unloading during mass flow, and groundwater seepage observed near the toe of the Oso landslide. These observations provide evidence to support the likelihood that strain localization may have occurred at Oso during failure during Stage 1 or Stage 2. However, further investigations and studies are required to support this claim.

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References Antonellini, M., & Aydin, A. (1994). Effect of faulting on fluid flow in porous sandstones: Petrophysical properties. AAPG Bulletin, 78, 355–377. Aydin, A. (1978). Small faults formed as deformation bands in sandstone. Pure and Applied Geophysics, 116, 913–930. Aydin, A., Borja, R. I., & Eichhubl, P. (2006). Geological and mathematical framework for failure modes in granular rock. Journal of Structural Geology, 28, 83–98. Aydin, A., & Johnson, A. M. (1983). Analysis of faulting in porous sandstones. Journal of Structural Geology, 5, 19–31. Baud, P., Klein, E., & Wong, T. F. (2004). Compaction localization in porous sandstones: Spatial evolution of damage and acoustic emission activity. Journal of Structural Geology, 26, 603–624. Baud, P., Vajdova, V., & Wong, T.-f. (2006). Shear-enhanced compaction and strain localization: Inelastic deformation and constitutive modeling of four porous sandstones. Journal of Geophysical Research - Solid Earth, 111.B12. Baud, P., Wong, T. F., & Zhu, W. (2014). Effects of porosity and crack density on the compressive strength of rocks. International Journal of Rock Mechanics and Mining Sciences, 67, 202–211. Bernabe, Y., & Brace, W. F. (1990). Deformation and fracture of Berea sandstone. In A. G. Duba, W. B. Durham, J. W. Handin, & H. F. Wang (Eds.), The brittle-ductile transition in rocks. Washington, DC: American Geophysical Union. Besuelle, P. (2001). Evolution of strain localisation with stress in a sandstone: Brittle and semi-brittle regimes. Physics and Chemistry of the Earth, A26, 101–106. Borja, R. I. (2004). Computational modeling of deformation bands in granular media. II. Numerical simulations. Computer Methods in Applied Mechanical Engineering, 193, 2699–2718. Borja, R. I., & Aydin, A. (1994). Computational modeling of deformation bands in granular media. I. Geological and mathematical framework. Computer Methods in Applied Mechanical Engineering, 193, 2667–2698. Borja, R. I., & Aydin, A. (2004). Computational modeling of deformation bands in granular media. I. Geological and mathematical framework. Computer Methods in Applied Mechanical Engineering, 193, 2667–2698. Cashman, S., & Cashman, K. (2000). Cataclasis and deformation-band formation in unconsolidated marine terrace sand, Humboldt County. Geology, 28, 111–114. Chase, B. R., Kehew, A. E., & Montgomery, W. W. (2001). Determination of slope displacement mechanisms and causes using new geometric modeling techniques and climate data. In R. S. Harmon, & W. W. Doe (Eds.), Landscape erosion and evolution modeling. New York: Kluwer Academic Publishers. 540p. Chauhan, F. A., Kjeldstad, A., Bjørlykke, K., & Høeg, K. (2002). Experimental compression of loose sands simulating porosity reduction in petroleum reservoirs during burial. Canadian Geotechnical Journal, 40, 995–1011. Chen, J., Yin, J.-H., & Lee, C. F. (2004). Rigid finite element method for upper bound limit analysis of soil slopes subjected to pore water pressure. Journal of Engineering Mechanics, 130(8), 886–893. Cilona, A., Baud, P., Tondi, E., Agosta, F., Vinciguerra, S., Rustichelli, A., et al. (2012). Deformation bands in porous carbonate grainstones: Field and laboratory observations. Journal of Structural Geology, 45, 137–157. Cooper, M. R., Bromhead, E. N., Petley, D. J., & Grant, D. I. (1998). The Selborne cutting stability experiment. Geotechnique, 48, 83–101. Crawford, B. R. (1998). Experimental fault sealing: Shear band permeability dependency on cataclastic fault gouge characteristics. M. P. Coward, H. Johnson, & T. S. Daltaban (Eds.), Structural geology in reservoir characterization (pp. 83–97). 127(pp. 83–97). London: Geological Society. Special Publications.

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Lothe, A. E., Gabrielsen, R. H., Bjørnevoll-Hagen, N., & Larsen, B. T. (2002). An experimental study of the texture of deformation bands; effects on the porosity and permeability of sandstones. Petroleum Geoscience, 8, 195–207. Main, I., Mair, K., Kwon, O., Elphick, S., & Ngwenya, B. (2001). Recent experimental constraints on the mechanical and hydraulic properties of deformation bands in porous sandstones: A review. R. E. Holdsworth, R. A. Strachan, J. F. Magloughlin, & R. J. Knipe (Eds.), The nature and significance of fault zone weakening (pp. 43–63). 186(pp. 43–63). London: Geological Society. Special Publications. Mair, K., Frye, K. M., & Marone, C. (2002). Influence of grain characteristics on the friction of granular shear zones. Journal of Geophysical Research, 107, 41–49. Mair, K., Main, I., & Elphick, S. (2000). Sequential growth of deformation bands in the laboratory. Journal of Structural Geology, 22, 25–42. Mandl, G., DeJong, L. N. J., & Maltha, A. (1977). Shear zones in granular material. Rock Mechanics, 9, 95–144. Menendez, B., Zhu, W., & Wong, T. F. (1996). Micromechanics of brittle faulting and cataclastic flow in Berea sandstone. Journal of Structural Geology, 18, 1–16. Mickelson, D. M., Edil, T. B., Bosscher, P. J., & Kendziorski, E. (1991). Landslide hazard assessment for shoreline erosion management. Landslides: (pp. 1009–1014). Rotterdam: AA Balkema. Montgomery, W. W. (1998). Groundwater hydraulics and slope stability analysis: Elements for prediction of shoreline recession. Unpublished Ph.D. dissertationDepartment of Geosciences, Western Michigan University. 256 p. Ng, C. W. W., & Shi, Q. (1998). A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Computers and Geotechnics, 22(1), 1–28. Nieuwenhuis, J. D. (1991). The lifetime of a landslide: Investigations in the French Alps. Rotterdam, The Netherlands: A.A. Balkema. 144 p. Ogilvie, S. R., & Glover, P. W. J. (2001). The petrophysical properties of deformation bands in relation to their microstructure. Earth and Planetary Science Letters, 193, 129–142. Okubo, C. H., & Schultz, R. A. (2005). Evolution of damage zone geometry and intensity in porous sandstone: Insight gained from strain energy density. Journal of the Geological Society, London, 162, 939–949. Okubo, C. H., & Schultz, R. A. (2006). Near-tip stress rotation and the development of deformation band stepover geometries in mode II. Geological Society of America Bulletin, 118, 343–348. Olsson, W. A. (1999). Theoretical and experimental investigation of compaction bands in porous rock. Journal of Geophysical Research, 104, 7219–7228. Olsson, W. A., & Holcomb, D. J. (2000). Compaction localization in porous rock. Geophysical Research Letters, 27, 3537–3540. Olsson, W. A., Lorenz, J. C., & Cooper, S. P. (2004). A mechanical model for multiply-oriented conjugate deformation bands. Journal of Structural Geology, 26, 325–338. Othman, M. A., & Benson, C. H. (1993). Effect of freeze–thaw on the hydraulic conductivity and morphology of compacted clay. Canadian Geotechnical Journal, 30(2), 236–246. Pittman, E. D. (1981). Effect of fault-related granulation on porosity and permeability of quartz sandstones, Simpson Group (Ordovician) Oklahoma. AAPG Bulletin, 65, 2381–2387. Regnet, J. B., David, C., Fortin, J., Robion, P., Makhloufi, Y., & Collin, P. Y. (2015). Influence of microporosity distribution on the mechanical behavior of oolithic carbonate rocks. Geomechanics for Energy and the Environment, 3, 11–23. Reid, M. E., Nielsen, H. P., & Dreiss, S. J. (1988). Hydrologic factors triggering a shallow hillslide failure. Bulletin of the Association of Engineering Geologists, 25(3), 349–361. Rudnicki, J. W. (1984). Effects of dilatant hardening on the development of concentrated shear deformation in fissured rock masses. Journal of Geophysical Research - Solid Earth, 89(B11), 9259–9270.

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C H A P T E R

11 Physical modeling of interaction problems in geotechnical engineering Subhamoy Bhattacharyaa, Hasan Emre Demircib, George Nikitasc, Ganga Kasi V. Prakhyad, Domenico Lombardie, Nicholas A. Alexanderf, M. Aleemc, Sadra Amanic, and George Mylonakisg,h,i a

Geotechnical Engineering, University of Surrey, Guildford, United Kingdom, b Geotechnical Engineering, Izmir Katip Celebi University, Izmir, Turkey, c SAGE Laboratory, University of Surrey, Guildford, United Kingdom, dSir Robert McAlpine Ltd, Hemel Hempstead, United Kingdom, eThe University of Manchester, Manchester, United Kingdom, fStructural Dynamics, University of Bristol, Bristol, United Kingdom, gUniversity of Bristol, Bristol, United Kingdom, hUniversity California Los Angeles, Los Angeles, CA, United States, iKhalifa University, Abu Dhabi, United Arab Emirates

Need for physical modeling The use of physical modeling in civil engineering design in general and geotechnical engineering, in particular, is not only well established but also highly valued in the engineering community. In geotechnical engineering, physical modeling is even more important as the main material under consideration is soil. Soil is a natural material that formed by various natural geophysical and biological processes over millennia. Soil is typically a multiphase, assembly of heterogeneous particles. Its bulk/ averaged constitutive characteristics are nonlinear, strain amplitudeand loading path-dependent and its cyclic/dynamic behaviors are challenging to characterize. As such there are large uncertainties in defining the soil response and its interaction with structural elements such as piled

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foundations. Even though soil is a particulate material made of an assembly of particles, it is conveniently modeled as a continuum medium that is characterized by macroscopic (averaged) parameters, such as strength and stiffness. The complicated stress/strain path history applied to soil in the realworld cases of large geophysical infrastructure projects introduces yet further large uncertainties in predicting coupled soil-structure behavior. Thus, physical modeling of geotechnical structures provides a vital tool for gaining insights into complex phenomena occurring within the soil and/or resulting from the interaction with embedded structures, such as foundations, tunnels, pipelines, etc. Therefore, testing physical models is necessary; however, full-scale tests are normally too expensive to justify. Thus, it becomes readily apparent that scaling down in size the real prototype into a small-scale model is required. This is to achieve a larger statistical sample, to be able to perform tests in a laboratory setting where sophisticated measurement techniques are permitted, and to optimize costs. However, the question remains that how does one design a scaled model such that results for it give credible insights into the behavior of the full-scale prototype geotechnical structure?

Why do we need scaled model tests? A nonexhaustive list of advantages in investigating geotechnical structures using physical modeling is provided below: (1) To gain new insights into unexplored complex phenomena that cannot be investigated in the real field or modeled in numerical simulations due to the lack of adequately accurate soil constitutive equations. One example includes the thermal conductivity of unsaturated clay which has a potential use in high-level nuclear waste repositories. A second example is understanding the cyclic behavior of foundations in shallow gassy sediments found in many offshore locations where wind turbines operate. (2) To verify the analysis/design methodologies of high-risk construction techniques and verify a particular (and novel) construction sequence. Examples include tunneling in soft clay in densely populated cities, offshore construction in challenging seabed conditions or pipeline mitigation techniques in seismic fault crossing zones. A specific example in the offshore category includes construction of suction caissons in deeper waters. (3) To validate a failure or collapse mechanisms (akin to forensic investigation) that can be used for settling liabilities, insurance claims or that can be used in arbitration or in a Court of Judgment. Fig. 1A shows the collapse of a pile-supported building in Shanghai and Fig. 1B shows the physical modeling of the problem. Here, the main idea is

Need for physical modeling

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FIG. 1 (A) Shanghai Building Collapse and (B) physical modeling of the problem, see Wang, Ng, Hong, Hu, and Li (2019).

(4)

(5)

(6)

(7)

to establish a chain or series of events that led to the collapse. In some cases, it is necessary to replicate the failure patterns. The readers are referred to Wang et al. (2019) for details on the cause of collapse of the depicted building. To validate a new hypothesis or a theoretical failure mode/ mechanisms. One example under this category is to verify if a pile can buckle (i.e., become laterally unstable and fail) under the axial load alone if the soil surrounding the pile liquefies during an earthquake. To verify different aspects of design for a new concept of foundation for which no codes of practice or guidelines exist. In other words, reducing the uncertainties in the design assumptions. It is often helpful to discuss by taking an example and is explained in the next paragraph using Fig. 2. Examples of particular design aspects for this specific problem of a new foundation can be modes of vibration or the modes of ultimate collapse. To understand the deformations of foundations under various loading cases [i.e., serviceability limit state (SLS) design] and linking the deformation to a soil element test. To develop design methods that may lead to the path of standardization.

FIG. 2 Transportation of the caisson and installation of the hybrid foundation. Source: https://www.offshore-energy.biz/first-monopile-caisson-hybrid-foundation-installed-at-chinese-off shore-wind-farm/.

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(8) To validate the constitutive models of soils by carrying our controlled scaled model tests. Example 1 Monopile-caisson hybrid foundation At the Fujian province offshore wind farm site (Putan Pinhhai Phase II) in China, engineers faced a daunting challenge of installing monopiles due to the presence of rock formations at a shallow depth. The presence of rock caused unexpected issues during monopile installation, including borehole collapse, jamming of a drilling tool and pile tip buckling. The engineers found a solution consisting of a hybrid shallow-deep foundation, which they named monopile-caisson hybrid foundation (see Fig. 2). The main concept is that the top plate of the caisson (i.e., shallow foundation) rests on the seabed while the monopile is fully embedded in the ground without touching the rock. There are no codes of practice or guidelines for such a hybrid foundation, and in these cases, scaled model tests become a necessity to validate numerical and analytical solutions.

Conventional geotechnical problems and complex interaction problems In routine physical modeling problems (such as consolidation of soft clays, settlement of foundations, liquefaction of different types of soils or bearing capacity problems, failure modes of shallow foundations under static loading) the model test can be designed using standard scaling laws available in textbooks in the form of tables. These problems may be characterized by fundamental soil properties, such as strength, stiffness, and permeability. In this context, it must be mentioned that there are broadly two types of small-scale testing: (a) scaled model testing at 1-g (single gravity) and (b) scaled model testing at multigravity in a geotechnical centrifuge. In the scaled model testing at 1-g (single gravity), the experiments are conducted on the laboratory floor under the action of the Earth’s gravity. In the scaled model (say 1:N where N is the scaling ratio) testing in a geotechnical centrifuge, the centrifugal force is used as a pseudo-gravitational force on the rotating model and thus it can be subjected to a multiple of gravity environment (typically N times Earth’s gravity for a 1:N model). Fig. 3A shows a schematic diagram of the operations of a geotechnical centrifuge test where a pile-supported structure is subjected to shaking under 50-g. Fig. 3B shows the photograph of the model. Further details can be found in PhD thesis of Bhattacharya (2003). In centrifuge physical modeling, the stresses in the soil are modeled correctly as far as practicable making it a very effective tool for studying various problems. On the other

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Need for physical modeling Package at 50-g

50-g

Arm of the centrifuge

1-g

Axis of the centrifuge Package at 1-g

Sw

in

gu

p

(A)

(B)

(C)

FIG. 3 (A) Schematic diagram of a centrifuge operation. (B) Photo of a centrifuge model studying single pile in liquefiable soils. (C) 1-g modeling of pile with mass at the top.

hand, Fig. 3C shows the 1-g modeling of the same pile problem as in Fig. 3A and B. The readers are referred to Lombardi and Bhattacharya (2014a) for further details. The scaling laws for 1-g testing for conventional geotechnical problems are summarized in Wood (2003), and the scaling laws for centrifuge modeling can be found in Schofield (1980) and Madabhushi (2014). Table 1 lists the scaling laws commonly used in centrifuge testing.

Interaction problems in geotechnical engineering In multidisciplinary problems and design issues (termed here interaction problems in the title of the chapter), physical modeling is often carried out to understand the hierarchy of importance of various physical effects for the problem under investigation. To describe these interaction

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11. Physical modeling of interaction problems in geotechnical engineering

TABLE 1 Scaling laws for centrifuge tests. Parameter

Model/prototype

Dimensions

Length

1/N

L

Mass

1/N3

M

Stress

1

ML
1 T
2

Strain

1

1

Force

1/N

MLT
2

Seepage velocity

N

LT
1

Time (seepage)

1/N2

T

Time (dynamic)

1/N

T

Frequency

N

1/T

Acceleration

N

LT
2

Velocity

1

LT
1

EI (bending stiffness)

1/N4

ML3 T
2

MP (Plastic moment capacity)

1/N3

ML2 T
2

2

problems in practical terms, we will consider a real problem consisting of a small-scale offshore wind turbine subjected to environmental loads (see Fig. 4). Offshore wind turbines can be considered as a long slender tower supporting a heavy mass (Rotor Nacelle Assembly) and three rotating blades. The foundation is subjected to a unique set of cyclic and dynamic loading conditions: (i) Loading history produced by the turbulence of the wind and the amplitude is function of the wind speed; (ii) Loading history caused by waves crashing against the substructure, the magnitude of which depends on the height and period of waves; (iii) Loading history caused by mass and aerodynamic imbalances of the rotor, whose forcing frequency equals the rotational frequency of the rotor (referred to as 1P loading in the literature); (iv) Loading history on the tower due to the vibrations caused by blade shadowing effect (referred to as 3P loading in the literature), which occurs as each blade passes through the shadow of the tower. The loads imposed by wind and wave are spatiotemporal, i.e., vary spatially across the structure and vary in time. Furthermore, the natural

211

Wind profile Blade passing frequency (3P)

Rotor frequency (1P)

Typical waveforms for the different type of loads:

Need for physical modeling

Wind load:

(time) Wave load:

(time) 1P load:

(time)

Sea level 3P load: Wave profile Mud-line (foundation level)

(time)

FIG. 4

Typical loads acting on an offshore wind turbine.

frequency of the whole system is very close to the predominant forcing frequencies making it dynamically sensitive. The number of cycles of loading applied to the model can be as high as 100 million in its lifetime. High fidelity numerical simulation of the problem involves Aero-ServoHydro-Dynamic Soil Structure-Interaction (ASH-DSSI). The behavior of offshore wind turbine structures depends on various interactions listed below: (1) Cyclic/dynamic aerodynamics loads on the foundation, i.e., Cyclic/ Dynamic Wind-Structure-Foundation-Soil-Interaction. The interaction frequencies are typically very low but the magnitude of loading can be high. It is noted that low and high are descriptive and problem-specific. In this context, “low” frequency is noninertial behavior (i.e., no dynamic amplification) and “high” means it is likely to drive the system into large soil strain behavior (i.e., strain level high enough to alter soil stiffness). (2) Hydrodynamic loads on the foundation, i.e., wave-structurefoundation-soil interaction. The interaction can be dynamic and the loads can be substantial which will necessitate high-cycle fatigue analysis of the system components. (3) There are dynamic loads due to the mass and aerodynamic imbalance of the blades.

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11. Physical modeling of interaction problems in geotechnical engineering

(4) Interaction due to wind and wave load misalignment. Due to the controlling action of the wind turbine (i.e., yaw action of the Nacelle), the wave and wind load may act in different directions. For a fully developed sea, the wave will mostly act in the direction of the wind (co-linear). However, to maintain a constant power, there will be controlling action of the turbine and the turbine may not face the wind directly causing a wind-wave misalignment. (5) Dynamic soil-structure interaction (SSI) with damping from many sources. A simplified loading pattern is shown in Fig. 4. Jalbi, Arany, Salem, Cui, and Bhattacharya (2019) demonstrate that when the effects of wind and wave are combined, the loading on the foundation can range between one-way to two-way and is a function of operating condition together with the environmental conditions. Thus, we are presented with a great challenge to physically model the whole system, even in a laboratory setting, with all its complex interaction problems. Aerodynamic problems are better modeled in a wind tunnel and hydrodynamic problems are studied in wave tanks. On the other hand, cyclic and dynamic SSI problems are best studied in a geotechnical centrifuge. Therefore, one may want to model the problem by placing a wind tunnel and a wave tank onboard the geotechnical centrifuge, but this is currently impossible. It has been pointed out in the literature that parasitic (unwanted) vibrations of a geotechnical centrifuge prevent modeling the cyclic and dynamic foundation-soil interaction. Furthermore, the damping due to the rotation of the whole package can add further uncertainties. In such cases, a physical modeling testing procedure needs to focus on “understanding the physics of the problem,” and this is schematically represented in Fig. 5 as the indirect route.

ASIDE Here the underlying assumption is the small-scale model is in its own right a small prototype. If one needs to imagine the prototype can think of wind turbines or buildings from Lilliput Island as depicted in Jonathan Swift’s novel Gulliver’s Travels. The focus

Experiments

FIG. 5

Scaling laws (standard tables)

Prediction of prototype

Understanding the physics/mechanics

Carry out numerical or analytical study

Physical modeling framework (Bhattacharya et al., 2011).

Need for physical modeling

213

of the experiments is to understand which underlying mechanics define the most significant/important system behavior and to find a mathematical expression of this “governing law.”

In Fig. 5, both types of scaling techniques are shown. In this context (the physical modeling of offshore wind turbine system such as the one in Fig. 2 for which the loading is shown in Fig. 4), the issues are as follows: (1) How is the load transferred from the structure to the ground through the foundation? This is essentially the load transfer mechanism covered in the ULS design. (2) What are the modes of vibration of the whole structure? This is a necessary component of a resonance avoidance strategy. (3) What are the long-term performance issues as more than 100 million cycles of loading of different frequencies/magnitudes will be applied to the model? This long-term performance issue requires further thoughts and is discussed in the next section. SSI can be cyclic as well as dynamic and will affect the following two main long-term design issues: (1) Whether or not the foundation will tilt progressively under the combined action of millions of cycles of one-way or two-way loads. The overall cyclic load on the foundation is asymmetric which depends on the site condition, i.e., relative magnitude of wind and wave components. It must be mentioned that if the foundation tilts more than the allowable, it may be considered failed based on SLS criteria and may also lose the warranty from the turbine manufacturer. (2) Repeated cyclic or dynamic loads on soil causes change in the properties which in turn can alter the stiffness of the foundation. A wind turbine structure derives its stiffness from the support stiffness and any change in natural frequency will have consequences leading to loss of years of service, which is to be avoided. Therefore, the physical modeling should focus on achieving understanding of the above issues and any other issues identified. However, it must be remembered that not all the mechanisms can be appropriately modeled or understood from a particular test and this calls for intelligent design of tests. The question that must always be remembered is “why are we doing this test?” What physics or mechanics are we trying to understand/confirm? Noting that tests are resource hungry in terms of human resources, time-consuming to set up with significant materials, fabrication, instrumentation, and laboratory costs. For example, the modes of vibration of a wind turbine structure is an important aspect of design and can be studied by making a scaled model

214

11. Physical modeling of interaction problems in geotechnical engineering

where the mass and stiffness ratio of the scaled model and the prototype are maintained. One must not assume that this same model will provide answers to the very different question on the long-term progressive tilt prediction. Define the question you are exploring, then define the appropriate scaling laws. Different questions may require different scaling laws. The example of mass similarity is explained through an example and is shown schematically in Fig. 6. To model the vibration of the tower, the mass distribution between the different components needs to be preserved and can be achieved through Eq. (1) where the terms are explained in Fig. 7. In other words, the M1:M2:M3:m in Fig. 7 needs to be maintained

FIG. 6

Asymmetric tripod (model and a prototype).

m (mass of one blade)

L

M3 (mass of the onboard machinery)

M2 (mass of the tower)

M1 (mass of the foundation frame)

b FIG. 7 2013).

Schematic diagram for multipod foundation wind turbines (Bhattacharya et al.,

Need for physical modeling

215

in the scaled-model and prototype. The following relationship based on mass similarity needs to be maintained: ðM1 : M2 : M3 : mÞmodel ¼ ðM1 : M2 : M3 : mÞprototype

(1)

However, for modes of vibration, mass similarity is also not the only criterion, and the stiffness terms must also be preserved. One way to understand the different terms that are required in physical modeling is to carefully note the terms in the governing equations of motion and they are mass, stiffness, and damping. For the particular problem shown in Fig. 7, the (L/b) ratio needs to be maintained and this is often known as Geometric Similarity. In other words, the dimensions of the small-scale model need to be scaled corresponding to the prototype to ensure that the similar modes of vibration will be excited. Rocking modes of vibration will govern the multipod (tripod or tetrapod suction piles or caissons) foundation and as a result relative spacing of individual pod foundations (b in Fig. 7) with respect to the tower height (L in Fig. 7) needs to be maintained. On the issue of general stability (i.e., the transfer of load from the structure to the supporting soil) one needs to look into the load transfer mechanism and (L/b) is important. The aspect ratio of the caisson (diameter to depth ratio) should also be maintained. By controlling this ratio, the relative load-carrying capacity of the end bearing and skin friction can be maintained. In addition, the flow path length of both model and prototype caissons will also be maintained to ensure the pore water flow is representative (if the soil porosity is maintained). This leads to the similitude relationship given by Eq. (2).

 L L ¼ (2) b model b prototype where L is the length of the tower and b is the spacing of the caissons

 D D ¼ (3) h model h prototype where D is the diameter of the caisson and h is the depth of the caisson.

Steps in physical modeling The design and interpretation of small-scale models require the assessment of a set of laws of similitude that relate the model to the prototype structure. These can be derived from differential equations and/or dimensional analysis from the assumptions that every physical process can be expressed in terms of nondimensional groups and the fundamental aspects of physics must be preserved in the design of model tests.

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11. Physical modeling of interaction problems in geotechnical engineering

The necessary steps associated with designing such a model can be stated as follows: (1) Step-1: What are the potential failure mechanisms or processes that are likely to occur? In other words, what are we trying to find? Care needs to be taken for the cases where ones a priori assumptions preclude certain system behavior of potential interest in the prototype. Using the example of offshore wind turbine—one vital aspect are the modes of vibrations. Which mode or modes are important to use for similitude scaling laws? Do we assume the flexible tower is coupled with an equivalent rocking foundation soil spring? Or do we neglect the soil rocking foundation soil spring (in our similitude laws) by assuming the foundation is just too rigid? This step is PHYSICS or MECHANICS based. (2) Step-2: Deduction of the relevant nondimensional groups for the identified mechanisms or processes in Step-1. This step can be carried out in many ways. We can analytically derive the approximate governing equations and then use a nondimensionalization procedure using simple algebraic manipulation. Alternatively, the Buckingham Pi theorem can be used to derive, conceptually, compatible dimensionless grouping spanning the dimensional space of the problem without the need to deriving the approximate governing equations explicitly. This step is MATHEMATICS based and often finite element analysis (FEA) may be used for sanity check of the analytical solutions. (3) Step-3: Ensure that the set of crucial scaling laws (which are essential) are simultaneously conserved between model and prototype through pertinent similitude relationships. (4) Step-4: Identify scaling laws that are approximately satisfied, and those which are violated and which therefore require special consideration. Examples of the latter are stiffness and dilatancy of soils in 1-g testing, and small amplitude vibration monitoring and damping in centrifuge modeling. Once the nondimensional groups are identified, scaled tests need to be designed to check the nonlinearity among those groups and compare with numerical results where applicable. These nondimensional groups can later be used to develop design charts. Fig. 8 shows a schematic diagram of the process discussed in this paragraph for a complex interaction problem.

ASIDE Spirit behind the scaled model tests: An example. to carry out the calculations: It was assumed that Earth was formed from a completely molten object and the age was determined by the amount of time it would

217

Need for physical modeling

take for the near-surface to cool to its present temperature. The governing mechanism here is the heat flow from inside to outside. A simple analogy is the boiling of an egg where the heat flows from outside to inside. Taking inspiration from a similar geotechnical problem of pile driving in clay (i.e., radial consolidation) and making bold simplifications, the governing nondimensional group is: CDh2t where Ch is coefficient of consolidation, t is the time and D is pile diameter.

Experimental modeling of dynamic soil-structure interaction of offshore wind turbines

Understand the physics/mechanism of the problem and derive nondimensional groups

Numerical or analytical modeling based on the understanding of the problem

Compare numerical/analytical with the experimental results and field case records (if available)

Do they agree?

Yes Carry out parametric study on nondimensional groups at all scales and find the practical range of the different parameters

Design charts for practical use

FIG. 8

Scaled model tests to design charts.

No

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11. Physical modeling of interaction problems in geotechnical engineering

If it is assumed similar fluid in egg and Earth, one can get the following similarity: t tearth ≡ egg D2earth D2egg

(Fig. 9).

If we use the values of Earth diameter and typical egg sizes and assume that it takes 3 min to boil an egg, one can get the following as time to cool the Earth.  2 D2 6371  106 mm tearth ≡tegg earth ¼ 3 min  ¼ 1:95  1017 min 25 mm D2egg The model tests predict about 371 billion years whereas the actual is about 4.54 billion years. It may be noted that the shape of the egg is not the same as the Earth and we used the same fluid coefficients for egg and Earth. Another interesting fact is that there was heat generation due to nuclear reactions within the Earth and therefore additional factors.

Physical modeling of collapse of pile-supported structures in liquefiable soils Engineered pile-supported structures still collapse after most major earthquakes despite large factors of safety being employed in their design. Postearthquake survey often shows that the superstructure (i.e., the part of the structure above the ground) is intact/undamaged and it tilts or rotates as a whole rendering it useless following an earthquake (Bhattacharya et al., 2018; Lombardi & Bhattacharya, 2014b). This suggests that the

FIG. 9

Spirit of a model test: modeling age of the Earth.

Physical modeling of collapse of pile-supported structures in liquefiable soils

219

FIG. 10 Collapse of pile-supported buildings. (A) Kandla Port Tower near the Arabian Sea and the damage following the 2001 Bhuj earthquake where lateral spreading was observed; (B) A building in Kobe close to a quay wall following the 1995 Kobe earthquake; (C) Tilting of a building in a level ground (no lateral spreading) in Kobe following the 1995 Kobe earthquake.

foundations may have been damaged. Fig. 10 shows three pile-supported buildings from two different earthquakes from two different countries and in all the cases the buildings tilted. One point to be noted that these pile-supported buildings tilted irrespective of the type of ground, i.e., level ground where no lateral spreading was observed or in a laterally spreading ground. Here, the aim is to demonstrate the application of physical modeling to understand the failure modes. Step 1 Postulating the different failure modes Following the steps discussed, Fig. 11 shows four critical stages of loading of the piles during the event with the mechanisms identified. The loadings at these discrete stages are as follows. Stage I: Before, or just at the onset of seismic shaking, the piles are predominantly subjected to static axial load denoted by Pgravity. The pile section is expected to have axial stresses. Pgravity (Stage I) represents the axial load on the piles in normal conditions and this axial compressive load may increase/decrease further due to inertial effects of the superstructure (Vinertial) as shown in Stage II. Pstatic can be obtained from the weight of the superstructure, i.e., simply dividing the weight by the number of piles. On the other hand, Vinertial and Hinertial can be estimated from the rocking motion of the superstructure and it will be transferred to the piles by push-pull (i.e., upwards tensile force on one side and downward compressive force on other side) action. Stage II: The pile will be subjected to inertial forces due to vibrations of the superstructure. The inertial load (Hinertial) is dependent on the flexible-base period of the structure. In a simplified way, the inertial load can be estimated using a design response spectral method. Axial statical gravity loads will continue to act on the pile. There will be

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11. Physical modeling of interaction problems in geotechnical engineering

Pgravity

Pgravity + Vinertial

Hinertial

Loose sand

Stage I Before earthquake on level ground

FIG. 11

Pgravity + Vinertial

Hinertial

Liquefied sand

Stage II Shaking starts. Soil yet to liquefy

Stage III Soil liquefies. Vertical inertial forces act with gravity. Piles may starts to buckle or settle

Pgravity + Vinertial

Hinertial

Liquefied sand

Stage IV On sloping ground Soil liquefies. Lateral spreading may combine with behaviour in stage III

Stages of loading.

additional vertical load on some piles due to inertia (Vinertial). However, the increase in dynamic axial load will be insignificant as the length of laterally unsupported pile is small. The bending moment in the pile can be estimated using Beam on nonlinear Winkler foundations model. Stage III: In this stage, the soil has liquefied and the pile has lost its axial and lateral resistance from the surrounding liquefied soil and acts like a long laterally unsupported column. The fundamental frequency of the structure and the damping of the soil have changed significantly. The inertia force on the pile will now be a function of the new fundamental period and damping of the structural system. There will also be a substantial increase in dynamic axial load in some piles. Stage IV: In some areas, due to the lateral flow of soils, piles will be subjected to passive soil pressure. Static axial load will continue to act on the pile foundation. Depending on the duration of the earthquake, there may be inertial load acting on the foundation. One of the crucial mechanisms identified is buckling instability of piles in liquefiable soil in Stage III and needs to be verified. Step 2 Derivation of nondimensional groups The readers are referred to Adhikari and Bhattacharya (2008) for the detailed mathematical treatment of the axial load-induced instability in

Physical modeling of collapse of pile-supported structures in liquefiable soils

221

FIG. 12 Combined pile-soil model using Euler Bernoulli beam with axial and lateral force resting against a distributed elastic support.

piles. Fig. 12 shows a pile embedded in elastic soil and this can be considered an Euler-Bernoulli beam having a flexural rigidity of EI and resting against a linear uniform elastic support of stiffness k. A constant static lateral force F is applied at the top of the beam (x ¼ L). The well-known equation of static equilibrium can be expressed as: EI

d4 wðxÞ d2 wðxÞ d + P + kwðxÞ ¼ F fδðx
LÞg 4 2 dx dx dx

(4)

where w(x) is the transverse deflection of the beam and x is the spatial coordinate along the length of the beam. It is assumed that the mechanical properties of the beam are constant along the length. Eq. (4) is a fourthorder differential equation and requires four boundary conditions for its solution. To comment on the pile head deflection, it is necessary to find the deflection at the top of the beam △0 in terms of k (soil support), P and EI. The Euler Critical load is denoted by Pcr. Let Δ0 be the deflection of the beam due to the lateral force F without the influence of the axial load and support stiffness (k). From basic mechanics,

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11. Physical modeling of interaction problems in geotechnical engineering

Buckling amplification

20

15

10

5

0 1

0 0.8

20 0.6

40

0.4

0.2

0

P/Pcr

60

h = kL 4/El

FIG. 13 Variation of Buckling Amplification factor with respect to support stiffness and axial load.

TABLE 2 piles. P Pcr

Similitude relationships for design model tests to study buckling instability of

Nondimensional axial load acting at the pile head. P is the axial load acting on the pile and Pcr is the critical load, i.e., the load at which the pile will become laterally unstable. 4

η ¼ kL EI

Nondimensional support stiffness.

Δ Δ0

Buckling Amplification Factor, i.e., this determines the extent of the P-delta effect.

Δ0 ¼

FL3 3EI

(5)

Eq. (4) is solved for the fixed-free cantilever case and is plotted in Fig. 13 in terms of nondimensional parameters. These nondimensional parameters are defined in Table 2. The above analysis shows that as soil liquefies progressively (i.e., k diminishes), the lateral pile head deflection increases. It may be noted that the amplification is purely due to the axial load and clearly does not consider the effect of dynamics of the problem. This is Static instability (i.e., Euler type) and is critical for the pile foundation when the soil remains liquefied and the depth of liquefaction is high enough. On the other hand, dynamic instability is critical, when the liquefaction starts and progresses in such a way that the fundamental frequency of the pile-soil system

Physical modeling of collapse of pile-supported structures in liquefiable soils

223

comes closer to the dominant earthquake frequency that may give a worse response to pile and Eq. (4) is not capable for such issues.

ASIDE The numerical modeling above provides us a clue about smart (intelligent) exper4 imentation. To have buckling of piles, one needs to have η ¼ kL EI close to zero and at the P same time Pcr close to one. This highlights the importance of analytical modeling and it will be later shown the significance of such a step in efficient cost-effective physical modeling.

Physical modeling for verification of pile buckling (simplest physical modeling) In this step, it is important to examine physically the nondimensional 4 relationship between PPcr and η ¼ kL EI which will clarify the role of soil in avoiding buckling of piles. Fig. 14 shows a conceptual modeling of column buckling and pile buckling were the figure in the left hand side (LHS) shows a column buckling using a fixed-fixed end boundary condition. On the other hand, on the right hand side (RHS) the column is surrounded by some soil. This test can be conducted using a Universal (tension/compression) Testing Machine (such as INSTRON) used for finding compressive strength of materials. Tests have been carried out using different

FIG. 14

Conceptual model of column buckling and pile buckling.

224

FIG. 15

11. Physical modeling of interaction problems in geotechnical engineering

A column is being subjected to axial load in an Instron Machine.

types of soils (soft clay, Bentonite, Rubber granules, loose sand, dense sand, etc.) and using different boundary conditions of the pile. Figs. 15 and 16 shows the test set up and Fig. 17 shows the different piles used and Fig. 18 shows a plot PPcr against the GL4/EI which is essentially analo4 gous to η ¼ kL EI where G is the Shear Modulus of the Soil. This physical modeling shows the importance of the soil support springs to avoid buckling and is strong corroborative evidence in favor of the analytical solution. It is beyond doubt that during soil liquefaction, G reduces to as low as 0.1% and the effect on pile can be catastrophic. This was investigated by Bhattacharya (2003) through a series of centrifuge tests and later re-affirmed by many researchers worldwide either through experiments or analytical or numerical. Figs. 19 and 20 shows photographs from physical modeling of pile buckling. There are other effects of soil liquefaction that may affect foundations. Liquefaction reduces the soil stiffness and the effect on foundations can (1) reduce the natural frequency of the system and may cause resonance type phenomenon (2) reduce the foundation stiffness may also increase the deformation of the structure. (3) increase the damping of the whole system will due to liquefaction Physical modeling was carried out and reported in Lombardi and Bhattacharya (2014a) to understand the effects of liquefaction. The test set-up is shown in Fig. 3C and the schematic of the test set-up is shown in Fig. 21 and the results are shown in Fig. 22.

Physical modeling of collapse of pile-supported structures in liquefiable soils

225

FIG. 16 The same bar has been surrounded by geo-material of different stiffness to see the load at which it fails.

FIG. 17

Piles of different length surrounded by different types of soils were tested.

226

11. Physical modeling of interaction problems in geotechnical engineering 1.8

1LB

1.7

1H 2LB

1.6

P/Pcrit

1.5 2H 1.4 5LB 3H 3LB 4LB

1.3 1.2

2R 5H 1.1

5R

1 0

0.01

0.02 0.03 (GL4)/EI

0.04

0.05

FIG. 18 Graph showing the Normalized critical load versus nondimensional support stiffness. In each of these cases, the boundary end conditions are Fixed-Fixed and the specimen is 420 mm long.

FIG. 19

Physical modeling in a centrifuge and 1-g.

FIG. 20 Observed buckling of piles in 1-g test at Harbin Institute of Technology (Wang et al., 2019). (A) overview of experimental set-up; (B) top-view of soil deposit; (C) failed pile and location of plastic hinge.

500 mm

PPT1

PPT2

PPT3

600 mm

Absorbing boundary

Absorbing boundary

BB--BB

2400 mm

GP1

AA

SP2

PPT4

200 mm

Strain gauges

Accelerometer

Pore water pressure trasducers (PPT)

PPT1

1300 mm

PPT1

1000 mm

PPT2

1600 mm

PPT2

1800 mm

PPT3

200 mm

PPT3

PPT4

2400 mm

GP2

BB

PPT3

Schematic Set-up of the tests shown in Fig. 3C. SP1 and SP2 are single piles and GP1 and GP2 are group piles.

2400 mm

GP2

1200 mm

200 mm

SP1

PPT

PPT2

FIG. 21

SP1

AA--AA

AA

GP1

2400 mm

BB

PPT4

2400 mm

PPT4

1800 mm

500 mm

Absorbing boundary

Absorbing boundary Absorbing boundary

SP2

1800 mm

200 mm

Absorbing boundary

500 mm

PPT1 1800 mm

1600 mm

1000 mm

1300 mm

600 mm

228

FIG. 22

11. Physical modeling of interaction problems in geotechnical engineering

Change in damping ratio and frequency as a function of the excess pore water

ratio ru.

As the main question is to understand the response as soil progressively liquefies, white noise testing was carried out for inverse system identification. As the amplitude of input motion was progressively increased (see Fig. 22) so that soil liquefies progressively. Further details can be found in Lombardi and Bhattacharya (2014a) and Lombardi (2014). Through the above tests, the overall understanding was achieved, and the important consequences on the performance of critical infrastructure were identified. One of them are highlighted here which is midspan collapse of pile-supported bridges and the readers are referred to Mohanty and Bhattacharya (2019) for further details.

ASIDE Midspan collapse of pile-supported bridges: Collapse of pile-supported river bridges in liquefiable soils are routinely observed after most major earthquakes, see Fig. 23 for a collage of bridge collapses from previous earthquakes.

Physical modeling of collapse of pile-supported structures in liquefiable soils

FIG. 23

229

Collapse of mid-span of bridge.

Fig. 24A shows a schematic diagram of a longitudinal section of a typical multispan bridge spanning across the river where piles of its abutments and piers pass through liquefiable deposits. Fig. 24B shows a section of the bridge to illustrate the difference between piles supporting a central pier and piers close to the abutments. Few points may be noted from the figures: (a) Due to the natural riverbed profile, water depth increases as we move from abutments toward the center of the river channel. This would lead to relatively higher unsupported length of piles for central piers. (b) Due to continuous scouring and in the absence of scour protection work, water depth may increase at the center of the river channel over time.

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11. Physical modeling of interaction problems in geotechnical engineering

Liquefiable soil

L1

Approach road

L2

Abutment

Nonliquefiable soil

Approach road

Abutment

P1, P2, P3, P4, P5:PILE FOUNDATIONS L1, L2:Unsupported length of pile-pier system at full liquefaction

P1

P2

P3

P4

P5

L1>L2

Lpile Hwater Hair

(A)

(B) FIG. 24 (A) Schematic diagram of a typical multi-span bridge showing the abutment, piers and foundation. (B) Figure to show the contrast between piles supporting central pier and piers close to abutments.

Also, due to scour, the topsoil in the central mudline usually consists of a very loose soil deposit. During liquefaction, it may be reasonable to expect a homogeneous ground that it will liquefy more or less equally to a given depth which is shown schematically in Fig. 24A. This postulation is in broad agreement with the simplified method of obtaining the depth of liquefaction prescribed in codes of practices. As the unsupported length of a pile is a function of water depth and depth of liquefaction, it may easily be derived from the above considerations that the piles supporting central piers will have higher unsupported length as soil liquefies. The effect of this higher unsupported length is enhanced elongation of natural period for the central bridge piers as compared to the neighboring piers. The impact of such elongation of natural period is the differential demand of pier-head displacement. Experiments were conducted to understand these effects and further details can be found in Mohanty (2020).

Physical modeling of collapse of pile-supported structures in liquefiable soils

231

Translating the understanding obtained from physical modeling to design methods In practice, piles are most often modeled as “Beams on Nonlinear Winkler Foundation” (also known as the “p-y spring” approach) where the soil is idealized as p-y springs. In Winkler method (Beam on Nonlinear Winkler Foundation) of analysis of piles, the pile-soil interactions are represented by a set of nonlinear soil springs: p-y springs (commonly known as curves that incorporate the lateral pile-soil interaction), t-z springs (models the shaft resistance, i.e., pile-soil friction) and q-z spring (models the end-bearing interaction). Fig. 25 shows a simple model of a pile which can be analyzed using any standard structural software and can incorporate advanced features such as P-delta effects, nonlinearity in the material of the pile. Therefore, the task in hand is to find the appropriate p-y curves for liquefied soil (Lombardi, Bhattacharya, Hyodo, & Kaneko, 2014). This can be carried out through physical modeling combined with element testing of the soil. Readers are referred to Dash, Bhattacharya, and Huded (2018) for a simplified method of constructing p-y curves from element testing of soils, the distinct shape of which is shown in Fig. 25.

Axial load Load (p) Lateral load Lateral soil Springs

t-z Liquefiable layer

p-y Displacement (y)

t-z p-y Load (p) t-z Nonliquefiable layer

p-y q-z

Displacement (y)

FIG. 25 BNWF model of piles in liquefiable soils.

Bottom of liquefiable layer

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11. Physical modeling of interaction problems in geotechnical engineering

For any load or displacement applied to the pile either at the pile head (represents inertia load from the superstructure) or along the pile, the required analysis outputs are pile deflection, rotation, bending moment, shear and soil reaction. However, undoubtedly the critical inputs for a realistic analysis are the springs that represent the interactions. A few important points regarding the shape of p-y curves: (1) A p-y curve appears self-similar to stress-strain curve of the soil. Essentially, a stress-strain curve is scaled to obtain a p-y curve and the readers are referred to Bouzid, Bhattacharya, and Dash (2013), Lombardi et al. (2017) and Dash, Rouholamin, Lombardi, and Bhattacharya (2017) for further details on scaling. (2) For fully liquefied soil, the shape is upward convex as shown in Fig. 25. Methods to construct p-y curves for liquefied soil are given in Bhattacharya, Orense, and Lombardi (2019), Dash et al. (2017), and Lombardi et al. (2017). Physical modeling using a geotechnical centrifuge has been used to verify the shape of the p-y curve and details can be found in the PhD thesis of Dash (2010). The physical model is shown in Fig. 26. Element tests of liquefiable soils is also designed to derive the p-y curves for liquefiable soils as shown in Fig. 25, see Rouholamin, Bhattacharya, and Orense (2017). This section of the chapter showed an alternative method of scaling whereby the mechanisms and processes are understood, and the method of design is based on soil element tests.

Example of physical modeling pipeline crossing seismic faults Pipeline systems are located over large geographical regions and they are often buried below ground for safety, environmental, economic, and aesthetic reasons. Consequently, they are subjected to a wide variety of soil profiles and hazards caused by earthquakes (Psyrras, Sextos, Crewe, Dietz, & Mylonakis, 2020). Past earthquake-related pipeline damage showed the vulnerability of buried pipelines to seismic hazards such as faults, landslides, liquefaction induced lateral spreading and liquefaction. Fig. 27 shows a few such scenarios. Different types of pipeline failure modes including local buckling, beam buckling, tension failure and joint failure were observed in past earthquakes, see Fig. 28. Physical model tests are widely used to investigate factors influencing the behavior of buried pipelines subjected to seismic hazards. Examples of 1 g physical model of buried continuous pipelines crossing reverse faults and landslides are shown in Figs. 29 and 30.

Example of physical modeling pipeline crossing seismic faults

Centrifuge facility, Shimizu corporation, japan

233

Container: Laminar shear box

Pile group Side-A

Pile group

Quay wall

Partition wall

FIG. 26 Physical modeling of pile-soil interaction in geotechnical centrifuge carried out at Shimizu Corporation.

Designing scaled model tests for pipelines crossing seismic faults One of the first steps is to identify the loads acting on buried pipelines crossing faults. For example, for a strike-slip fault, Fig. 31 schematically explains the problem together with the standard way of modeling using the Winkler approach. (1) Pipelines bend at either side of faults (see Fig. 31) and curved pipeline sections that are shown with red color are therefore developed. Lateral soil reaction (Pu) for strike-slip faults and vertical soil reaction (Qu and Qd) for reverse and normal faults along soil-pipeline interface due to relative displacements between soil and pipeline (Fig. 31A). (2) Relative displacements between soil and pipeline develop in the axial direction due to pipelines pulling away from the soil surrounding them under faulting. As a result, axial soil-pipe friction force (Tu) along

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11. Physical modeling of interaction problems in geotechnical engineering

FIG. 27 Principal effects of landslides on pipelines depending on their orientation: (A) perpendicular crossing, (B) oblique crossing, (C) parallel crossing.

FIG. 28 The schematic illustration of failure modes of buried continuous pipelines: (A) tensile failure, (B) local buckling and (C) beam buckling.

soil-pipeline interface occurs (Fig. 31A). The axial soil-pipe friction becomes zero at anchor points. (3) The axial, lateral, and vertical soil-pipe interactions are modeled by using elasto-plastic axial, lateral, and vertical soil springs as shown in Fig. 31B and C. Anchor points are specified by taking into account unanchored length (La) and the points are pinned. The soil reactiondisplacement relationships for lateral, axial, and vertical soil springs are shown in Fig. 31D. The terms of kz, kx and ky in Fig. 31D are the lateral, axial, and vertical spring stiffness, respectively. Buried pipelines crossing strike-slip fault may experience (tension + bending) or (compression+ bending) depending on the fault crossing angle.

Example of physical modeling pipeline crossing seismic faults

235

FIG. 29 1-g physical model of reverse fault, see Demirci, Bhattacharya, Karamitros, and Alexander (2018).

A buried pipeline experiences tension and bending in the case of normal faulting whereas compression and bending forces develop within the pipeline under reverse faulting.

Experimental modeling One needs to consider two different critical aspects at the very first stage of physical modeling of soil-pipe interaction problems: (a) Scaling laws and mechanics based nondimensional groups. Two methods are widely used to obtain governing nondimensional groups

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11. Physical modeling of interaction problems in geotechnical engineering

FIG. 30

(A) Photograph of 1-g model of strike-slip fault rupture or landslide. (B) Schematic diagram of the test set-up shown in (A).

(1) governing differential equations and (2) Buckingham-π theorem. Pipelines crossing active faults and landslides can be modeled as a beam on elastic Winkler foundation, see Fig. 31. Field steel pipelines have relatively small sectional dimensions compared to the distance between support points. Thus, they can be considered as slender beams and they are therefore modeled by using Euler-Bernoulli beam approach. The soil surrounding the pipelines is also assumed to be uniform along pipelines. The governing equation of the problem is very similar to the laterally loaded beam on elastic uniform support and is given in Eq. (6) (Fig. 32).
 2 ðx d4 w dw dw EI 4 + P
f ðxÞdx + kw ¼ FðxÞ (6)
f ð xÞ 2 dx dx dx 0 where EI ¼ bending flexural rigidity of the pipe w ¼ transverse deflection of the pipe f(x) ¼ the friction per length (Tu)

Example of physical modeling pipeline crossing seismic faults

237

FIG. 31 (A) Forces acting on buried pipelines crossing active faults, (B) and (C) the schematic illustration of a simple numerical model of the problem, and (D) force-displacement relationships for lateral, axial and vertical soil springs.

238

11. Physical modeling of interaction problems in geotechnical engineering

x w

EI k

FIG. 32

Analytical modeling.

k ¼ soil stiffness (in compression) P ¼ external axial load on pile/beam head (for pipelines crossing active faults, P ¼ 0) F(x) ¼ external loads which may be present at the surface level, e.g., roadways It is convenient to express the equation of motion Eq. (6) in terms of nondimensional parameters by elementary re-arrangements as:


4 d 4 W ðξÞ Px D2 d2 W ðξÞ f ðxÞD3 dW ðξÞ kD + +
W ðξÞ ¼ FðξÞ 4 2 dξ EI EI EI dξ dξ

(7)

where Ð Px is axial force in the pipeline at location x and formulated as P
x0f(x)dx and D is pipe diameter and ξ is nondimensional length (x/D). The nondimensional groups derived and their physical meaning are summarized in Table 3. It is important to note that the nondimensional groups derived are strictly applicable in the elastic range. Similitude relationships/scaling laws The rules of similitude (similarity) between the physical model and the prototype that need to be maintained are (1) (2) (3) (4) (5) (6)

Relative soil-pipe stiffness (kD4/EI), Normalized soil-pipe friction (TuD3/EI), Geometric similarity, Grain size effect, Scaling of fault movement and Scaling of anchorage length.
4
4 kD kD Relative soil
pipe stiffness : ffi EI model EI field

Example of physical modeling pipeline crossing seismic faults

TABLE 3

239

Scaling laws for studying soil-pipe interaction under faulting.

Name of the nondimensional group

Physical meaning

Remarks

(kD /EI)

Flexibility of the pipeline to have similar soil-structure interaction

Small (kD4/EI): rigid pipe behavior Large (kD4/EI): flexible pipe behavior

(D/t)

Slenderness of the pipeline (affects pipeline failure mode)

Large (D/t): shell buckling failure mode Small (D/t): beam buckling failure mode

(H/D)

Nondimensional burial depth (affects soil failure type)

Small (H/D): wedge type of soil failure Large (H/D): soil flow around the pipe

(La/D)

Nondimensional anchor length

Providing anchor length results in no boundary effects at both end sides of the pipe (La ¼ σ yA/Tu)

(d50/D)

Nondimensional average soil grain size

Grain size effects on soil-pipe interaction

(δ/D)

Nondimensional fault displacement (strain field in the soil around the pipeline)

Similar strain field will control soil-pipe interaction

4





 Tu D3 Tu D3 ffi EI model EI field



 D D H H Geometric similarity : ffi and ffi t model t field D model D field
 D Soil grain size effect :  48 d50

 δ δ ffi Scaling of fault movement : D model D field

 La L Scaling of anchorage length : ffi D model D model Normalized soil
pipe friction :

More details of similitude relationships and scaling laws for buried continuous pipelines crossing active faults can be found in the work of Demirci et al. (2017, 2018).

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11. Physical modeling of interaction problems in geotechnical engineering

Limitations of physical model tests It is important to highlight the known limitations of physical modeling and some possible remedies. (1) Stress dependency and nonlinearity of the soils: Soils have a nonlinear stress-strain behavior and their behavior is also stress-dependent. Sands may experience contractive or dilative behavior depending on the relative density and confining pressure. This issue can be dealt with pouring sand into model containers at lower relative densities and is widely used in practice. Fig. 33 shows a method of pluviation of loose sand samples whereby the height of fall and rate of flow can determine the relative density. Another issue related to 1 g physical model tests is soil stiffness (e.g., Shear Modulus). Shear modulus of the

FIG. 33

Technique of pluviation.

Example of physical modeling pipeline crossing seismic faults

241

soil increases with increasing mean confining stress. The stress level in physical model tests is much smaller than that in the field and this problem can be tackled by considering appropriate values of kD4/EI and performing tests within the domain of interest. Centrifuge models don’t have such issues as the stress levels are appropriately scaled. (2) Pipe material: Steel pipelines are often used for conveying gas and oil. However, different pipe materials such as HDPE, brass, and aluminum are commonly used for model tests. It is very clear that the differences in stress-strain behavior of these materials affect pipeline response to faults or landslides especially in the zone of plastic deformations and local buckling. kD4/EI term can be used for small deformation problems however it should be cautiously used for large deformation problems. (3) Pipe end conditions: The unanchored length for field pipelines crossing active faults is equal to several hundreds of pipe diameter length as suggested in Kennedy, Chow, and Williamson (1977). Therefore, very long boxes are necessary to avoid boundary effects due to pipe end conditions. However, this is not practical. When pipe ends are fixed or pinned to the boxes, unwanted axial pipe strains develop within the pipeline. Therefore, a new type of end connections needs to be developed to realistically simulate pipe end boundaries under faulting or landslide movement. (4) Pipe diameter to wall thickness (D/t): The ratio of D/t govern the structural behavior of buried continuous pipelines. For instance, pipelines with small D/t ratios and buried at shallow trench tend to experience beam buckling while pipelines with large D/t ratios and buried at deeper trenches tend to experience local (shell) buckling under compression and bending. Fig. 34 shows two types of pipeline collapse. However, pipelines used in model tests often have relatively small D/t ratios compared to field pipelines due to practical reasons. Therefore, the interpretation of observations needs further thoughts while scaling up the results. It must be stated that experiments are

Euler beam buckling mechanism

Shell (local) buckling mechanism (wrinkling)

FIG. 34

Types of pipeline buckling.

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11. Physical modeling of interaction problems in geotechnical engineering

most often carried out to clarify mechanisms and observe trends of behavior. (5) Pipe bending zone: As seen in Fig. 31A, a pipe crossing active faults bends and curved parts of pipelines occur at either side of the fault. To avoid the effects of pipe end conditions on pipe bending response, these curved parts of model pipelines need to stay within the soil container box. (6) Small-scale models: Uncertainties are bound to arise in small-scale models due to measurement errors (e.g., measuring in miniature sensors or transducers), manufacturing defects (e.g., when specimens are too small there can be issues with surface roughness or ovalization of a circular section) and scaling issues. These are lot more pronounced in centrifuge tests than in 1-g tests.

Physical modeling of offshore wind turbine foundations Foundations typically cost 25%–35% of an overall offshore wind farm project and to reduce the levelized cost of energy (LCOE) new innovative foundations are being proposed. However, before any new type of foundation can be used in a project, a thorough technology review is often carried out to de-risk it. The European Commission defines this through technology readiness level (TRL) numbering starting from 1 to 9 (see Table 4 for different stages of the process together with the meanings). One of the early studies that needs to be carried out is technology validation in the laboratory environment (TRL 3 and 4) and in this context of foundations, it would mean carrying out tests to verify the failure mechanism, modes of vibration and long-term performance under the action of cyclic loads. It must be realized that it is very expensive and operationally challenging to validate in a relevant environment and therefore laboratory-based evaluation has to be robust to justify the next stages of investment. From the point of view of assessment, the main issues are: (a) Verification of safe load transfer from the superstructure to the supporting ground, (b) Modes of vibration of the structural system adopted (c) Long-term change in dynamic characteristics, i.e., change in natural frequency and damping (d) Long-term deformation so that SLS requirements are not violated.

Physical modeling of offshore wind turbine foundations

TABLE 4

243

TRL (technology readiness level).

TRL level as European Commission

Interpretation of the terminology and remarks

TRL-1: Basic principles verified

In this step, the requirement is to show that mechanics principles are obeyed. For example, in the case of foundation, it must be checked whether the whole system is in equilibrium under the action of environmental loads.

TRL-2: Technology concept formulated

In this step, it is necessary to think about the whole technology starting from fabrication to methods of installation and finally operation, maintenance (O&M) and decommissioning. In this step, it is expected that method statements will be developed.

TRL-3: Experimental proof of concept

In this step, small-scale models will be developed to verify steps in TRL 1 and TRL 2. In terms of foundation, this would correspond to checking the modes of failure in ultimate limit state (ULS) and identifying the modes of vibration.

TRL-4: Technology validated in lab

Once TRL-3 is satisfied and business decision is taken to go ahead with the development/design, it is necessary to check the technology for further details. This may correspond to long term performance under millions of cycles of loading and checking the dynamic performance over the lifetime in relation to Fatigue Limit State (FLS).

TRL-5 Technology validated in relevant environment

Relevant environment may mean numerical simulation whereby close to reality analysis can be carried out. In the context of foundation, this step may use advanced soil constitutive models to verify the performance under extreme loading.

TRL-6: Technology demonstrated in relevant environment

In this step, a prototype foundation is constructed and tested in an offshore environment. Critical aspects are verified.

TRL-7: System prototype demonstration in an operational environment

In this step, the foundation is subjected to operational loads and the performance is monitored.

TRL-8: System complete and qualified

Based on the results in TRL 7, the system can be classified as qualified or not-qualified or changes are required.

TRL-9: Actual system proven in an operational environment

Technology may be used in energy generation with contingency plans.

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Suitability on different methods of testing The behavior of offshore wind turbines involves complex DynamicWind-Wave-Foundation-Structure Interaction and the control system onboard the RNA hub adds further interaction. There are different established methodology, see Table 5 for carrying out testing for some part of the problem to a scientifically acceptable level (1) Wind Tunnel can model the aerodynamics and aeroelasticity of the problem. (2) Wave tank can model the hydrodynamics part. (3) Geotechnical centrifuge testing can SSI problem. (4) Shaking table at 1-g or in a centrifuge can model the seismic SSI In wind tunnel tests, aerodynamic effects are modeled efficiently and correctly (as far as practicable) and as a result, the loads on the blade and towers can be simulated. On the other hand, in the wave tank, the hydrodynamic loads on the substructure and scouring on the foundation can be modeled. In a geotechnical centrifuge, one can model the stress level in the soil, but the model package is spun at a high RPM which will bring in unwanted vibrations in the small-scale model. Ideally, a tiny wind tunnel together with a tiny wave tank onboard a geotechnical centrifuge may serve the purpose, but this is not viable and will add more uncertainty to the models than it tries to unearth. Each of the techniques has TABLE 5 Different forms of testing for offshore wind turbines. Types of testing

Remarks on the understanding

Wind tunnel testing

Example: Blades can be tested to show the importance of profile

Wave tank testing

Wave tank of different forms can be used to study scour, hydrodynamic loading, tsunami. In this type of testing, the wave loads on the foundations can be understood.

Geotechnical centrifuge

In a geotechnical centrifuge, the stress levels in a soil can be modeled accurately (as far as practicable). However, the whole model is spun at a high rate which creates unwanted small vibrations. Therefore, the subtle dynamics of the problem are difficult to study when filtering signals are inevitable during the processing of data.

Whole system modeling

Small-scale whole system modeling which can be considered as a very scale prototype pioneered by Bhattacharya, Lombardi, and Wood (2011) is one of the ways to study the overall system. This type of modeling was used to carry out TRL of self installing wind turbine (SIWT), a-symmetric tripod and details are provided in Bhattacharya, Cox, Lombardi, and Muir Wood (2013) and Bhattacharya et al. (2013). As the system is tested on a stable floor, the dynamics of the problem can be studied very well.

245

Physical modeling of offshore wind turbine foundations

1:150

1:200

Power spectral amplitude

Scale 1:100

11

13

15

17

19

15

17

19

21

23

17

19

21

23

25

Frequency (Hz)

FIG. 35

Modes of vibration for asymmetric types of foundation.

its own limitations and these aspects must be taken into consideration while scaling the observations. Therefore, the focus of the experiments needs to be on the governing laws or mechanics or process (Fig. 35).

Examples of understanding from scaled model tests This section of the chapter highlights the lessons learned from scaled model tests: Example 1 Asymmetric multipod foundation as shown in Fig. 6 Tests were carried out to understand the modes of vibration of the Wind Turbine Support Structure as shown in Fig. 6. The structure is characterized by shallow foundations. Three tests were carried out at different scales (1:100, 1:150, and 1:200) and it was found that the structure exhibit

246

11. Physical modeling of interaction problems in geotechnical engineering

rocking modes of vibration arising from two closely spaced natural frequencies. This is a repeated observation and is verified by analytical and numerical studies. The readers are referred to Chapter 3 of Bhattacharya (2019) for detailed discussion. Example 2 Physical model tests to under what loading conditions provide long term tilt Based on the discussion on the loads on the foundations (Fig. 4) in “Need for physical modeling” section one can envisage the loads on the foundation and Fig. 36 shows a simplified schematic of the loads and Fig. 37 shows a schematic of the over-turning moment in the foundation. The nature of the overturning moment is characterized by: (a) Maximum moment (Mmax) which is a combination of wind and wave load (b) One-way asymmetric moment (Mmin/Mmax) (c) Number of cycles of loading over the lifetime It is clear that there is bias in the foundation load and one of the challenges is to predict the long-term tilt due to this type of loading for the entire design lifetime of 25–30 years. The performance requirement is shown in Fig. 38 and all original equipment manufacturer (OEM) prescribes a limit on the rotation. If the rotation exceeds the limit, the

×10−4

Force (MN)

Average 1P load *:

Force (MN)

Wind load during design*:

0

0.2

0.4 0.6 Time (min)

0.8

1.0

0

0.2

0.4 0.6 Time (min)

0.8

×10−3

Force (MN)

Average 3P load *:

0

0.2

0.4 0.6 Time (min)

0.8

1.0

Force (MN)

Dynamic wave load during design*:

0

0

0.2

0.4 0.6 Time (min)

0.8

1.0

*These values are just indicating the magnitude of the different quantities

FIG. 36

Simplified load on foundations.

1.0

Physical modeling of offshore wind turbine foundations

247

FIG. 37

Simplified mudline bending moment in the monopile.

FIG. 38

Prediction of long-term tilt over 30 years lifetime for a complex load scenario.

warranty of the turbine may be lost, and the turbine will be withdrawn from power generation. Design of physical modeling Considering all the loads and load effects, it is extremely challenging to carry out numerical work. One of the ways to gather understanding in a relatively cheap way is to carry out physical modeling (Bhattacharya et al., 2013; Guo et al., 2015; Lombardi, Bhattacharya, & Wood, 2013; Nikitas, Arany, Aingaran, Vimalan, & Bhattacharya, 2017; Nikitas, Vimalan, & Bhattacharya, 2016; Yu et al., 2015). One of the overarching intellectual tasks of the physical modeler is to design experiments (preferably within the mechanics based nondimensional framework) so that tilting of such structures MUST occur in the lab. Furthermore, it is necessary to check which nondimensional parameter is highly sensitive. In other words,

248

FIG. 39

11. Physical modeling of interaction problems in geotechnical engineering

Observed tilting of wind turbine structure.

which nondimensional group controls the tilt. Fig. 39 shows observed tilting from scaled model tests. Discussion is beyond the scope of this chapter and the readers are suggested to keep an eye on upcoming publications. Example 3 Study of failure mechanisms using physical modeling This example shows the foundation failure mechanism of a Hywind Type floating wind turbine system. Fig. 40 shows one of the failure modes and here is the task to understand the failure modes for such anchors (Figs. 41–44). Physical modeling was conducted to understand the optimum location of the pad-eye, i.e., where the chain will be attached to the anchor piles and what failure mechanism may be invoked, see Fig. 41. A second purpose was to see the deformation mechanism of the soil around the foundation. Fig. 43 notes the observed modes of failure. Knowledge from this understanding was used (without scaling any numbers) to develop a design method and can be found in Arany and Bhattacharya (2018). Example 4 Controlling the deformations of floating wind turbine systems One of the main innovations that is necessary for floating systems is to control the deformations. Fig. 44 shows a physical model of floating systems (Photograph and the associated schematic diagram) and this can be considered as a proof-of-concept study at lower TRL level.

Model container requirements for seismic tests

FIG. 40

Schematic diagram of Hywind Type Floating Wind Turbine System.

FIG. 41

To verify the above hypothesis of failure mechanism.

249

Model container requirements for seismic tests In seismic SSI tests, there are some additional requirements. Whether the tests are carried out using shaking tables at normal gravity or geotechnical centrifuges, the soil stratum needs to be confined in a container with relatively small dimensions. A significant challenge encountered when performing geotechnical physical modeling consists of minimizing the boundary effects created by the model confinement, therefore to simulate free-field seismic conditions, see Fig. 45 (Bhattacharya et al., 2012; Lombardi, Bhattacharya, Scarpa, & Bianchi, 2015). Over the past decade, researchers have developed different types of model container to minimize the effects introduced by artificial boundaries. One example is represented by flexible soil containers. Assuming that the soil layer and the adjacent end-walls behaves as an assembly of equivalent shear beams, the container is designed to mimic the shear beam response.

250

FIG. 42

11. Physical modeling of interaction problems in geotechnical engineering

Schematic of the test set-up.

Another type of container is the so-called laminar box (Durante, Di Sarno, Mylonakis, Taylor, & Simonelli, 2016; Kloukinas et al., 2015). The working principle of a flexible laminar box consists of minimizing the lateral stiffness of the container to match one of the liquefied soil columns. This can be achieved by using a stack of aluminum rings supported individually with bearings, which permit a relative movement between the rings with minimal friction. Several studies have confirmed that the laminar container is compatible with the large soil deformation expected during the simulation of earthquake-induced liquefaction. Model containers with rigid ends have also been used. To increase the volume of soil subjected to the free-field condition, soft material can be placed on the inner sides of the model container (Bhattacharya, Krishna, Lombardi, Crewe, & Alexander, 2012; Hall, Lombardi, & Bhattacharya, 2018; Lombardi & Bhattacharya, 2016), which in turn diminishes the reflection of body waves (P and S waves) from the boundaries (Snell’s law). Duxseal material (a putty-like, pipe sealant rubber mixture compound) has been extensively used in the past decade for centrifuge modeling whereas conventional foam has successfully been used at normal gravity to minimize generation and reflection of body waves from the artificial boundaries. The readers are referred to Bhattacharya et al. (2012) for a summary and discussion on the model container.

Summary and concluding remarks Scaling or developing constitutive relations is one of the challenging steps in a successful scaled test. If the scaling for a particular problem is incorrect, the experiment carried out will not show the dominant mechanism.

Summary and concluding remarks

251

FIG. 43

Examples of different failure mechanisms observed in anchors with different locations of pad-eye.

The example of pile-supported structures described in “Physical modeling of collapse of pile-supported structures in liquefiable soils” section is taken: Most piles in seismically liquefiable areas are long and slender (length to Diameter ratio a high as 100) and carries a large axial load. If the physical modeling of collapse of pile-supported structures during the earthquake was conducted with no axial load (or pile head mass) the buckling instability mechanism of slender piles will not be invoked. Therefore, the identification of the correct mechanism is necessary for the right

252

FIG. 44

11. Physical modeling of interaction problems in geotechnical engineering

Example of different concepts of floating systems.

FIG. 45 Shear-beam idealization of infinite lateral extent stratum overlying a bedrock subjected to one-dimensional shaking.

scaling. It may often be argued that if experiments are not carried out— how will be mechanism be understood. The issue in hand is analogous to which comes first the chicken or the egg? Or a Catch-22 situation. This points out to the fact that Scaled Experiments in relation to understanding is an iterative process. Clearly, what is necessary is more than one scaled test, an iterative loop and analytical models to explore parametric configurations.

Summary and concluding remarks

253

What is suggested as a foolproof method is: (1) List all the failure mechanisms (however improbable they are) and then RANK them based on simple idealistic mathematical idealization. For the pile-supported structures (Eq. 4) which is a fourth order differential equation taking the necessary terms. Carry out an analytical solution and this involves rigorous Mathematics, Mechanics/Physics to develop understanding. Fig. 13 shows the results of the analysis based on analytical modeling. The danger or 4 no-go zone can be identified, i.e., Higher (P/Pcr) and lower η ¼ kL EI . (i) Carry out FEA of the same problem for sanity check as not all the parameters can be accounted for in simple idealistic analysis given by Eq. (4). (ii) Following the above two steps, one can do the scaling and derive the similitude relationships. Now design and carry out the experiments (iii) Verify if the experiments showed behavior that was predicted. They may be a need for Rounds 2 and 3 of the experiments. There may be new lessons which we will learn and were thought to be unimportant and vice-versa. (iv) Iterate the above procedure till convergence of understanding. This chapter advocates the use of mechanics-based scaling laws to design scaled model tests and also the interpretation of the results as opposed to black-box-type scaling laws as shown in Table 1. While simplistic scaling laws may be acceptable for simple problems, it is not advisable for complex interaction problems. An example of offshore wind turbine is taken to illustrate the point. Offshore Wind Turbines is a complex system with uncertain material behavior (ground supporting the foundation), aero-hydro-servo dynamic SSIs, and therefore large nonlinearities are possible.

Reductionist approach One of the approaches often used is “Reductionist” which means the whole problem is broken down into individual bits and each of the bits is tested individually. At the end, all the understandings are superimposed. In the context of offshore wind turbine, an individual bit may mean studying only the foundation and the supporting soil. This approach may fail for offshore wind turbine problem (or any problem with a large number of interactions) as it may not be possible to separate out the individual bits due to complex loading histories and nonlinear material and as well as geometries. Reductionist approach works well for linear problem, i.e.,

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11. Physical modeling of interaction problems in geotechnical engineering

where superposition principles apply. Designing scaled model tests involving many interactions is therefore challenging. Engineering solution for a particular problem is to iterate to converge to the best scaling laws and this requires multiscale testing, state-of-the-art numerical FEA, and insightful reduce order mechanics/physics models. Predictions require a fusion of (i) Reduced order models and FEA models (ii) Small-scale tests (Centrifuge, 1-g shake tables, material element testing) and (iii) full-field monitoring observations. This section of the chapter summarizes the do’s and don’t in physical modeling: DO’s

(1) Try to replicate the mechanisms (for example failure mechanism) in small-scale tests. Example include verifying ULS design, see the example of failure of anchor foundation for floating wind turbines. (2) Repeat the tests in few scales (at least 3)—such as 1:200, 1:100, 1:50 [Modeling of models] to remove artifacts (if any) and to establish reliability. Nonlinearity is defined by amplitude and this step helps to identify. (3) Identify or clarify physical mechanisms or processes (essentially the laws of physics!) that governs the observation or control the behavior of interest. This can guide SLS. (4) If possible—write the governing equation and using mathematical techniques—find the nondimensional groups. (5) Check nonlinearity of the nondimensional groups DON’Ts

(6) Unless a very simple problem, do not scale the numerical values of the observations to predict the prototype consequences.

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Lombardi, D. (2014). Dynamics of pile-supported structures in seismically liquefiable soils. Doctoral dissertation, University of Bristol. Lombardi, D., & Bhattacharya, S. (2014a). Modal analysis of pile-supported structures during seismic liquefaction. Earthquake Engineering and Structural Dynamics, 43(1), 119–138. Lombardi, D., & Bhattacharya, S. (2014b). Liquefaction of soil in the Emilia-Romagna region after the 2012 Northern Italy earthquake sequence. Natural Hazards, 73(3), 1749–1770. Lombardi, D., & Bhattacharya, S. (2016). Evaluation of seismic performance of pile-supported models in liquefiable soils. Earthquake Engineering & Structural Dynamics, 45(6), 1019–1038. Lombardi, D., Bhattacharya, S., Hyodo, M., & Kaneko, T. (2014). Undrained behaviour of two silica sands and practical implications for modelling SSI in liquefiable soils. Soil Dynamics and Earthquake Engineering, 66, 293–304. Lombardi, D., Bhattacharya, S., Scarpa, F., & Bianchi, M. (2015). Dynamic response of a geotechnical rigid model container with absorbing boundaries. Soil Dynamics and Earthquake Engineering, 69, 46–56. Lombardi, D., Bhattacharya, S., & Wood, D. M. (2013). Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil. Soil Dynamics and Earthquake Engineering, 49, 165–180. Lombardi, D., Dash, S. R., Bhattacharya, S., Ibraim, E., Muir Wood, D., & Taylor, C. A. (2017). Construction of simplified design p–y curves for liquefied soils. Geotechnique, 67(3), 216–227. Madabhushi, G. (2014). Centrifuge modelling for civil engineers. CRC Press. Mohanty, P., (2020). A study towards collapse of pile supported bridge foundations in seismically liquefiable soils (Doctoral dissertation, University of Surrey). Mohanty, P., & Bhattacharya, S. (2019). Reasons for mid-span failure of pile supported bridges in case of subsurface liquefaction. In Civil Infrastructures Confronting Severe Weathers and Climate Changes Conference (pp. 148–164). Cham: Springer. Nikitas, G., Arany, L., Aingaran, S., Vimalan, J., & Bhattacharya, S. (2017). Predicting long term performance of offshore wind turbines using cyclic simple shear apparatus. Soil Dynamics and Earthquake Engineering, 92, 678–683. Nikitas, G., Vimalan, N. J., & Bhattacharya, S. (2016). An innovative cyclic loading device to study long term performance of offshore wind turbines. Soil Dynamics and Earthquake Engineering, 82, 154–160. Psyrras, N., Sextos, A., Crewe, A., Dietz, M., & Mylonakis, G. (2020). Physical modeling of the seismic response of gas pipelines in laterally inhomogeneous soil. Journal of Geotechnical and Geoenvironmental Engineering, 146(5)04020031. Rouholamin, M., Bhattacharya, S., & Orense, R. P. (2017). Effect of initial relative density on the post-liquefaction behaviour of sand. Soil Dynamics and Earthquake Engineering, 97, 25–36. Schofield, A. N. (1980). Cambridge geotechnical centrifuge operations. Geotechnique, 30(3), 227–268. Wang, W. D., Ng, C. W. W., Hong, Y., Hu, Y., & Li, Q. (2019). Forensic study on the collapse of a high-rise building in Shanghai: 3D centrifuge and numerical modelling. Geotechnique. https://doi.org/10.1680/jgeot.16.P.315. Wood, D. M. (2003). Geotechnical modelling. CRC Press. Yu, L., Wang, L., Guo, Z., Bhattacharya, S., Nikitas, G., Li, L., et al. (2015). Long-term dynamic behavior of monopile supported offshore wind turbines in sand. Theoretical and Applied Mechanics Letters, 5(2), 80–84.

C H A P T E R

12 SPH modeling for soil mechanics with application to landslides G.R. Liua, Zirui Maoa,b, and Yu Huangc a

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH, United States, bDepartment of Material Science and Engineering, Texas A&M University, College Station, TX, United States, cDepartment of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai, People’s Republic of China

Introduction Geotechnics often concerns about the problems related to mechanical behaviors of geo-materials such as soil and rock. Theoretical analysis, experimental investigation, and numerical modeling are the most widely used approaches. Since most practical problems in geotechnical engineering are multi-physics, multi-phases, and even multi-scale, the traditional analytical methods based on layers of assumptions and simplifications have limits. Experimental approaches can be time-consuming, with heavy cost, often dangerous, and may not feasible in extreme environments. Numerical modeling has become an increasingly important approach for the study of geotechnical problems. The mature finite element method (FEM) (Liu, 2003), for example, has played an important role in the fields of nonlinear analysis of soil and rock mechanics problems. It is a desirable and powerful numerical method for a wide class of finite deformation problems. However, the traditional FEM faces many challenges in handling large deformation problems in geotechnical engineering, like flow-like landslides. This is because its grid structure moving along with deforming and moving material causes severe distortion of the mesh structure. Since accuracy of FEM solution highly depends on quality of mesh, the extreme distortion of mesh can lead to inaccurate numerical solutions.

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00004-6

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© 2021 Elsevier Inc. All rights reserved.

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12. SPH modeling for soil mechanics with application to landslides

For large deformation problems in geotechnical engineering, the meshfree particle methods (Liu, 2010, 2016), including the smoothed particle hydrodynamics (SPH) (Huang & Dai, 2014; Liu & Liu, 2003), discrete element method (DEM) (Li, Li, Dai, & Lee, 2012; Tavarez, Plesha, & Bank, 2002), and material point method (MPM) (Bolognin, Martinelli, Bakker, & Jonkman, 2017; Dong, Wang, & Randolph, 2017), have been found advantageous over the traditional grid-based methods. The key idea of meshfree methods is to employ nodes and particles for representing the media/material rather than element mesh. The particles in meshfree methods move under the control of governing equations just like in Lagrangian grid-based methods. Since the distribution of particles does not influence the accuracy of numerical solution, meshfree methods can thus be much more accurate for problems under extremely large deformation conditions, such as flow-like landslides. Of all the meshfree methods, the SPH method is one of the most mature and widely used Lagrangian particle methods, thanks to its special advantages of easy implementation and excellent applicability to extremely large deformation. It was originally invented to solve astrophysical problems (Gingold & Monaghan, 1977; Lucy, 1977) and was then extended and applied to a range of engineering applications, such as molecular dynamics in multiple scale (Monaghan & Kocharyan, 1995; Yan, Jiang, Li, Martin, & Hu, 2016), explosion and underwater shocks (Liu, Feng, & Guo, 2013; Swegle & Attaway, 1995), penetration (Kulak, 2011; Seo, Min, & Lee, 2008), high-velocity impact (Aktay & Johnson, 2007; Liu, Zhou, & Wang, 2013), and geophysics (Bui, Fukagawa, Sako, & Ohno, 2008; Huang & Dai, 2014; Mao & Liu, 2018a). In geotechnical applications, SPH has been successfully used for slope stability analysis (Bui, Fukagawa, Sako, & Wells, 2011), soil cracking analysis (Bui, Nguyen, Kodikara, & Sanchez, 2015), run-out analysis of flow-like landslides with either Navier-Stokes (Hu, Liu, Xie, & Liu, 2015; Huang, Zhang, Xu, Xie, & Hao, 2012) or depth-integrated equations (Haddad, Pastor, Palacios, & Mun˜oz-Salinas, 2010), and other geo-disasters (Huang, Dai, & Zhang, 2014). It has been demonstrated frequently that SPH is capable and ideal in describing the complex fluidization characteristics of flow-like landslides. This chapter discusses the essentials of SPH method, key ingredients, and implementation of SPH modeling for geotechnical problems, effects of these key ingredients on the numerical results. Our discussion is organized as follows. “SPH theory and formulation” section presents the fundamentals of SPH method. The overall SPH framework for modeling geotechnical problems is presented in detail in “SPH framework of geotechnical problems” section. “Applications of SPH to geotechnical engineering” section gives the application of SPH to granular column collapse and earthquake-induced landslide. Some numerical experiments have also been done and presented in “Applications of SPH to geotechnical engineering” section to highlight some key features and

SPH theory and formulation

259

ingredients of SPH modeling. Finally, the advantages and limitations of SPH relative to the traditional grid-based methods and other meshfree methods are analyzed and compared in “Advantages and limitations of SPH modeling” section.

SPH theory and formulation In SPH method, the problem domain is represented by a set of arbitrarily distributed particles that carry field variables such as mass, density, stress tensor, etc. and move along with the material. The movement of SPH particles is governed by the strong-form (differential form) of Navier-Stokes equations in Lagrangian form as presented in Eq. (1). 8 Dρ ∂vβ > > ¼ ρ > > ∂xβ > < Dt α Dv 1 ∂σ αβ (1) ¼ +g > Dt ρ ∂xβ > > α > > : Dx ¼ vα Dt where α and β denote the Cartesian components x, y, z with the Einstein convection applied to repeated indices. ρ, σ, x and v are the scalar density, total stress tensor, spatial coordinates, and velocity, respectively. g is the gravity acceleration.

SPH gradient approximation formulations The SPH method approximates the derivatives of velocity and stress/ pressure in the governing PDEs by translating them into solvable SPH formulations. This is realized by two key steps: “integral representation” and “particle approximation.” In the first step, the integral form of a spatial derivative rf is achieved by integrating the multiplication of function f and the gradient of a welldesigned kernel function W over the overall supporting domain Ω as shown in Eq. (2). ð (2) hrf ðxÞi ¼  f ðx0 ÞrW ðx  x0 , hÞdx0 Ω

The SPH convention marks the kernel approximation form of functions by the angle bracket. Ω is the local supporting domain with a circular shape in 2D or sphere volume in 3D defined by the radius of κh, where h is named as smoothing length in SPH. The radius of the local supporting domain Ω locates within the range of [2 Δ r, 3 Δ r] with Δ r being the initial spacing among particles. Eq. (2) is achieved based on the assumption that

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12. SPH modeling for soil mechanics with application to landslides

the kernel function W ¼ 0 beyond the local supporting domain Ω. For the theoretical deductions, please refer to (Liu & Liu, 2003; Mao, Liu, & Dong, 2017). In the second step, the continuous integral representation in Eq. (2) is discretized as a summation operation over all the particles locating within the local supporting domain Ω, hrf ðxi Þi ¼ 

N X   mj   f xj r i W xi  xj , h ρ j¼1 j

(3)

This process is carried out with an assumption, i.e., each SPH particle possesses individual mass and density and occupies individual space. The representative area or volume of a certain particle is determined by dividing the mass with the instant density which is obtained from the governing equations in Eq. (1). Under this circumstance, the infinitesimal volume dx0 in Eq. (2) can be replaced with m/ρ, and the integral can be approximated by the summation on the overall neighboring particles within the supporting domain of the target particle. Eq. (3) is the standard SPH gradient approximation formulation, which is applied to approximate gradient of variables in the governing PDEs in Eq. (1).

Accuracy of SPH gradient operator The accuracy condition of the SPH gradient approximation formulation in Eq. (3) can be analyzed easily through applying Tylor series expansion of f(xj),       (4) f xj ¼ f ðxi Þ + xj  xi f 0 ðxi Þ + r Δx2 Substituting into Eq. (3) gives hrf ðxi i ¼

N m h     X  i j f ðxi Þ + xj  xi f 0 ðxi Þ + r Δx2 ri W xi  xj , h ρ j¼1 j

¼ f ðxi Þ

N m        X  j ri W xi  xj , h + f 0 ðxi Þ xj  xi ri W xi  xj , h + r Δx2 ρj ρ j¼1 j

N m X j j¼1

(5)

Therefore, it requires that N X mj j¼1

and

ρj

  r i W xi  xj , h ¼ 0

(6)

SPH theory and formulation N X    mj  x j  xi r i W xi  xj , h ¼ 1 ρ j¼1 j

261 (7)

It is easy to get the conclusion that the SPH gradient formulation is at least first order accurate once the requirements in the Eqs. (6)–(7) are satisfied. To guarantee the first-order consistency of SPH gradient formulation, many groups of kernel functions W have been proposed (Liu & Liu, 2003). As one of the most widely used kernel functions, the cubic spline function is presented and employed here as an example. 8 2 2 1 3 > > <  rij + rij 0  rij  1   3 2 (8) Wi, j rij , h ¼ αd  >   > : 1 2  rij 3 1  rij  2 6 where rij ¼

   xj  xi  h

(9)

Basic algebraic deductions yield that the cubic spline function above satisfies the consistency requirements represented by Eqs. (6) and (7). Accordingly, the SPH gradient formulation in Eq. (3) associated with the cubic spline function in Eq. (8) is of consistency under any condition.

Nearest neighboring particles searching algorithms The particles locating within the supporting domain Ω of a certain target particle are named as “supporting particles” or “nearest neighboring particles.” To approximate the gradient of field variables accurately with the SPH gradient formulation, it is necessary to search supporting particles or nearest neighboring particles at every time step. Several nearest neighboring particles searching NNPS algorithms have been proposed, e.g., the all-list searching algorithm and the link-list algorithm (Liu & Liu, 2003). Comparatively, the all-list algorithm searches all the particles in the entire problem domain while the link-list algorithm pre-assigns a background grid and only examines those particles locating in the adjacent pre-assigned grids as illustrated in Fig. 1. Hence, the linklist algorithm is always more efficient in computation than the all-list algorithm especially when a large number of particles are employed in simulation (Mao et al., 2017), while the all-list algorithm has the apparent advantage of implementation ease over the link-list.

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FIG. 1 Sketch of all-list (A) and link-list (B) searching algorithms in SPH. h is the smoothing length generally defined as 1.2Δ x, while κ ¼ 2 is widely adopted to specify the size of supporting domain in SPH.

SPH framework of geotechnical problems This section presents all the essential treatments in SPH simulations when applied to geotechnical problems.

Constitutive model In the following examples of this chapter, the soil media is employed in SPH simulations. Since the development of constitutive model for geomaterial is beyond the scope of this chapter, the final form of soil’s constitutive model is presented directly. Please refer to Bui et al. (2008) and Mao et al. (2017) for the detail. The elastic-perfectly plastic constitutive model associated with Drucker-Prager yield criterion is adopted. The constitutive model has the final form of

  Dσ αβ G βγ γβ αγ γγ αβ αβ αβ i _ _ ffiffiffiffi p _ _ _ ¼ σ αγ e + σ + K δ + 2G  λ 3K α δ + ταβ (10) ω ω ε i i i i i i ψi i i i i Dt J2 i i _ the change rate of plastic multiplier λ, being with λ, 8 pffiffiffiffi αβ αβ γγ pffiffiffiffi < 3αϕi Ki ε_ i + ðG= J2 Þi τi ε_ i if f ¼ αϕ I1 + J2  kc ¼ 0 λ_ i ¼ 9Ki αϕi αψi + Gi pffiffiffiffi : 0 if f ¼ αϕ I1 + J2  kc < 0

(11)

SPH framework of geotechnical problems

263

where α, β, and γ denote Cartesian components x, y, and z with Einstein convention applied to repeated indices; δαβ is the Kronecker delta; K and G are the elastic bulk and shear moduli, respectively; τ is the devia_ ε_ , and e_ are the spin rate tensor, total strain rate tensor toric stress; ω, and deviatoric shear strain rate tensor, respectively, with the form of  1 ∂vα ∂vβ ω_ αβ ¼  (12) 2 ∂xβ ∂xα  1 ∂vα ∂vβ αβ ε_ ¼ + (13) 2 ∂xβ ∂xα 1 e_ αβ ¼ ε_ αβ  ε_ γγ δαβ 3

(14)

In Eq. (11), I1 is the first invariant of the stress tensor and J2 is the second invariant of the deviatoric stress tensor, αϕ and kc are Drucker-Prager constants that are calculated from the Coulomb material constant c (cohesion) and ϕ (internal friction angle). The Drucker-Prager constants and invariants of stress tensors are computed by: tan ϕ αϕ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 + 12tan 2 ϕ I1 ¼

pffiffiffiffi  3 σ xx + σ yy  3α J2 2

3c kc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 + 12 tan 2 ϕ

    σ xx + σ yy 2 2 J2 ¼ + σ xy = 1  3α2 2

(15)

(16)

in 2D plane strain cases. The dilatancy factor αΨ in Eq. (11) is related to the dilatancy angle Ψ in a fashion similar to that between αϕ and friction angle ϕ as shown in Eq. (15). The dilatancy angle Ψ is assumed as zero in the following simulation works. The above constitutive model represented by Eq. (10) is employed in the simulation of the following geotechnical examples, and it contains six necessary material parameters, including density (ρ), cohesion (c), internal friction angle (ϕ), dilatancy angle (ψ), elastic modulus (E), and Poisson’s ratio (ν). Accuracy of elastic-perfectly plastic constitutive model It should be noted that the stress state of material pffiffiffiffi should either locate on or below the yield surface defined byf ¼ αϕ I1 + J2  kc ¼ 0 as illustrated in Fig. 2 for elastic-perfectly plastic soil. However, due to the unavoidable numerical errors during computation, the stress state of soil may go beyond the yield surface unphysically. Under this condition, two return mapping algorithms are always necessary to ensure the soil’s behavior

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12. SPH modeling for soil mechanics with application to landslides

J2 Tension

Compression

f=0

Scaling back C

Tension cracking treatment

af

D

kc

E F Apex

FIG. 2

B

Elastic –I1

A kc / af

Yield surface

O

Stress state response in tension cracking treatment (D ) E) and scaling back (C ) B).

consistent with this elastic-perfectly plastic constitutive model. One is the tension cracking treatment, which pulls the point D back to E as illustrated in Fig. 2 by correcting the normal stress components, the other one is the stress scaling back algorithm, which drives the point E to F or C to B by correcting the shear stress components. This stress corrective treatments can be dated back the work in FEM simulation (Chen & Mizuno, 1990), and should also be considered in the SPH simulations. For the detail, please refer to the literature Bui et al. (2008), Chen and Mizuno (1990), and Mao et al. (2017).

Stability-related corrections In SPH modeling, there exist some instability-related corrections which are always indispensable to ensure the stability of SPH model. As for the application to geotechnical problems, the artificial viscosity and artificial stress are always added necessarily in the momentum equation. Numerical oscillation and artificial viscosity It is well known that SPH is a zero-energy method, which means that it does not contain any intrinsic dissipation. Under this circumstance, when the initial conditions are released in the earlier stage, the sudden shocks and discontinuity cannot be smeared out immediately without a viscous force. As a consequence, it has often been observed significant oscillations or fluctuations unphysically as shown in Fig. 3 in SPH simulations. Hence, an artificial viscosity term must be inserted explicitly in the momentum equation to prevent the unphysical oscillations and to keep the numerical solution stable. Three widely used forms of artificial viscosity were developed by Monaghan et al. (Monaghan & Gingold, 1983), Hernquist et al. (Hernquist & Katz, 1989), and Lattanzio et al.

SPH framework of geotechnical problems

FIG. 3

265

Numerical oscillation of pressure in SPH modeling of soil column collapse.

(Lattanzio, Monaghan, Pongracic, & Schwarz, 1986), respetively. In the applications of geotechnical problems, the form of artificial viscosity designed by Monaghan (Monaghan, 1992; Monaghan & Pongracic, 1985) is the most widely used and also adopted in this work. The artificial viscosity has a form of 8 2 > * < αΠ cij ϕij + βΠ ϕij , * v ij  x ij < 0 Πij ¼ (17) ρij > : * * 0, v ij  x ij  0 *

*

hij v ij  x ij ϕij ¼  2  2 *   x ij  + 0:1hij ρij ¼

(18)

ρi + ρj si + sj hi + hj , sij ¼ and hij ¼ 2 2 2

(19)

* v ij

(20)

*

*

¼ vi  vj

N Dvi α X ¼ mj Dt j¼1

* x ij

*

*

¼ xi  xj ! σ αβ σ αβ j αβ ∂Wij i + 2  Πij δ +fα ρ2i ρj ∂xβi

(21)

where s is sound speed in soil, whichpcan be ffiffiffiffiffiffiffiffiffiffi ffi calculated by the material elastic modulus and density, i.e., si ¼ Ei =ρi . αΠ and βΠ are two parameters in the artificial viscosity formulation. • The βΠ term handles high Mach number shocks, which is roughly equivalent to the Von Neumann-Richtmyer viscosity used in the finite difference method (Monaghan & Gingold, 1983). In the applications of geotechnical problems, this term contributes little to removing the unphysical oscillations since the velocity of particles is always very small compared to the sound speed. So βΠ is set as 0.

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12. SPH modeling for soil mechanics with application to landslides

• The αΠ term is designed by Monaghan to approximate the bulk viscosity in flows, and its viscous effect is controlled directly by the value of αΠ . In applications, the smallest αΠ should be selected which can remove the numerical oscillation completely. Generally, for geotechnical problems, αΠ ¼ 0.1 works well. Tensile instability and artificial stress “Tensile instability,” which means the SPH method is unstable as the material is in tension, is another instability problem of SPH resulting from the intrinsic features of SPH method. It was first proven by Swegle et al. (Swegle, Hicks, & Attaway, 1995) that the instability condition of SPH method is that the product of the kernel’s second derivative W00 and the total stress T must be greater than zero (W00 T > 0). The sign of the kernel’s second derivative W00 can be determined by checking the slope of first derivative curve in Fig. 4 (Liu & Liu, 2003). In light of the fact that the least distance among SPH particles is around a smoothing length h, i.e., at the position of R ¼ 1, the second derivative of the kernel is always positive. Therefore, the SPH method associated with cubic B-spline function is conditionally stable when the stress is in compression (T < 0) but unconditionally unstable in tension (T > 0). In simulations, the tensile instability 1

0.75

Functions x αd

0.5

0.25

0

–0.25

–0.5

–0.75 Smoothing function Derivative of the smoothing function –1 –2

–1.5

–1

–0.5

0

0.5

R

FIG. 4

The cubic B-Spline kernel and its first derivative.

1

1.5

2

SPH framework of geotechnical problems

FIG. 5

267

The tensile instability problem in cohesive soil column collapse problem.

could result in unphysical particle clumping and large voids as shown in Fig. 5, or even complete blowup in the computation. Since the tensile instability problem results from the tensile stress of soil, it can be improved by reducing the tensile stress explicitly. In specific, the key idea of “artificial stress” is to introduce an artificial compression stress into the stress of soil whose principle stress is positive (in tension) such that the tensile stress becomes compressive. This is equivalent to introducing a small repulsive force between two approaching particles to prevent them from getting closer. Please noted that this correction is only made in the momentum equation to avoid the unphysical motion of particles but not change the actual stress of material. This is the way that the “artificial stress” term resolves the tensile instability issue in SPH. This repulsive force, according to Monaghan (Monaghan, 2000), must increase as the separation between two particles decreases, which can be realized by the factor fijn in Eq. (24). Fig. 6 gives how fijn changes over

FIG. 6

fijn(q) with n ¼ 1.0/2.55/4.0 respectively.

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12. SPH modeling for soil mechanics with application to landslides

the distance between two particles. It shows that the repulsive force will increase by a factor of 11 (which depends on the exponent n’s value) as q decreases from initial space ratio Δ x to zero, and will decrease rapidly in the domain h  q  2 h when 2.55 is given to n as this chapter. Moreover, the strength of artificial stress is controlled by the parameter ε in Eq. (23). Since the artificial stress term will introduce errors into numerical results though it has been shown negligible by Gray and Monaghan (Gray, Monaghan, & Swift, 2001), the first choice of ε is the smallest value which is large enough to remove the tensile instability completely. ! N   ∂W σ αβ σ αβ Dvαi X ij j αβ αβ αβ n i ¼ mj + 2  Πij δ + fij Ri + Rj +fα (22) 2 β Dt ∂x ρ ρ i j j¼1 αβ R0 i

¼

8 > < > :

ε

σ 0 αβ i ρ2i

0

αβ

if σ 0 i > 0

(23)

otherwise

80 12:55 > > 2 1 > > B C  q2 + q3 > > B > 3 2 C > 0q  n > < @2 1 1 1 A W ð q, h Þ ij  + fijn ¼ ¼ 3 1:2 2 1:2 > Wij ðΔx, hÞ > > !2:55 > > > > ð2  qÞ3 > > 1q : ð2  1=1:2Þ3

(24)

xy

tan 2θi ¼

2σ i yy σ xx i  σi

(25)

xy

yy

xy

yy

2 xx 2 σ 0 xx i ¼ cos θ i σ i + 2sin θ cosθσ i + sin θσ i yy

2 σ 0 i ¼ sin 2 θi σ xx i  2 sin θ cos θσ i + cos θσ i 9 0 xx 2 0 yy 2 Rxx i ¼ R i cos θ i + R i sin θ i > = yy 2 0 yy 2 Ri ¼ R0 xx sin θ + R cos θ i i i >   i ; xy 0 yy sin θ Ri ¼ R0 xx  R cosθ i i i i

) (26)

(27)

Boundary treatments The boundary condition is one of the most vital treatments for ensuring the accuracy of SPH solution. In implementation, the no-slip and free-slip boundary treatments proposed by Bui et al. (2008), which can be dated back to Morris et al. (Morris & Monaghan, 1997) and Takeda et al. (Takeda, Miyama, & Sekiya, 1994), are referred.

SPH framework of geotechnical problems

FIG. 7

269

Initial particles arrangement and treatment of solid boundaries.

To simulate the no-slip boundary condition, three layers of virtual particles (boundary particles) are generated parallel to the solid wall. The virtual particles have the same spacing ratioΔd as the real particles and their density is set equal to the reference density ρ0 of real particles. As in the work of Morris et al. (Morris & Monaghan, 1997) and Bui et al. (Bui et al., 2008), boundary particles contribute to the SPH expression for velocity gradient and stress gradient in the governing PDEs in Eq. (1). The velocity of boundary particle B is extrapolated as vB ¼ (dB/dA)vA assuming zero velocity on the solid wall, where dA and dB are the normal distance of real particle A and virtual particle B from the solid wall, respectively, as illustrated in Fig. 7. This artificial velocity vB is then used for the evaluation of the velocity gradients of real particles near or on the boundary. Moreover, when real particles get too close to the solid boundary the artificial velocity will be extremely large. Hence, the following limit approach is applied to vB, vB ¼ ð1  βÞvA + βvwall  dB β ¼ min βmax , 1:0 + dA

(28) (29)

where vwall ¼ 0 m/s is the velocity of solid wall. According to Morris et al. (Morris & Monaghan, 1997) and Bui et al. (Bui et al., 2008), βmax ¼ 1.5 is adopted in this study. Only the velocity assignment to boundary particles cannot provide sufficiently large repulsive force preventing real particles from penetrating through the solid boundary in the applications of geotechnical engineering. Accordingly, some stress is necessarily assigned to the boundary particles meanwhile. For simplicity, the uniform local stress treatment for the no-slip boundary is adopted. That is, for a given real particle A, if a

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12. SPH modeling for soil mechanics with application to landslides

boundary particle B locates in its supporting domain, the stress tensor of real particle A will be assigned to the boundary particle B. The testing results (Bui et al., 2008; Mao et al., 2017), showed that this uniform local stress treatment gives a smooth stress contour in the simulations with small computational efforts. To simulate the free-slip boundary condition, the same treatment is applied as the no-slip boundary treatment including the boundary stress assumption, except for the velocity assignment in the direction of parallel to the solid boundary. Specifically, the tangent velocity of boundary particles is assumed to equal to that of the closest real particles to simulate the free-slip boundary condition (Mao et al., 2017).

Time integration Similar to other explicit hydrodynamic methods, the explicit SPH scheme can be integrated with the standard methods such as the second order accurate Leap-Frog (LF), the predictor-corrector and Runge-Kutta (RK) schemes etc. In this work, the LF scheme is applied considering its lower memory storage and better efficiency. The field variables are updated at each step as follows:  Dρ ρn + 1=2 ¼ ρn1=2 + Δt (30) Dt n  αβ Dσ αβ αβ Δt (31) σ n + 1=2 ¼ σ n1=2 + Dt n  Dv vn + 1=2 ¼ vn1=2 + Δt (32) Dt n xn + 1 ¼ xn + vn + 1=2 Δt

(33)

The explicit LF scheme is subject to the Courant-Friedrichs-Levy (CFL) condition for stability which requires the time step to be proportional to the smallest spatial resolution,  Δx Δt ¼ λ min (34) s with s being the sound speed in the media. Furthermore, due to the introduction of the artificial viscosity term, by referring to the work for SPH (Monaghan, 1989, 1992), the critical time step is subjected to the more rigid criterion shown in Eq. (35) instead of Eq. (34). If the material suffers from external force as well, the critical time step should also be subject to expression (36) (Monaghan, 1989, 1992).

271

Applications of SPH to geotechnical engineering

0 Δtcv ¼ min @

1 Δxi

  A si + 1:2 αΠ si + βΠ max ϕij

Δxi 1 2 Δtf ¼ min fi   Δt ¼ λ min Δtcv , Δtf

(35)



(36) (37)

where αΠ and βΠ are parameters in the artificial viscosity term in Eq. (17), f is the acceleration due to external force, and the safety factor λ ¼ 0.2 is adopted.

Flowchart The flowchart of SPH framework can be represented in Fig. 8.

Applications of SPH to geotechnical engineering This section presents the performance of SPH modeling applied to geotechnical engineering, e.g., soil column collapse and earthquake-induced landslides. Within these applications, some numerical results aiming at investigating the effects of artificial viscosity and artificial stress are presented as well. From these numerical experiments, readers can expect a deeper comprehension on the essential artificial terms in SPH modeling.

Soil column collapse Non-cohesive soil column collapse In this section, the SPH method is first applied to simulate the collapse process of non-cohesive soil. The soil properties employed in the corresponding experiment (Bui et al., 2008) are adopted in simulation as listed in Table 1. A total of 3200 real particles are employed to form an initial rectangular soil area 0.2 m in length and 0.1 m in height with an initial particle spacing of 0.0025 m. Three layers of boundary particles are used to generate the no-slip bottom wall and the free-slip left wall. The collapse process of non-cohesive soil simulated by SPH is shown in Fig. 9, and the free surface profile and failure line of soil column when the collapsing soil becomes standstill are compared with those in the experiment conducted by Bui et al. (Bui et al., 2008). It shows that the SPH simulation agrees well with the experimental data, which validates the

272

FIG. 8

12. SPH modeling for soil mechanics with application to landslides

Flowchart of SPH framework.

273

Applications of SPH to geotechnical engineering

TABLE 1

Soil properties in non-cohesive soil’s column problem.

Δx [m]

m [kg]

ρ [kg/m3]

K [MPa]

ν [unit]

c [Pa]

θ [°]

ψ [°]

0.0025

0.01656

2650

0.7

0.3

0

19.8

0

FIG. 9 Flow process of non-cohesive soil by LL-GSM. vx is the horizontal velocity with a m/s unit. The red (dark gray in print version) dashed curve represents the final profile while the green (light gray in print version) one represents the failure line in the experiment.

effectiveness of the SPH framework in handling granular flows with material strength in geotechnical engineering. Effect of artificial viscosity To explain the effect of the artificial viscosity term on the flow of soil, a parametric study is conducted with respect to αΠ , the controlling parameter in the artificial viscosity formulation as shown in Eq. (17). αΠ ¼ 0.05/0.1/0.5/1.0 are tested, respectively, and the soil properties listed in Table 2 are employed in simulation. Fig. 10 shows the stress fields under different αΠ values. It is seen that the numerical oscillation can be improved greatly by the artificial viscosity term and will be removed totally as αΠ is sufficiently large (αΠ > ¼ 0.1). To study how the artificial viscosity affects the flow process of granular column collapse, the velocity contours of collapsing soil column at an earlier stage (0.109 s) and a later stage (0.235 s) are presented in Fig. 11. As expected, we can see that the simulated soil column with larger αΠ (meaning stronger damping effect) moves stiffer and yields a smaller run-out distance than that with smaller αΠ . Finally, the flow process of soil column collapse by the numerical approach with αΠ ¼ 0.1 is compared carefully with the experiment results (Nguyen et al., 2017) as shown in Fig. 12. It shows that the numerical model associated with αΠ ¼ 0.1 in the artificial viscosity term works very well in granular flow simulation, and a very similar result is obtained as the experimental work by Nguyen et al. (2017).

274

12. SPH modeling for soil mechanics with application to landslides

TABLE 2

FIG. 10

Soil properties adopted in this study.

Name

Value

Unit

Density

20.4

kN/m3

Elastic modulus

5.84

MPa

Poisson’s ratio

0.3

unit

Cohesion

0

Pa

Internal friction angle

21.9

degree

Dilatancy angle

0

degree

Vertical stress contour using different αΠ for artificial viscosity.

In sum, the artificial viscosity term can eliminate the numerical oscillation issue completely in SPH simulation. Meanwhile, a too large αΠ value will cause an unphysically stiffer flow of soil and lead to a relatively shorter run-out distance.

Cohesive soil column collapse Then, the SPH method is applied to the cohesive soil column collapse process. In this simulation, the numerical model in “Non-cohesive soil column collapse” section is taken with cohesion of 300 Pa and internal friction angle of 40° as listed in Table 3. Due to the existence of cohesion, the soil column will experience tensile stress in the necking region. Under this condition, the SPH results without any special correcting technique will suffer from the tensile instability issue as shown in Fig. 13A. From Fig. 13B, we can see that the artificial stress treatment can effectively remove the “tensile instability” issue in SPH.

Applications of SPH to geotechnical engineering

275

Effect of αΠ in the artificial viscosity term on the flow process at time t ¼ 0.109 s (left) and t ¼ 0.235 s (right). Red (dark gray in print version) curves are the corresponding profiles in experiment (Nguyen, Nguyen, Bui, Nguyen, & Fukagawa, 2017).

FIG. 11

To study the influence of the artificial stress term on the flow of soil, the final profile of the collapsed cohesive soil column simulated by SPH is compared to that by another SPH-like meshfree method, the Lagrangian Gradient Smoothing Method or L-GSM (Mao & Liu, 2018b), as plotted in Fig. 14. In the SPH and L-GSM simulations, the same configurations of numerical model are employed. Advantageously, L-GSM is free from the “tensile instability” issue, and therefore the artificial stress correction is unnecessary. From Fig. 14, one can easily observe that, compared to the L-GSM result, the front edge of the final profile by the stable SPH method is 3% shorter. This is mainly because the artificial stress in SPH leads the soil column stiffer.

276

12. SPH modeling for soil mechanics with application to landslides

FIG. 12 Comparison between the numerical simulation and the experiment for soil column collapse (Nguyen et al., 2017) (αΠ ¼ 0.1).

TABLE 3 Soil properties in cohesive soil’s natural failure problem. Δx [m]

m [kg]

ρ [kg/m3]

K [MPa]

ν [unit]

c [Pa]

θ [°]

ψ [°]

0.0025

0.01656

2650

0.7

0.3

300

40

0

FIG. 13

Tensile instability conditions of SPH results without (A) and with (B) artificial stress (ε ¼ 0.5) in cohesive soil column collapse.

Applications of SPH to geotechnical engineering

277

FIG. 14 Comparison of final profiles by SPH with artificial stress and L-GSM without artificial stress for the cohesive soil column collapse.

Earthquake-induced landslides The earthquake-induced landslide is one of the most dangerous hazards in intense earthquake disasters. Its secondary geological hazard and casualty, such as blocking the roads and rivers, destroying the buildings, may cause even severe deaths of people and economic losses than the earthquake itself. The earthquake-induced landslide is a high-velocity flowing process of granular media with significantly large deformation. Hence, the meshfree methods have huge advantages in handling the earthquake-induced landslides over the traditional grid-based methods, e.g., FEM which can only address finite deformation problems. In this section, the SPH method is applied to a series of earthquakeinduced landslides in the 2008 Ms8.0 Wenchuan earthquake in China to evaluate its effectiveness and reliability in predicting the flowing process and run-out distance of landslides. The flowing process of landslides under the seismic loadings measured by GPS stations (Mao, Liu, Huang, & Bao, 2019) is simulated by using the SPH framework.

Daguangbao landslide The Daguangbao landslide, with an estimated volume of 800 million m3, is the largest one triggered by the Wenchuan earthquake. It is about 15 km far from the nearest MZQP observation station. According to the satellite map and post-earthquake onsite survey, the principal sliding direction locates in the direction of N60°E. The GPS recorded pre-earthquake topography (dashed curve) and onsite surveyed sliding slope (real curve) as shown in Fig. 15 were followed to generate the initial distribution of soil particles and boundary particles. In specific, 3566 real or soil particles are employed and distributed uniformly above the sliding curve with a spacing of d ¼ 16 m while the sliding slope is represented by 1420 boundary or virtual particles horizontally ranged from 0 to 5000 m with a spacing of d ¼ 16 along the sliding boundary. The sliding slope is assumed no-slip solid boundary in

278

FIG. 15

12. SPH modeling for soil mechanics with application to landslides

Record of pre-failure and post-failure topographies in Daguangbao landslide.

TABLE 4 Soil parameters in Daguangbao landslide simulations (Zhang, Zhang, Chen, Zheng, & Li, 2015). Δx [m]

ρ [kg/m3]

E [GPa]

ν [unit]

c [MPa]

ϕ [°]

ψ [°]

16

2500

1.86

0.2

1.276

10.8

0

simulation by adopting the no-slip boundary treatment in “Boundary treatments” section. αΠ ¼ 0.1 are selected in the artificial viscosity. The other necessary parameters used in L-GSM and SPH simulations of Daguangbao landslide are listed in Table 4. The velocity and pressure fields at four representative time points by SPH are presented in Figs. 16 and 17, respectively. It is shown by Figs. 16 and 17 that the SPH method associated with the adopted solid boundary treatment and artificial terms can generate smooth and

FIG. 16 Flow process of DaGuangBao landslide simulated by SPH modeling. vx represents the horizontal velocity.

Applications of SPH to geotechnical engineering

279

FIG. 17 Pressure contour of Daguangbao landslide at the four representative time points by SPH modeling.

desirable local variable fields even in the region nearby boundary. To evaluate the reliability of the SPH method in predicting the run-out distance and final topography of landslides, the SPH result is compared to the corresponding L-GSM result and the post-earthquake GPS satellite record as plotted in Fig. 18. From Fig. 18, one can easily observe the good agreement between the SPH result and the L-GSM result. The numerical results also show a fair agreement with the recorded data. The difference between the numerical results and the recorded data is mainly from the limitation of 2D numerical models in simulating 3D engineering problems owning bad 2D characteristics. Tangjiashan landslide Tangjiashan Mountain, on the right bank of the Tongkou River and 6 km upstream from Beichuan County, failed during the 2008 Wenchuan MS 8.0 earthquake. The subsequent landslide was composed of weathered

FIG. 18 Final profiles comparison of Daguangbao landslide by L-GSM (circle), SPH (square) and record.

280

12. SPH modeling for soil mechanics with application to landslides

schist, slate, and sandstone sliding along rock. The height difference between the landslide toe and the main back scar was 650 m, and the horizontal dimension of the landslide was 900 m (Xiewen, Runqiu, Yubing, Xiaoping, & Haiyong, 2009). The landslide formed an extremely large barrier lake, with water storage capacity 250 million m3 (Cui, Zhu, Han, Chen, & Zhuang, 2009). This lake is a serious threat to the city of Mianyang and other towns downstream. An SPH simulation of Tangjiashan landslide was conducted to study the forming mechanism of the impounded lake. Similar to the initial particle configuration in Daguangbao landslide, a total of 3986 real particles and 1820 boundary particles with a spacing of d ¼ 4 m are employed and uniformly distributed in initial by following the recorded and surveyed sliding slope (real curve in black) and pre-failure topography (dashed curve) as shown in Fig. 19. The other necessary soil parameters in SPH simulation are derived from Luo et al. (Luo, Hu, Gu, & Wang, 2012) as listed in Table 5. The flowing process of Tangjiashan landslide by SPH is presented in Fig. 20 and the final topographies by SPH, L-GSM and GPS record are compared in Fig. 19. We can observe from Fig. 19 that the SPH solution and L-GSM agree very well with each other, while they both show a good agreement with the recorded data. The SPH method yields a desirable variable field as shown in Fig. 20. This confirms the validation of the SPH framework in

FIG. 19

Final profiles comparison of Tangjiashan landslide by SPH, L-GSM and record.

TABLE 5 Soil parameters in Tangjiashan landslide simulations (Luo et al., 2012). Δx [m]

ρ [kg/m3]

K [GPa]

G [GPa]

c [kPa]

ϕ [°]

ψ [°]

4

2000

1.3

0.8

30

35

0

Applications of SPH to geotechnical engineering

281

FIG. 20

Simulated run-out process of Tangjiashan landslide at the four representative time points by SPH. vx represents the horizontal velocity.

handling large-scale granular flows with material strength and complex solid boundary profiles in the applications of earthquake-induced landslides.

Wangjiayan landslide As one of the most serious landslide disasters during the 2008 Wenchuan earthquake, the Wangjiayan landslide in Beichuan County is a typical high-speed flow-like landslide. It killed about 1600 people and destroyed hundreds of houses. The landslide was only 300 m away from the rupture zone of a main central fault, and was composed of Cambrian sandstone, shale, and schist. The landslide occurred on an anti-dip slope and had a volume of 4.8 million m3. The height difference between the front and rear edge was 350 m, with a sliding distance of 550 m (Yin, Wang, & Sun, 2009). A total of 3691 real particles and 1605 boundary particles with a spacing of d ¼ 2 m are employed and uniformly distributed in SPH simulation. The sliding slope and initial distribution of granular media are plotted in Fig. 21. Other necessary soil parameters can be found in Table 6. The Wangjiayan landslide is simulated by the SPH method. The flowing process of Wangjiayan landslide is given in Fig. 22, and the final profile by SPH is compared to the corresponding L-GSM solution and the onsite record as shown in Fig. 21. The numerical solutions agree well with the record topography and L-GSM solution.

282

12. SPH modeling for soil mechanics with application to landslides

FIG. 21 Topographies of Wangjiayan landslide by SPH, L-GSM, and the record topography.

TABLE 6 Soil parameters in Wangjiayan landslide simulations (Huang et al., 2012). Δx [m]

ρ [kg/m3]

K [GPa]

G [GPa]

c [kPa]

ϕ [°]

ψ [°]

2

2000

1.3

0.8

30

35

0

FIG. 22 Flow process of Wangjiayan landslide simulated by SPH. vx represents the horizontal velocity.

Advantages and limitations of SPH modeling

283

FIG. 23 Tensile instability conditions of SPH without (in A) and with (in B) artificial stress (ε ¼ 0.5) in Wangjiayan landslide simulation.

Due to the existence of protrusion in the sliding curve as shown in Fig. 21, the soil media will experience significant tensile stress when flowing through the protrusion, which causes apparent “tensile instability” issue in SPH simulation without special treatment as illustrated in Fig. 23A. It can be easily seen from Fig. 23B that the artificial stress term can effectively remove the “tensile instability” phenomenon and generate a more regular distribution of particles.

Advantages and limitations of SPH modeling From the knowledge about the SPH method provided above, we can see that the SPH framework owns the following key features relative to other numerical methods: (a) Good adaptability to large deformation problems: Without the limitations of grid quality or particles’ distribution to the accuracy condition of numerical solution, the SPH method can handle the large deformation problems very naturally and straightforwardly. Even for those engineering problems with extremely irregular distribution of particles such as explosion, the accuracy of SPH solution is still desirable. (b) Easy implementation procedure: Compared to the complicated grid generation process in the traditional grid-based methods, the distance-oriented neighboring particles searching algorithm for SPH is very straightforward and simple to implement. (c) Explicit solution of governing equations: As a kind of strong-form numerical method, the SPH method solves the governing PDEs directly through approximating the gradient of field variables with the

284

12. SPH modeling for soil mechanics with application to landslides

SPH gradient approximation formulation. The equation of state or constitutive equation can be integrated with the governing PDEs explicitly without the necessity of additional algebraic deductions. (d) Rigid conservation of mass and momentum: In the SPH framework, on the one hand, the mass conservation is automatically satisfied by taking the Lagrangian particle assumption, i.e., each SPH particle contains a constant mass without mass transfer among particles. On the other hand, the momentum conservation is guaranteed by assuring the anti-symmetric form of the SPH-discretized momentum equation represented by Eq. (22). Dvαi, j Dt

¼

Dvαj,i Dt

where the left-hand term is the acceleration of particle I due to the force from the particle j. The above relationship is true considering the fact of Wij ¼ Wji derived from Eq. (8). Even, the antisymmetric feature of the SPH-discretized momentum equation will not be broken by the symmetric artificial viscosity and artificial stress terms. (e) Easy development of parallel computing algorithms. Considering the explicit time integration scheme and the independent gradient approximation of field variable on each SPH particle, the corresponding parallel computing algorithm can be developed easily to achieve a desired computational efficiency. Nevertheless, it cannot be ignored that the SPH framework suffers from the following limitations: (a) Lower accuracy for small/finite deformation problems: It should be noted that the excellent adaptability of SPH to large deformation problems is achieved with the cost of lower-order accuracy and weaker connecting mechanism among particles. For small/finite deformation problems where the traditional second-order accurate grid-based methods are applicable, the first-order accurate SPH method has relatively lower accuracy. (b) Instability issue: Generally, the distance-oriented neighboring particle mechanism makes the SPH more subjective to the instability issue than the traditional grid-based methods. For instance, the “numerical oscillation” and “tensile instability” discussed in this study are the most widely existed instability issues in the standard SPH framework. Although many corrective techniques have been proposed, e.g., the adaptive kernel selection based on the stress state of each particle (Sirotkin & Yoh, 2012; Yang, Liu, & Peng, 2014) for removing the “tensile instability,” they will either lead to a much complicated implementation procedure or cause a very accuracy-sensitive numerical solution to the selection of controlling parameters.

Conclusions

285

(c) Lower computational efficiency: Because of the necessary updating of kernel functions and neighbors searching at every time step, the computational efficiency of the SPH framework is incomparable to the grid-based methods. The computation task of a series SPH algorithm with >10,000 particles will become very challenging for the traditional computer machines. (d) Difficult boundary treatments: Due to high dependence on wellbounded neighboring particles of SPH particles’ interaction, the boundary treatments are never easy for SPH modeling. Under most conditions, the virtual particles or ghost particles are also indispensable to mimic the effect of solid boundaries. Even a smooth field variable is achieved eventually, the rigid accuracy of boundary conditions is usually much harder to control than the grid-based methods. To overcome the above limitations in SPH, some major progresses have been made recently. For instance, to hold both the good accuracy of gridbased methods and the good adaptability of SPH to large deformation problems, the SPH method has been coupled with the grid-based methods for geotechnical problems (Mardalizad, Manes, & Giglio, 2017; Tan & Chen, 2017). Specifically, the grid-based methods are utilized to simulate the region of material under finite deformation while the SPH method is only applied to the region of material under large deformation. The tensile instability issue can be resolved by constructing a special kernel function (Dehnen & Aly, 2012) which can restore the stability condition of SPH formulation even the material is under tension. Alternatively, it can be prevented by replacing the SPH gradient formulation with another robust gradient smoothing operator (Mao & Liu, 2018b, 2019; Mao, Liu, Dong, & Lin, 2019; Mao, Liu, Huang, & Bao, 2019), which is inspired by its advantageous performance in Eulerian methods (Liu et al., 2008; Liu & Xu, 2008; Zhang, Liu, Lam, Li, & Xu, 2008). The computational efficiency of the SPH framework can also be significantly improved by developing parallel computing algorithms ( Ji, Fu, Hu, & Adams, 2019), GPU-acceleration (Chen, Lien, Peng, & Yee, 2020), or using an alternative gradient approximating strategy without the necessity of neighbors searching frequently (Mao & Liu, 2018b; Mao, Liu, & Huang, 2019).

Conclusions The SPH method, as a meshfree, Lagrangian particle method, has huge advantages of adaptability over the traditional grid-based methods in handling the practical problems with large deformation in geotechnical

286

12. SPH modeling for soil mechanics with application to landslides

engineering. The SPH modeling of geotechnical problems can be implemented in a much easier fashion than traditional grid-based methods. In SPH modeling of geotechnical problems, the artificial viscosity and artificial stress are essential to guarantee the stability of the SPH scheme. For specific cases, it may be necessary to carry out some exclusive numerical tests to find out the most desirable values for the controlling parameters in those artificial terms. The SPH framework is a reliable numerical approach for modeling the large deformation problems with arbitrary free surfaces in geotechnical engineering, e.g., slope stability simulation and landslide. The mechanical behaviors of geo-material can be accurately simulated by adopting an appropriate constitutive model.

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Mao, Z., & Liu, G. R. (2018b). A Lagrangian gradient smoothing method for solid-flow problems using simplicial mesh. International Journal for Numerical Methods in Engineering, 113 (5), 858–890. https://doi.org/10.1002/nme.5639. Mao, Z., & Liu, G. (2019). A 3D L-GSM framework with an adaptable GSD-constructing algorithm for simulating large deformation free surface flows. International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.6265. Mao, Z., Liu, G. R., & Dong, X. (2017). A comprehensive study on the parameters setting in smoothed particle hydrodynamics (SPH) method applied to hydrodynamics problems. Computers and Geotechnics, 92, 77–95. https://doi.org/10.1016/j.compgeo.2017.07.024. Mao, Z., Liu, G. R., Dong, X., & Lin, T. (2019). A conservative and consistent Lagrangian gradient smoothing method for simulating free surface flows in hydrodynamics. Computational Particle Mechanics. https://doi.org/10.1007/s40571-019-00262-z. Mao, Z., Liu, G. R., & Huang, Y. (2019). A local Lagrangian gradient smoothing method for fluids and fluid-like solids: A novel particle-like method. Engineering Analysis with Boundary Elements, 107, 96–114. https://doi.org/10.1016/j.enganabound.2019.07.003. Mao, Z., Liu, G., Huang, Y., & Bao, Y. (2019). A conservative and consistent Lagrangian gradient smoothing method for earthquake-induced landslide simulation. Engineering Geology. 260, https://doi.org/10.1016/j.enggeo.2019.105226. Mardalizad, A., Manes, A., & Giglio, M. (2017). The numerical modelling of a middle strength rock material under Flexural test by Finite Element method-coupled to-SPH. Procedia Structural Integrity, 3, 395–401. https://doi.org/10.1016/j.prostr.2017.04.050. Monaghan, J. J. (1989). On the problem of penetration in particle methods. Journal of Computational Physics, 82(1), 1–15. Monaghan, J. J. (1992). Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 30(1), 543–574. Monaghan, J. J. (2000). SPH without a tensile instability. Journal of Computational Physics, 159(2), 290–311. Monaghan, J. J., & Gingold, R. A. (1983). Shock simulation by the particle method SPH. Journal of Computational Physics, 52(2), 374–389. Monaghan, J. J., & Kocharyan, A. (1995). SPH simulation of multi-phase flow. Computer Physics Communications, 87(1–2), 225–235. Monaghan, J. J., & Pongracic, H. (1985). Artificial viscosity for particle methods. Applied Numerical Mathematics, 1, 187–194. Morris, J. P., & Monaghan, J. J. (1997). A switch to reduce SPH viscosity. Journal of Computational Physics, 136(1), 41–50. Nguyen, C. T., Nguyen, C. T., Bui, H. H., Nguyen, G. D., & Fukagawa, R. (Feb. 2017). A new SPH-based approach to simulation of granular flows using viscous damping and stress regularisation. Landslides, 14(1), 69–81. Seo, S., Min, O., & Lee, J. (2008). Application of an improved contact algorithm for penetration analysis in SPH. International Journal of Impact Engineering, 35(6), 578–588. Sirotkin, F. V., & Yoh, J. J. (2012). A new particle method for simulating breakup of liquid jets. Journal of Computational Physics, 231(4), 1650–1674. https://doi.org/10.1016/J. JCP.2011.10.020. Swegle, J. W., & Attaway, S. W. (1995). On the feasibility of using smoothed particle hydrodynamics for underwater explosion calculations. Computational Mechanics, 17(3), 151–168. Swegle, J. W., Hicks, D. L., & Attaway, S. W. (1995). Smoothed particle hydrodynamics stability analysis. Journal of Computational Physics, 116(1), 123–134. Takeda, H., Miyama, S. M., & Sekiya, M. (1994). Numerical simulation of viscous flow by smoothed particle hydrodynamics. Progress in Theoretical Physics, 92(5), 939–960. Tan, H., & Chen, S. (2017). A hybrid DEM-SPH model for deformable landslide and its generated surge waves. Advances in Water Resources, 108, 256–276. https://doi.org/10.1016/j. advwatres.2017.07.023.

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C H A P T E R

13 Use of coupled finite-infinite element in modeling of liquefaction Sunita Kumaria,b and V.A. Sawanta,b a

Department of Civil Engineering, National Institute of Technology Patna, Patna, Bihar, India b Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

Introduction The interaction of pore pressure with soil skeleton during earthquake results in the “weakening” of a soil-fluid composite which reduces the effective stress in soil mass, causing liquefaction. Liquefaction takes place often in saturated loose sands under earthquake and impact loadings. The pioneering experimental work associated with the liquefaction phenomenon and cyclic mobility was proposed by Seed and Lee (1966), Seed and Idriss (1971), Castro and Poulos (1977), Seed (1979) and Seed, Tokimatsu, Harder, and Chung (1985). The physical phenomenon is well defined, whereas the analytical modeling of soil liquefaction and computer simulation remains a challenge. Therefore, soil behavior has been analyzed by considering the effects of the transient flow of the pore-fluid through voids. Hence, it requires a two-phase continuum formulation for saturated porous media which is also known as coupled formulation. In coupled numerical analysis, the analysis domain, such as a liquefiable soil deposit, is expressed by coupled field equations. These equations are solved by considering coupling between solid and fluid phase under dynamic loading (inertial coupling is an added advantage). A fully coupled effective stress analysis accounts for the dynamic interaction between the solid and fluid phases. Resulting dynamic equilibrium

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00009-5

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© 2021 Elsevier Inc. All rights reserved.

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13. Use of coupled finite-infinite element in modeling of liquefaction

equation and flow continuity equation are solved simultaneously. Various researchers have studied the effectiveness of different fully coupled finite element codes for predicting the response of saturated soil under dynamic loading. Sandhu and Wilson (1969) started the finite element method (FEM) to study flow through saturated porous media. A variational principles approach has been incorporated into the dynamic stress equilibrium and continuity equation for a fully saturated soil. Ghaboussi and Wilson (1972) also applied a variational approach to the Biot’s dynamic field equations for saturated porous elastic media in the regime of finite element formulation. This formulation has considered the compressible nature of solid and fluid phases to simulate dynamic soil-structure interaction and wave propagation studies in saturated soils. Prevost (1980, 1982) discussed the multiphase formulation of porous media. The extended Biot’s theory with nonlinear elastic model was integrated to investigate the transient response of soil deposits. The two-phase saturated porous medium is defined by the stress, acceleration, velocity, and displacement fields within each phase. Zienkiewicz and Mroz (1984) and Zienkiewicz, Chan, Pastor, Schrefler, and Shiomi (1999) summarized different analyzing methods on numerical simulation of the Biot-type formulation. Later, the numerical solution has been applied to study the undrained, drained, and dynamic behavior of saturated porous media. Popescu and Prevost (1993) presented a validation of a numerical model based on multiyield plasticity theory using results of centrifuge model soil tests (VELACS). The model parameters were estimated from the results of conventional laboratory tests with numerical correlation formulae. A unique set of constitutive parameters were suggested to be used in the numerical simulation. Oka, Yashima, Shibata, Kato, and Uzuoka (1994) formulated an easy and realistic numerical approach for prediction of liquefaction response using u-p formulation (u-displacement of the solid phase and p-pore pressure). The FEM is applied for the stress equilibrium equation over the domain represented by a number of elements. An elasto-plastic constitutive model which incorporates nonlinear kinematic hardening rule is developed to illustrate the stress-strain behavior of loose sand subjected to cyclic loading. The validation of the proposed model is done by comparing the numerical response with analytical solutions for the transient response of saturated porous solids. Madabhushi and Zeng (1998) simulated the response of gravity quay walls subjected to seismic load using coupled finite element code SWANDYNE. A new numerical technique is used to model the absorbing boundaries, used for simulation of the free field condition in the centrifuge experiments. Cooke (2000) presented the fully coupled analyses for considering the fluid-solid interaction and highlighted the

Introduction

293

importance of some key parameters. These parameters along with the overall soil mass behavior might be difficult to evaluate in the analysis. Oka, Kodaka, Moromoto, and Kita (2002) incorporated finite deformation theory considering a cyclic elasto-plastic model for prediction of liquefaction response. Both u-p formulation and u-w-p formulation (displacement—relative fluid displacement—pore pressure) were assumed with an efficient Lagrangian FEM method to study the efficiency of u-w-p formulation. It was observed that the u-w-p formulation is essential for highly permeable soils during the earthquake with comparatively high frequency. Zhang (2003) proposed an advanced nonlinear computational model and implemented with OpenSees software to analyze the soilstructure-foundation domain of Middle Channel Bridge. The model demonstrated a simplified and better explanation of the seismic excitation along with the boundaries of the soil domain. Byrne et al. (2004) presented a numerical model that is based on an effective stress approach. This model is used to validate the centrifuge test results. A lack of full saturation and densification at any depth due to the application of the highacceleration level in centrifuge study were mostly responsible for the apparent limitation on liquefaction at that particular depth. Snieder and Beukel (2004) introduced the liquefaction cycle as a framework to define the coupled phenomena usually takes place in fluid-saturated granular soil, resulting in liquefaction. The numerical implementation of the liquefaction cycle evaluated that the scale analysis is unreliable in case of the presence of strong spatial variations in the permeability, which restrain fluid migration. The numerical model was applied to enumerate the effect of a relatively low-permeability layer on the liquefaction behavior of soil deposit. Jafari-Mehrabadi (2006) used DYNAFLOW for the successful simulation of eight centrifuge tests for finding the seismic liquefaction countermeasures of waterfront slopes. Popescu, Prevost, Deodatis, and Chakrabortty (2006) provided a finite element implementation of the extension of Biot’s theory of porous media for the dynamic behavior of saturated loose sand considering nonlinear behavior. The dynamic interaction between structure and liquefying soil was examined in the first application, whereas the second numerical application considered the liquefaction of stochastically spatially variable soils. The dependency of the characteristic frequency of the domain was highlighted. Taiebat, Shahir, and Pak (2007) developed a fully coupled dynamic algorithm based on u
P formulation to evaluate the liquefaction potential of saturated sandy deposits. Coupled equations were integrated into the time domain using a generalized Newmark method. A critical state two-surface plasticity model and a densification model were considered to characterize soil behavior. Jeremic, Cheng, Taiebat, and Dafalias (2008), Taiebat, Jeremic, Dafalias, Kaynia, and Cheng (2010) presented a simulation of pore fluid and soil skeleton responses using fully coupled dynamic field equations

294

13. Use of coupled finite-infinite element in modeling of liquefaction

with u
p
U formulation. This model also takes into account water accelerations (U) in the analysis. This model is functional in modeling dynamic interaction between media of different stiffnesses such as in soilfoundation-structure interaction. Dewoolkar, Chan, Ko, and Pak (2009) simulated the two centrifuge experiments on a similar type of cantilever retaining wall model having liquefiable backfill with using DIANASWANDYNE II program. The code is based on Biot’s formulation. The parameters of the Pastor-Zienkiewicz Mark III constitutive model used in the dynamic simulations of the soil were obtained from the backanalysis of the centrifuge test on the horizontal-ground model. Shahir and Pak (2010) presented a three-dimensional coupled dynamic analysis to discuss the dynamic behavior of shallow foundations on liquefied soils. The simulation of the proposed numerical model is verified and validated with the results of centrifuge experimental measurements. A sensible relationship was proposed for the assessment of liquefaction-derived displacement of a rigid footing resting on homogeneous loose to medium fine sand. Bao and Sture (2010) proposed a fuzzy set plasticity theory based cyclic constitutive model, which can model the dilatancy effect during seismic loading. The suggested model efficiently captures the features of a rise in pore pressure and consequent reduction in strength during cyclic loading. Shan, Ling, and Ding (2012) developed the exact solutions to the one-dimensional transient analysis of incompressible saturated single-layer porous media under four types of boundary conditions based on Biot’s theory of porous media. Taiyab, Alam, and Abedin (2010) presented a numerical simulation of granular soil under cyclic loading using a finite element code based on the coupled analysis. Based on the available literature it has been observed that partially or fully coupled finite element computer codes have been able to predict the measured behavior of liquefiable soils successfully. Liquefaction and spreading problems frequently relate to semiinfinite soil domain. However, in general, the FEM only satisfies the boundary displacement conditions of finite domains. For spatially unbounded seismic problems, the finite outer boundaries are problematic because undesired spurious reflections may be generated due to the reflection of waves from boundary. The problem is more critical when material damping considered in the analysis is not significant. Undesired reflections affect the numerical simulation results and should be avoided from the formulation. In the case of strong ground motion, this can be accomplished easily the material damping is usually substantial and relatively small amounts of energy are radiated away from the structure-soil system. Several techniques have been developed for modeling unbounded domains. For a geometrically complex continuum material, the finite element coupled with the infinite element approach seems to be a rational way to deal with the unbounded region problems.

Development of infinite element

295

The objective of the present study is to develop a coupled finite-infinite element based numerical model which simulates semiinfinite two-phase soil media during cyclic loading. The finite element formulation is based on Biot’s coupled equations of porous media. The infinite element has been used to deal with the unbounded region problems for a geometrically complex continuum material. Generalized plasticity, bounding surface, a nonassociated-type model known as Pastor-Zienkiewicz Mark III has been used to model nonlinearity of soil mass.

Development of infinite element The infinite element formulation is described here to model the unbounded domain for computational efficiency. Infinite elements extending to infinity are placed at the boundary of the computational domain. A mapped 2-D infinite element is developed for simulating the response of the unbounded domain. Infinite element shape functions are constructed for the infinite element. The unknown displacement in the infinite element varies in the infinite direction from the edge of the computational domain according to the selected decay function. The constructed element maintains compatibility between the finite domain and the infinite domain. The coupled 2-D finite element-infinite element model contains mixed 8-4 node displacement and pore pressure elements in the finite domain (as discussed in the previous chapter) and mixed 5-4 node displacement and pore pressure infinite element for simulating the infinite boundary in vertical and horizontal directions. At the corner mixed 3-3 node displacement and pore pressure infinite element is used to model unbounded nature in both directions. The displacement and excess pore pressure (EPP) at the infinity is assumed zero. To model the unbounded domain, 1-D infinite element is developed (Bettess, 1977; Patil, Sawant, & Deb, 2010; Sawant, Patil, & Deb, 2011, 2012). Then this technique can be extended to developing 2-D (Patil, Sawant, & Deb, 2013a) and 3-D infinite elements (Patil, Sawant, & Deb, 2013b). Shape functions of this element should be derived to incorporate its unbounded nature at one end. The 1-D infinite element is shown in Fig. 1, where X0 is the mapping origin, X1 is at the endpoint of the boundary of the finite domain, X2 is a selected point and X3 extends to the infinity. X2 is selected so that the distance between the mapping origin and the finite domain boundary is equal to the distance between the selected point X2 and the finite domain boundary. Then the coordinate of the selected point can be calculated by the coordinates of the mapping origin and the finite domain boundary.

296

13. Use of coupled finite-infinite element in modeling of liquefaction

X3 = ∞

(Mapping origin) X0

X1

1 x = –1

FIG. 1

X2

2 x=0

3

ξ

x=1

1-D infinite element.

X1
X0 ¼ X2
X1 ) X2 ¼ 2X1
X0

(1)

The shape functions of a 1-D infinite element are given by: N1∞ ðξÞ ¼


2ξ 1+ξ ; N ∞ ðξÞ ¼ 1
ξ 2 1
ξ

(2)

The coordinate for an arbitrary point within the infinite element can be written as: 2 X X¼ Ni∞ ðξÞXi (3) i¼1

In which, Xi represents the nodal coordinates. Apparently, from Eqs. (1) and (2), have X ¼ X1 at ξ ¼
1, X ¼ X2 at ξ ¼ 0 and X ¼ ∞ at ξ ¼ 1. The unknown (displacement or stress) in the infinite element is assumed to decay from the boundary of the finite domain. Therefore, a decay function should be specified in the infinite direction. The most frequently used decay functions include exponential function, reciprocal function, and so on. Now, shape functions derived from 1-D element are included in normal 2-D finite element to model unbounded domain in a specified direction. In the present study, mixed 5-4 node displacement and pore pressure infinite elements are employed for simulating the infinite boundary in vertical and horizontal directions (Figs. 2 and 3). At the corner mixed 3-3 node displacement and pore pressure infinite element is used to model unbounded nature in both directions. The shape functions for describing displacements and pore pressure within 2-D infinite elements at the left-hand side (LHS) boundary, right-hand side (RHS) boundary, vertical, and both corner elements used in the present study are presented in Table 1. For mixed 3-3 corner, infinite elements shape functions for displacement and pore pressure are given by the same expressions.

297

Development of infinite element

FIG. 2 2-D infinite elements showing displacement nodes: (A) horizontal LHS, (B) horizontal RHS, (C) left bottom corner, (D) vertical, and (E) right bottom corner.

g

g 3

2

S

S

4

3

w

w

w=0

(A)

g = +1

w = –1

(B) 4

w=0

2

3

1

g=0

2

1

w

1

w

g g

S

S

(D)

g = –1

(D)

S

g

S

(C)

2

S

3

w = +1

3

S

w

1

4

2

w = +1

S

S

1

w = –1

(E)

FIG. 3 2-D infinite elements showing pore pressure nodes: (A) horizontal LHS, (B) horizontal RHS, (C) left bottom corner, (D) vertical bottom, and (E) right bottom corner.

298

13. Use of coupled finite-infinite element in modeling of liquefaction

TABLE 1 Shape functions for displacement and pore pressure nodes of infinite elements. Horizontal LHS Displacement

Horizontal RHS Pore pressure

Þð1
ηÞ N1 ¼ ð1
ξ 2ð1 + ξÞ

p N1

Þ N2 ¼
ξðη1ð1
η + ξÞ

N3 ¼

2ξð1
η2 Þ ð1 + ξÞ

N4 ¼ ξηð1ð1++ξηÞ Þ

Þ ð1
ηÞ ¼ ð1
ξ 2ð1 + ξÞ

Displacement

Pore pressure

ð1
ηÞ N1 ¼ ξηð1
ξ Þ

1
ηÞ N1 ¼
ξðð1
ξ Þ

Þ N2 ¼ ξðð11
η + ξÞ

Þð1
ηÞ N2 ¼ ð1 +2ðξ1
ξ Þ

Þ ð1
ηÞ N2 ¼ ð1 +2ðξ1
ξ Þ

N3 ¼ ξðð11++ξηÞÞ

Þð1 + ηÞ N3 ¼ ð1 +2ðξ1
ξ Þ

Þ ð1 + ηÞ N3 ¼ ð1 +2ðξ1
ξ Þ

Þ ð1 + ηÞ N4 ¼ ð1
ξ 2ð1 + ξÞ

1 + ηÞ N4 ¼
ξηðð1
ξ Þ

1 + ηÞ N4 ¼
ξðð1
ξ Þ

p

p

p

Þð1 + ηÞ N5 ¼ ð1
ξ 2ð1 + ξÞ

N5 ¼


Vertical bottom

p

p

p

p

2ξð1
η2 Þ ð1
ξÞ

Displacement

Pore pressure

Left bottom corner Displacement and pore pressure

Þ ð1
ηÞ N1 ¼ ð1
ξ 2ð1 + ηÞ

Þ ð1
ηÞ N1 ¼ ð1
ξ 2ð1 + ηÞ

Þ N1 ¼ ð1 +2ðξ1
η Þð1 + ηÞ

Þ N2 ¼ ð1 2+ðξ1Þ+ð1
η ηÞ

Þ N2 ¼ ð1 2+ðξ1Þ+ð1
η ηÞ

Þ N2 ¼ ξηð1++3ξðÞξð+1 η
1 + ηÞ

N3 ¼ ξηð1ð1++ηÞξÞ

N3 ¼ ηðð11++ηξÞÞ

Þ N3 ¼ ð1 +2ðξ1
ξ Þð1 + ηÞ

Þ N4 ¼ ηðð11
ξ + ηÞ

Right bottom corner Displacement and pore pressure

N4 ¼

2ð1
ξ2 Þη ð1 + ηÞ

Þ N5 ¼
ξηð1ð1
ξ + ηÞ

p

p

p

p

2ð1
ηÞ N1 ¼ ð1
ξ Þð1 + ηÞ 2ð1 + ξÞ N2 ¼ ð1
ξ Þð1 + ηÞ + 3ð
ξ + η
1Þ N3 ¼
ξηð1
ξ Þð1 + ηÞ

Definition of problem In the present study, the FEM is used to solve numerically the coupled equation of mixture’s theories (Kumari, Sawant, & Sahoo, 2016). Plane strain condition is assumed to reduce the computational efforts. The saturated loose sand layer of thickness 10 m, underlain by 4 m depth of gravel had been considered for numerical simulation (Fig. 4). The unbounded soil domain in the XZ plane is discretized into 196 finite and infinite element mesh as shown in Fig. 4. Free drainage was assumed at the top surface only, while the lateral boundaries and the base were considered to be of infinite extent.

299

Definition of problem

22 m –8

+8

14 m

2m

Infinite element

–8

Infinite element

1m

+8

–8

–8

+8

Infinite element

Infinite element –8

10 m

Displacement + pore pressure

+8

–8

Kelvin element

Displacement

FIG. 4

Soil deposit considered for numerical simulation.

The transmitting boundary is approximated horizontally within the range of 10 m on each side from the center of the soil domain and vertically 12 m in the downward direction. Kelvin elements are connected in vertical and horizontal directions to the nodes of the transmitting boundary. Material properties and model parameters are taken from Sadeghian and Manouchehr (2012). The variation of displacement and excess pore pressure with time at different nodes had been calculated using finite element code written in FORTRAN-90. The variation of both parameters with time was considered for comparing the response. Analyses were performed in two steps: (1) static analysis and (2) dynamic analysis. A static analysis was performed to apply the gravitational forces due to self-weight of the soil before cyclic excitation. The resulted hydrostatic pressures of fluid and the stress state along with a soil column were used as initial conditions for the subsequent dynamic analysis. The coupled equations for static analysis were considered. When equilibrium condition was achieved for initial stress condition, a nonlinear analysis was performed for the cyclic/seismic excitation. Soil behavior under cyclic loading is complex. Hence, the constitutive relation used for the numerical prediction should be able to model the soil behavior during seismic loading considering permanent settlement, dilatancy, and hysteresis loops to predict the reasonable value of

300

13. Use of coupled finite-infinite element in modeling of liquefaction

displacements and excess pore pressure. Hence, a generalized plasticity, bounding surface, nonassociated type model known as PastorZienkiewicz Mark III (Pastor, Zienkiewicz, & Chan, 1990; Pastor, Zienkiewicz, & Leung, 1985; Zienkiewicz et al., 1999) has been used to model nonlinearity of soil mass. This model includes both volumetric and deviatoric plastic strains during loading and unloading in the hardening parameter of the bounding surface. The dynamic analyses were performed using a Generalized Newmark scheme (Katona & Zienkiewicz, 1985) with nonlinear iterations using an initial linear elastic tangential global matrix. The numerical integration parameters of the generalized Newmark’s method were selected as α ¼ 0.60 and β ¼ 0.3025 for the dynamic analysis. The time step used usually depends on time of cyclic loading and frequency of the input. Void ratio, permeability, and other geometric properties were kept constant during the analysis. Rayleigh damping of 5% is applied at the dominant frequency in the earthquake-like motion input to enhance the energy dissipation characteristic of the constitutive model.

Numerical study The model is analyzed for seismic excitation of El-Centro ground motion is considered for seismic loading in the parametric study. Response in the form of resultant displacements, liquefaction tendencies, excess pore pressure, and other parameter are studied. The time step used is the same as the time interval of input motion. Variation in predicted horizontal and vertical displacement at different depth with respect to time for El-Centro earthquake (k ¼ 6.6  10
5 m/s and G ¼ 8 MPa) has been shown in Figs. 5 and 6 respectively. The maximum values of a horizontal settlement of 8.79 cm are predicted at the top of soil layer; whereas maximum values of a vertical settlement of 9.64 cm are predicted at 8 m depth because of higher confining stress at the bottom. It has been observed that the maximum settlements occur after 15 s of loading. After attaining maximum value, settlement again decreased and almost become constant during the rest of the shaking period. A similar trend in displacement was reported by Taiebat et al. (2007) under seismic loading. Generally, the horizontal settlement is less than vertical settlement at different depths. Fig. 7 displays the computed excess pore pressure at different depths. The first maximum peak in EPP is attained at 5.73 s for a very small duration. At a depth of 2, 4, 6, and 8 m, EPP is higher than initial vertical stress at this particular time, resulting in the phenomenon of liquefaction. At 10 m depth of soil domain, an increase in excess pore pressure is less due to the gravel layer of higher permeability, hence no liquefaction is visible. After this time a sudden decrease in EPP is noticed. Again after

301

Numerical study

0

10

Time (S) 20

30

40

0.00

Displacement (m)

–0.02

–0.04

–0.06 Disp-X Top

–0.08

Disp-X 4 m Disp-X 8 m

–0.10 Horizontal displacement

FIG. 5

Variation in computed horizontal displacement at different depth.

0

10

Time (S) 20

30

40

0.00

Displacement (m)

–0.03

–0.06

–0.09

Disp-Z Top Disp-Z 4 m Disp-Z 8 m

–0.12

Vertical displacement FIG. 6

Variation in computed vertical displacement at different depth.

8.38 s of loading a maximum peak in EPP is noticed. But this high value of EPP is retained for a longer duration of time, showing complete liquefaction. The numerical analysis is very much efficient in dissipating energy. After attaining a maximum value of EPP, the model is efficient in dissipation the extra pore pressure. This trend is the same at all the depth of the

302

13. Use of coupled finite-infinite element in modeling of liquefaction

180 2m 4m

150

6m 8m

EPP (kPa)

120

10m 90

60

30

0 0

FIG. 7

5

10

15 Time (s)

20

25

30

Computed EPP at depths 2, 4, 6, 8, and 10 m with respect to time.

soil stratum. A similar trend of EPP was reported by Taiebat et al. (2007) under seismic loading. The stress paths presented in Fig. 8 show the characteristic mechanism of cyclic decrease in effective stress due to excess pore pressure build-up, captured using the Pastor-Zienkiewicz Mark III model. It is observed that the maximum stress ratio q/p is 1.05 at a depth of 0.5 m and 0.64 at 11.5 m depth. The trend indicates a decreases value with depth mainly due to the effect of overburden pressure. Figs. 9 and 10 show the computed horizontal and vertical acceleration time histories at different depths. It has been observed that the peak value of these parameters is found to be about 2.8 and 4.4 m/s2 at the top surface, 30

8

25 20 q (kPa)

q (kPa)

6 4

15 10

2 5

Depth 0.5 m

Depth 11.5 m

0

0 0

(A) FIG. 8

2

4 p (kPa)

6

0

8

(B)

5

10

15

p (kPa)

Computed effective stress path at different depths (A) 0.5 m (B) 11.5 m.

20

303

Numerical study

Acceleration (m/s2)

3.0 Accel-X Top 1.5 1.0 –1.5 –3.0 0

(A)

10

20

30

40

Time (S) 4.0

Acceleration (m/s2)

Accel-X 4 m 2.0 0.0 –2.0 –4.0 0

10

(B)

20

30

40

Time (S)

Acceleration (m/s2)

1.5 Accel-X 8 m

1.0 0.5 0.0 –0.5 –1.0 –1.5 0

(C)

10

20

30

40

Time (S)

FIG. 9

Computed horizontal acceleration time histories at different depths. (A) Top, (B) 4 m, (C) 8 m.

resulting in higher settlement. A relatively less value of accelerations is seen at 4 m depth, corresponding to lesser excess pore pressure. A sudden peak in acceleration response is noticed at 5.73 s. At this particular time, a sudden increase in displacement, as well as EPP, was also noticed because soil deposits show peak amplification in acceleration response at this point of time. This trend occurs only for a very small span of time. A less value of acceleration is reported in both directions after 15.62 s of shaking. Results indicate amplification of earthquake input

304

13. Use of coupled finite-infinite element in modeling of liquefaction

Acceleration (m/s2)

4.0 Accel-Z Top 2.0 0.0 –2.0 –4.0 0

(A)

10

20

30

40

Time (S)

Acceleration (m/s2)

5.0 Accel-Z 4 m

3.0 1.0

–3.0 –5.0 0

(B)

10

20

30

40

Time (S)

Acceleration (m/s2)

2.5 Accel-Z 8 m

1.5 0.5 –0.5 –1.5 –2.5 0

(C) FIG. 10

10

20 Time (S)

30

40

Computed vertical acceleration time histories at different depths. (A) Top, (B) 4 m,

(C) 8 m.

motion from base to the top surface showing maximum amplification at 4 m depth. Sharp spikes are observed at 15 s of shaking at all the depth of soil stratum. The settlement and EPP are also very high at this particular time, marking the occurrence of liquefaction phenomena. In acceleration

305

Concluding remarks

time history involving the soil densification, a significant decrease is observed in asymmetric spiky response after liquefaction. This can be explained by lesser lateral displacements after 15.62 s. The acceleration values are comparable in shape and magnitude to the input ground motion, showing an attenuation-spiky behavior. This behavior may be defined as the gradual EPP-induced strength degradation, with stresspath excursions along the phase transformation line. Finally, full attenuation of ground motions occurs at the time when shear strength is almost lost due to liquefaction. It is found that the behavior of liquefaction response progress from the surface downwards, in due course of time affecting the entire soil domain. Fig. 11 depicts the variation of computed EPP along with the depth for the case k ¼ 6.6  10
5 m/s and G ¼ 8 MPa. It has been observed that the value of EPP is increasing along with the depth but at 10 m depth, a reducing trend has been predicted due to the presence of gravel deposit having higher permeability. Here also, it is observed that up to 9 m (Z-coordinate 5 m), EPP is increasing with depth and after that reduction in EPP is observed.

Concluding remarks The present model developed is able to predict reasonable changes in excess pore pressure occurring during seismic load, which can be useful for analyzing earth structures situated in the regions of moderate to high seismic zone. It allows the distribution of pore pressure and the effects that drainage and internal flow have on the time of liquefaction to be

EPP (kPa) 50

75

100

125

150

0 k = 6.6 × 10–5 m/s and G = 8 MPa 2

Depth (m)

4 6 8 10 12

FIG. 11

Variation of EPP along depth.

175

306

13. Use of coupled finite-infinite element in modeling of liquefaction

determined quantitatively. A maximum vertical settlement of 9.64 cm at 8 m depth and horizontal displacement of 8.79 cm at the top surface are observed. It is observed that maximum stress ratio q/p is 1.05 at the depth of 0.5 m, which is decreasing with depth as overburden pressure increases with depth. This results in the development of higher excess pore pressure at shallow depth. In general, it is observed that vertical displacement is higher than horizontal displacement. Due to shaking, soil will try to compact and settlement will occur in a vertical direction.

References Bao, Y., & Sture, S. (2010). Application of a kinematic-cyclic plasticity model in simulating sand liquefaction. International Journal of Advances in Engineering Sciences and Applied Mathematics, 2(3), 119–124. Bettess, P. (1977). Infinite element. International Journal for Numerical Methods in Engineering, 11, 53–64. Byrne, P. M., Park, S., Beaty, M., Sharp, M., Gonzalez, L., & Abdoun, T. (2004). Numerical modeling of liquefaction and comparison with centrifuge tests. Canadian Geotechnical Journal, 41, 193–211. Castro, G., & Poulos, S. J. (1977). Factors affecting liquefaction and cyclic mobility. Journal of Geotechnical Engineering, ASCE, 103(6), 501–516. Cooke, H. G. (2000). Ground improvement for liquefaction mitigation at existing highway bridges. (Ph.D. thesis), Virginia Polytechnic Institute and State University. Dewoolkar, M. M., Chan, A. H. C., Ko, H., & Pak, R. Y. S. (2009). Finite element simulations of seismic effects on retaining walls with liquefiable backfills. International Journal for Numerical and Analytical Methods in Geomechanics, 33, 791–816. Ghaboussi, J., & Wilson, E. L. (1972). Variational formulation of formulation of dynamics of fluid-saturated porous elastic solids. Journal of the Engineering Mechanics Division, 98(4), 947–963. Proceedings of the American Society of Civil Engineers. Jafari-Mehrabadi, A. (2006). Seismic liquefaction countermeasures for waterfront slopes. (Ph.D. thesis), Memorial University of Newfoundland. Jeremic, B., Cheng, Z., Taiebat, M., & Dafalias, Y. (2008). Numerical simulation of fully saturated porous materials. International Journal for Numerical and Analytical Methods in Geomechanics, 32, 1635–1660. Katona, M. G., & Zienkiewicz, O. C. (1985). A unified set of single step algorithms Part 3: The Beta-m method, a generalization of the Newmark scheme. International Journal for Numerical Methods in Engineering, 21(7), 1345–1359. Kumari, S., Sawant, V. A., & Sahoo, P. P. (2016). Assessment of effect of cyclic frequency and soil modulus on liquefaction using coupled FEA. Indian Geotechnical Journal, 46(2), 124–140. Madabhushi, S. P. G., & Zeng, X. (1998). Seismic response of gravity quay walls II: Numerical modeling. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(5), 418–427. Oka, F., Kodaka, T., Moromoto, R., & Kita, N. (2002). An elasto-plastic liquefaction analysis method based on finite deformation theory. In: Proc. 15th KKCNN 2002, NUS, Dec. 19–20, pp. G81–G88. Oka, F., Yashima, A., Shibata, T., Kato, M., & Uzuoka, R. (1994). FEM-FDM coupled liquefaction analysis of a porous soil using an Elasto-Plastic Model. Applied Scientific Research, 52, 209–245.

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Pastor, M., Zienkiewicz, O. C., & Chan, A. H. C. (1990). Generalized plasticity and the modeling of soil behavior. International Journal for Numerical and Analytical Methods in Geomechanics, 14(3), 151–190. Pastor, M., Zienkiewicz, O. C., & Leung, K. H. (1985). Simple model for transient soil loading in earthquake analysis. II: Non-associative models for sands. International Journal for Numerical and Analytical Methods in Geomechanics, 9(5), 477–498. Patil, V. A., Sawant, V. A., & Deb, K. (2010). Use of finite and infinite elements in static analysis of pavements. Interaction and Multiscale Mechanics, 3(1), 95–110. Patil, V. A., Sawant, V. A., & Deb, K. (2013a). 2-D finite element analysis of rigid pavement considering dynamic vehicle pavement interaction effects. Applied Mathematical Modelling, 37, 1282–1294. Patil, V. A., Sawant, V. A., & Deb, K. (2013b). 3-D finite element dynamic analysis of rigid pavement using infinite elements. International Journal of Geomechanics, 13(5), 533–544. Popescu, R., & Prevost, H. (1993). Centrifuge validation of a numerical model for dynamic soil liquefaction. Soil Dynamics and Earthquake Engineering, 12, 73–90. Popescu, R., Prevost, J. H., Deodatis, G., & Chakrabortty, P. (2006). Dynamics of nonlinear porous media with applications to soil liquefaction. International Journal of Soil Dynamics and Earthquake Engineering, 26(6–7), 648–665. Prevost, J. H. (1980). Mechanics of continuous porous media. International Journal of Engineering Science, 18(5), 787–800. Prevost, J. H. (1982). Nonlinear transient phenomena in saturated porous media. Computer Methods in Applied Mechanics and Engineering, 20, 3–18. Sadeghian, S., & Manouchehr, L. N. (2012). Using state parameter to improve numerical prediction of a generalized plasticity constitutive model. Journal Computers & Geosciences, 51, 255–268. Sandhu, R. S., & Wilson, E. L. (1969). Finite element analysis of seepage in elastic media. Journal of the Engineering Mechanics Division, ASCE, 95, 641–652. Sawant, V. A., Patil, V. A., & Deb, K. (2011). Effect of vehicle-pavement interaction on dynamic response of rigid pavements. Geomechanics and Geoengineering, 6(1), 31–39. Sawant, V. A., Patil, V. A., & Deb, K. (2012). Finite element analysis of rigid pavement on a nonlinear two parameter foundation model. International Journal of Geotechnical Engineering, 6(3), 275–286. J. Ross Publication, U.S.A. Seed, H. B. (1979). Soil liquefaction and cyclic mobility evaluation for level ground during earthquakes. Journal of the Geotechnical Engineering Division, ASCE, 105(2), 201–255. Seed, H. B., & Idriss, I. M. (1971). Simplified procedure for evaluating soil liquefaction potential. Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, 92(6), 1249–1273. Seed, H. B., & Lee, K. L. (1966). Liquefaction of saturated sands during cyclic loading. Journal of Geotechnical Engineering, ASCE, 92(6), 105–134. Seed, H. B., Tokimatsu, K., Harder, L. F., & Chung, R. M. (1985). Influence of SPT procedures in soil liquefaction resistance evaluations. Journal of the Geotechnical Engineering Division, ASCE, 111(12), 1425–1445. Shahir, H., & Pak, A. (2010). Estimating liquefaction-induced settlement of shallow foundations by numerical approach. Journal of Computers and Geotechnics, 37(3), 267–279. Shan, Z., Ling, D., & Ding, H. (2012). Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media. Applied Mathematics and Mechanics, 34(1), 75–84. Snieder, R., & Beukel, A. V. D. (2004). The liquefaction cycle and the role of drainage in liquefaction. Granular Matter, 6(1), 1–9. Taiebat, M., Jeremic, B., Dafalias, Y. F., Kaynia, A. M., & Cheng, Z. (2010). Propagation of seismic waves through liquified soils. Soil Dynamics and Earthquake Engineering, 30(4), 236–257.

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Taiebat, M., Shahir, H., & Pak, A. (2007). Study of pore pressure variation during liquefaction using two constitutive models for sand. Soil Dynamics and Earthquake Engineering, 27, 60–72. Taiyab, M. A., Alam, M. J., & Abedin, M. Z. (2010). Numerical simulation of level ground loose sand deposit subjected to cyclic loading. In: Proc., Bangladesh geotechnical conf., Bangladesh society for geotechnical engineering in association with ISSMFE, Dhaka, Bangladesh, 346–352. Zhang, Y. (2003). Treatment of seismic input and boundary conditions in nonlinear seismic analysis of a bridge ground system. In: 16th ASCE engineering mechanics conference, (July 16–18, Seattle), 1–11. Zienkiewicz, O. C., Chan, A. H. C., Pastor, M., Schrefler, B. A., & Shiomi, T. (1999). Computational geomechanic: With special reference to earthquake engineering. New York: Wiley. Zienkiewicz, O. C., & Mroz, Z. (1984). Generalized plasticity formulation and applications to geomechanics. In C. S. Desai, & R. H. Gallagher (Eds.), Mechanics of engineering materials (pp. 655–679). New York: Wiley.

C H A P T E R

14 Assessment of undrained shear strength using ensemble learning based on Bayesian hyperparameter optimization Wu Chongzhia,b,c, Wang Lina,b,c, and Wengang Zhanga,b,c a

Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, People’s Republic of China, b National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas, Chongqing University, Chongqing, People’s Republic of China, cSchool of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China

Introduction Soft sensitive clays are generally characterized by low shear strength and high compressibility and widely distributed in a near marine environment. Therefore, geotechnical design involving soft sensitive clays is rather challenging (D’Ignazio, Phoon, Tan, & L€ansivaara, 2016). Nowadays, the need for the building on soft sensitive clays is increasing considering the recent tendency to infrastructure development and connectivity projects at the port. Hence, an accurate assessment of undrained shear strength by a reliable method is a critical issue for geotechnical engineers. The undrained shear strength of soft sensitive clays can be assessed by various laboratory tests as well as in situ tests. Most commonly, it is tested from situ, such as typically field vane shear testing (FVT) and piezocone cone penetration testing (CPTU). Generally, the in situ test is costly and time-consuming. Besides, there is a growing consensus that there is

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00014-9

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© 2021 Elsevier Inc. All rights reserved.

310

14. Assessment of undrained shear strength

inevitable uncertainty of correlation factors between the undrained shear strength and primarily CPTU parameters. As for laboratory tests, it requires relatively sophisticated and expensive sampling and laboratory testing techniques (Ching & Phoon, 2018; D’Ignazio et al., 2016; Zhang, Han, et al., 2020). The necessary geotechnical design parameters are not always directly measured from laboratory and in situ tests, rather often estimated from empirical or numerical correlations that are developed from regression fitting to a prescribed dataset. Since the pioneering work of previous researchers, the transformation model was proposed based on empirical or semiempirical regression analyses to evaluate the related design parameters of soil according to its properties (e.g., Jamiolkowski, 1985; Kulhawy & Mayne, 1990). However, the transformation model usually contains a significant amount of uncertainties from the soil properties, soil behavior, and site geology, etc. These may differ from the data source, against which the transformation models are calibrated (D’Ignazio et al., 2016; Phoon & Kulhawy, 1999). Therefore, the limitation of such models needs to be recognized or else may result in bias as a direct consequence. To augment engineer’s judgment, many the probabilistic transformation models based on Bayesian methods have been proposed by scholars including Ching and Phoon (2012, 2018), Ching, Phoon, & Chen, 2014, Wang and Cao (2013), Cao and Wang (2014), Wang and Aladejare (2016), Wang, Cao, Li, Phoon, and Au (2018), Zhang, Wu, Zhong, Li, and Wang (2020), and Wang, Tang, Wang, Liu, and Zhang (2020). Artificial intelligence (AI) techniques have developed rapidly in the past near years, many novel machine learning (ML) algorithms have been proposed and widely used in various fields and may provide a new perspective in this regard. This is in alignment with the International Society for Soil Mechanics and Geotechnical Engineering’s (ISSMGE’s) latest initiative to explore machine learning methods in geotechnical engineering (Ching & Phoon, 2018). Machine learning algorithms, such as artificial neural network (ANN) (Goh, 1995; Teh, Wong, Goh, & Jaritngam, 1997; Yousefpour et al., 2011; Zhang, Wu, Li, Wang, & Samui, 2019; Zhang et al., 2019), Bayesian network (BN) (Koduru, 2019; Li, Zhang, & Zhang, 2018), support vector machine (SVM) (Rodriguez-Galiano, Sanchez-Castillo, Chica-Olmo, & Chica-Rivas, 2015; Zhang, Wu, Ji, & AbouRizk, 2016; Zhou, Li, & Mitri, 2016; Zhou et al., 2016), and multivariate adaptive regression splines (MARS) (Zhang & Goh, 2013; Wang et al., 2020; Zhang, Goh, Zhang, Chen, & Xiao, 2015), etc., have gradually become the alternative resolution for geotechnical problems. However, a single sophisticated algorithm may be not a consideration to build design soil parameters’ surrogate model, since database derived from different sites have different soil properties, soil behavior, and site geology. Consequently, there is a necessary need to

Introduction

311

apply ensemble methods in geotechnical engineering. Ensemble learning combines multiple algorithms that process different hypotheses to form a better hypothesis, so its predictions perform well (Nascimento, Coelho, & Canuto, 2014; Xia, Liu, Li, & Liu, 2017). Ensemble methods are better than single ML and other statistical methods as reported by Nanni and Lumini (2009) and Lessmann, Baesens, Seow, and Thomas (2015). Therefore, it is promising to encourage using it for further application in the geotechnics. Ensemble methods can be broadly categorized into two methods according to their structures: bagging (parallel) and boosting (sequential). The bagging ensemble method is the combination of different learning algorithms, each of which generates an independent model in parallel (Xia et al., 2017). As a famous bagging ensemble learning method, random forest (RF) is understandable because it can be regarded as a process of consensus decision trees making, each decision tree decides independently on a specific issue, and then the results are combined to make final decisions (Breiman, 2001). Several researchers have also applied RF to solve geotechnical engineering problems, such as Zhou, Li, and Mitri (2016), Zhou, Shi, et al. (2016) investigated the feasibility of using RF to forecast surface movements induced by tunnel construction. Zhang, Wu, et al. (2019), Zhang, Zhang, et al. (2019) constructed RF and MARS models for the prediction of pile drivability with numerous data. In boosting, the decision trees are built boosting such that each subsequent decision tree aims to reduce the errors of the previous decision tree. Each decision tree learns from its predecessors and updates the residual errors. Hence, the tree that grows subsequently is apt to learn from updated residuals (Friedman, 2001). Extreme Gradient Boosting (XGBoost), which is an advanced supervised algorithm proposed by Chen and Guestrin (2016) under the boosting framework, which has been widely recognized in Kaggle machine learning competitions due to its advantages of high robustness and sufficient flexibility. So far, the XGBoost-based model is rarely used in geotechnical engineering. Motivated by the benefits mentioned above, XGBoost-base and RF-based ensemble learning methods are developed for capturing the relationships between the undrained shear strength and various basic soil parameters in this chapter. To reduce the dependence on the rule of thumb and brute-force search, the Bayesian optimization method is used to find the appropriate models hyperparameters. The remainder of this chapter starts with the introduction of two efficient ensemble learning algorithms, followed by the introduction of the datasets investigated. Then, four performance measures were employed to analyze and assess the predictive results. The last part gives the developed models are used as a contrastive analysis with two comparison machine learning methods and two transformation models under 5-fold CV for better persuasive results.

312

14. Assessment of undrained shear strength

Methodology Extreme gradient boosting XGBoost, which is an advanced supervised algorithm proposed by Chen and Guestrin (2016) under the Gradient Boosting framework, which has been widely recognized in Kaggle machine learning competitions due to its advantages of high efficiency and sufficient flexibility. For the objective function, XGBoost’s loss function adds additional regularization term, which helps to smooth the final learnt weights to avoid overfitting (Chen & Guestrin, 2016). It also uses the first- and secondorder gradient statistics to optimize the loss function. Moreover, in addition to adding regular terms to prevent overfitting, XGBoost also supports row and column sampling to solve this issue. Quicker model exploration is possible as the parallel and distributed computing ensures faster learning. In the following paragraphs, the framework of XGBoost is briefly introduced. The estimated output ^ yi of the gradient boosting tree model can be expressed as the sum of the prediction score fk(xi) of all trees: ^ yi ¼

K X

fk ðxi Þ, fk  Γ

(1)

k¼1

where Γ is the space of regression trees, and K is the number of regression trees, xi represents the features corresponding to sample i. For a given dataset, there is a prediction score fk(xi), also known as leaf weight, for each leaf node j. The leaf weight ωj is the regression value of all samples at this leaf node j in this tree, in whichj  {1, 2, …T}, and T is the number of leaves in the tree. Objective functions are the most basic expression in ML problems, and the process of boosting continues until the objective functions reduction becomes limited. To approximate the set of functions adopted in the model, which define the following regularized objective function as: Φ¼

n  T X  1 X l yi , ^ yi + γT + λ ω2j 2 i¼1 j¼1

where n is given data samples, and

(2)

n   P l yi , ^ yi is the training loss function

i¼1

describing the model how well fit with training data. γT + 12 λ

T P j¼1

ω2j is a reg-

ularization term for penalizing the complexity of the model. In the regularization term, γ is the complexity cost by introducing additional leaf, λ is a regularization hyperparameter, and ω2j is the L2 norm of leaf node j weights.

313

Methodology

All the trees are built sequentially in the additive learning processes, each newly added tree learns from its former trees and updates the residðk1Þ uals in the prediction values. So ^ yi has been already included the iter-

ðkÞ ation results of all the trees. Therefore, for the k-th iteration, y^i can

^ði k1Þ + fk ðxi Þ, and the objective function Φ(k) is written as: represent y ΦðkÞ ¼

n  T  X 1 X ðk1Þ l yi , ^ yi + fk ðxi Þ + γT + λ ω2j 2 i¼1 j¼1

(3)

To efficiently optimize the objective in the general setting for the first term loss training function, we approximate it using the second-order Taylor expansion.  n   T  X 1 1 X ðk1Þ ΦðkÞ ’ l yi , ^ yi ω2j (4) + gi fk ðxi Þ + hi fk2 ðxi Þ + γT + λ 2 2 i¼1 j¼1     yðk1Þ and hi ¼ ∂2^yðk1Þ l yi , ^ yðk1Þ are first- and secondwhere gi ¼ ∂^yðk1Þ l yi , ^ order gradient statistics the loss function, respectively. The constant terms can be removed to get the following approximate objective in step k:  n  T X 1 1 X ΦðkÞ ’ gi fk ðxi Þ + hi fk2 ðxi Þ + γT + λ ω2j (5) 2 2 i¼1 j¼1 We define tree by a vector of scores in leaves, and a leaf index mapping function that maps an instance to a leaf j, and this process can be expressed n T P P as fk ðxÞ ¼ ωj , and Eq. (5) can be rewritten as follows: i¼1

j¼1

Φ ð kÞ ¼

T X j¼1

20 4@

X iIj

1

0 1 3 X 1 g i A ωj + @ hi + λAω2j 5 + γT 2 iI

(6)

j

Given a fixed tree structure, the optimal leaf weight scores on each leaf ∗ node ω∗j , and extreme value of Φ(k) is solved by simple quadratic programming: X gi iIj

ω∗j ¼  X

hi + λ

(7)

iIj

0

X

12

@ gi A T X iIj 1 X Φ∗ðkÞ ¼  + γT 2 j¼1 hi + λ iIj

(8)

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14. Assessment of undrained shear strength

Eq. (8) can be regarded as a structure scoring function that measures how suitable a given vector of leaf scores is. A smaller value is preferred because it better fits the data. To avoid an infinite number of possible tree structures, a greedy algorithm has been adopted to find an optimal tree structure in practical applications. More detailed explanations of the XGBoost algorithm are referred to Chen and Guestrin (2016).

Random forest The RF is a powerful ensemble learning method based on classification and regression trees (CART). The first algorithm for random decision forests was created by Ho (1995), and the extension of the algorithm was developed by Breiman (2001). It has been widely used in many aspects and exhibits satisfactory performance, which can apply for classification and regression. RF is a statistical learning theory that uses the bootstrap resampling method to extract multiple samples from the original sample, model the decision tree for each bootstrap sample, and then combine the predictions of multiple decision trees to average for the final forecast results. The model increases the diversity of the decision trees by having a put-back sample and randomly changing the combination of predictors in different tree evolutions. The primary intent of this research is to predict regression, so only the regression tree (RT) will be introduced in this section. At each branching of RT, the mean of the samples on the leaf nodes and the mean square error (MSE) formed between each sample were calculated. Pursuing the minimum the leaf node MSE as branching condition, until no more features are available, or the overall MSE is optimal, the regression tree will stop growing. It consists of two critical custom parameters: the number of regression trees (N estimators) and the number of random variables of nodes (Max depth) (Zhou et al., 2019). These parameters must be optimized to minimize the errors that occur during data processing. The modeling process of the RF algorithm is as follows: adopt a bootstrap sampling technique to extract N estimators training sets from the original data set. Each training set is about two-thirds of the size of the original data set. Each bootstrap sample of the RF during the training process will have about one-third of the data not drawn. This part of the data is called out-of-bag data. The following step is creating a regression tree for each bootstrap training set. A total of N estimators’ regression trees are formed to form a “forest,” but these regression trees are not pruned. In the growth process of each tree, all optimal attributes are not selected as the internal node for branching, alternatively, the optimal attribute is selected from the randomly selected Max depth attributes for branching.

Methodology

315

Thus, RF algorithm increases the difference between the regression models by constructing different training sets, thereby improving the extrapolation prediction ability of the combined regression model. Through n-time model training, a regression model sequence {t1(x), t2(x), …, tk(x)} is obtained, which is used to form a multiregression model system (Forest), collect the prediction results of the N estimators’ regression tree, and then adopt a simple average strategy to calculate the value of the new sample. The ultimate regression decision formula is as follows: ^f K ðxÞ ¼ 1 rf K

K X

t i ð xÞ

(9)

k¼1

where ^f rf ðxÞ represents the combined regression model, ti is a single decision tree regression model, and K is the number of regression trees (N estimators). K

Bayesian hyperparameter optimization In machine learning, hyperparameter is parameter whose value need to be preset before the learning process, and algorithms are rarely hyperparameter free. For XGBoost, RF and comparative ML algorithms, all have several hyperparameters that have huge impact on the predictive accuracy of models. Therefore, reasonable tuning of these hyperparameters, i.e., hyperparameter optimization, is important. However, hyperparameter optimization is a combinatorial optimization problem, which cannot be optimized by gradient descent method like general parameters. In addition, due to individual adjustment of the hyperparameter requires retraining to evaluate the effect, computation for evaluating a set of hyperparameter configurations is very demanding. Grid search (GS) and random search (RS) methods are commonly used for tuning hyperparameters in the machine learning practice. GS is a way that searches a suitable set of optimal hyperparameter configurations by trying combinations of all the hyperparameter. However, GS is apt to suffering from the dimensionality constraint, i.e., the number of times required to evaluate the model during hyperparameter optimization grows exponentially with consideration of more parameters, which has been proved to be infeasible to XGBoost for substantial number of hyperparameters. Additionally, it is not even guaranteed to find the optimal solution, often aliasing over the optimal configuration. RS is a way that selects a suitable set of optimal hyperparameter configurations by trying random combinations of hyperparameters. The drawback of RS lies in the

316

14. Assessment of undrained shear strength

unnecessarily high variance. The method is, entirely random and uses no intelligence in selecting the trial points. Neither GS nor RS utilizes correlations between different combinations of hyperparameters. More and more hyperparametric tuning processes are now done through automated methods that aim to find optimal hyperparameters in less time using a strategy-based informed search. In addition, no additional manual operations are required except for the initial setup. Bayesian optimization is the top choice for optimizing objective functions (Ghahramani, 2015; Snoek, Larochelle, & Adams, 2012; Xia et al., 2017). Bayesian optimization finds the value that minimizes the objective function by building a surrogate reconstruction (probability model) based on the past evaluation results of the target. It has been widely used for machine learning hyperparameter tuning recently. The results show that the method is more generalized on the testing set and requires fewer iterations than GS and RS (Bergstra, Yamins, & Cox, 2013).

Database This study employed the datasets of F-CLAY/7/216 and S-CLAY/ 7/168 (D’Ignazio et al., 2016), which are available in the TC304 database. The first one F-CLAY/7/216 was compiled from 216 samples of field vane tests, which was conducted in 24 different test sites in Finland. The second clay dataset S-CLAY/7/162 consisted of 168 data points of field vane tests from 12 sites in Sweden and 7 sites in Norway. Each data points in the two datasets include six parameters, including undrained shear strength (USS), vertical effective stress (VES), preconsolidation stress (PS), liquid limit (LL), plastic limit (PL), and natural water content (W). The two datasets were combined and studied together herein. Table 1 lists the statistics with the mean value, the maximum (Max), minimum (Min), coefficient of variation (COV), and the sample numbers (n). It should be noted that VES and PS have the largest variations.

TABLE 1 Basic statistics of the five feature variables and label. LL (%)

PL (%)

W (%)

VES (kPa)

PS (kPa)

USS (kPa)

Mean

68.37

28.49

76.47

48.72

79.82

19.21

COV

0.35

0.28

0.31

0.56

0.61

0.52

Max

201.81

73.92

180.11

212.87

315.64

75.00

Min

22.00

2.73

17.27

6.86

15.20

5.00

n

384

384

384

384

384

384

317

Implementation procedure

Implementation procedure Comparison models To evaluate the performance of the proposed XGBoost-based and RF-based design soil parameters’ surrogate models, it is compared with two comparison ML algorithms, including SVM (Cortes & Vapnik, 1995) and multilayer perceptron (MLP), a class of feedforward artificial neural network) (Gardner & Dorling, 1998). To be fair, all comparison ML models here also will use the Bayesian optimization method for tuning hyperparameters. Besides, two relatively good transformation models presented in D’Ignazio et al. (2016), Eqs. (10), (11) also present as a baseline model compared with the above models. USS ¼ 0:296OCR0:788 W 0:337 VES

(10)

USS ¼ 0:319OCR0:757 LL0:333 VES

(11)

Performance measures The following performance measures were employed to analyze and assess the predictive results of the machine learning models: Root mean square error (RMSE) value closer to 0 indicates that the error in prediction is less. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn RMSE ¼ ðy  y^i Þ2 (12) i¼1 i n R2 (Coefficient of determination) values closer to 1 indicate that this model is also better for data fitting. X ðyi  y^i Þ2 2 R ¼1 X (13) ð yi  yÞ 2 Bias factor b is the sample mean of (the actual target value)/(the predicted target value), and if b ¼ 1, the model prediction is unbiased (Ching & Phoon, 2014; D’Ignazio et al., 2016). b¼

1 X n yi i¼1 y ^i n

(14)

Mean absolute percentage error (MAP) (De Myttenaere, Golden, Le Grand, & Rossi, 2016) value closer to 0 shows high prediction accuracy. 100% Xn yi  y^i MAPE ¼ (15) i¼1 y n i

318

14. Assessment of undrained shear strength

where n is the total number of data; yi and y^i are the actual undrained shear strength and the predicted undrained shear strength, respectively; and y is the mean of the actual undrained shear strength.

Calculation and results In this section, the developed models were systematically compared with two comparison machine learning methods and two transformation models with respect to prediction accuracy and robustness under 5-fold CV.

Predictive comparisons among different models Figs. 1 and 2 show the RMSE curves of the training and testing sets of ML models under 5-fold CV, respectively. It is well recognized that the RMSE value closer to 0 indicates that the error in prediction is marginal. It can be observed that the two ensemble learning models’ RMSE are smaller than the other two ML models’ RMSE, indicating that XGBoost-based and RF-based models’ prediction performance is better than others in the training set. The curves are more fluctuant in Fig. 2. In addition, all models performed the same trend and get the worst at k ¼ 2 and best at k ¼ 3, indicating that models are obviously influenced by the dataset’s quality. In addition, it can be noticed that XGBoost-based model’s predictive performance is obviously better than the others, while the advantages of the

FIG. 1

The RMSE curves of training patterns under 5-fold CV.

Calculation and results

FIG. 2

319

The RMSE curves of testing patterns under 5-fold CV.

RF-based model cannot be clearly observed. By calculating the RMSE average values for these models, the RF-based model’s RMSE average value is lower than that of comparison models, implying that the RF-based model is much better. The average values of RMSE, R2, MAPE and bias factor b under the 5-fold CV for all models are summarized in Table 2. It should be stressed that the two transformation models, Eqs. (10), (11), directly used the entire database instead of 5-fold CV. The bias factor b derived from all models above are closer to 1, indicating the unbiased property of these models. The mean values of RMSE, R2, MAPE for the testing set of the XGBoostbased model is 4.40 kPa, 0.73% and 19.23%, respectively, and it outperformed the other ML-based models and two transformation models. Furthermore, the overall order of prediction accuracy followed by XGBoost, RF, SVM, MLP, Eqs. (10), (11). In a word, the ensemble models have advantages in prediction accuracy over the single sophisticated algorithm in the case of multisource data. In addition, it also outperformed the models presented in D’Ignazio et al. (2016).

Fitting performance of XGBoost and RF at k 5 3 Figs. 3 and 4 show the fitting performance of XGBoost-based and RF-based model at k ¼ 3. The fitting results of k ¼ 3 are selected as a representative to show fitting performance since this set of testing data is the closest to overall data’s distribution. The cyan line represents the actual USS values with sample number, and the upper boundary of the orange area is 1.2 times the actual value of USS and the lower boundary of the

6.11

4.19 4.42

MLPR MARS

Eq. (16)

3.60

SVR

5.88

2.51

RFR

4.91 4.89

4.82

4.60

4.40

Testing

RMSE (kPa)

Eq. (15)

2.38

Training

XGBoost

Evaluation index

0.75 0.73

0.82

0.91

0.92

Training

0.48

0.52

TABLE 2 Comparisons among models predictive modeling results.

0.66 0.66

0.67

0.70

0.73

Testing

R2

19.41% 20.23%

13.69%

10.93%

10.85%

Training

23.60%

23.39%

21.38% 22.43%

20.56%

19.63%

19.23%

Testing

MAPE (%)

1.00 1.01

1.02

0.99

0.99

Training

1.04

1.07

1.02 1.04

1.04

1.00

1.01

Testing

Bias

Calculation and results

FIG. 3

Prediction performance of ensemble learning training pattern (k ¼ 3).

FIG. 4

Prediction performance of ensemble learning testing pattern (k ¼ 3).

321

orange area is 0.8 times the actual value of USS. Furthermore, the blue square and red circle represent the prediction values of the XGBoostbased model and RF-based model, respectively. It is apparent that most of the data points fall into the orange area, and the MAPE values of the two models for testing set are 16.92% and 18.19%, respectively. These

322

FIG. 5

14. Assessment of undrained shear strength

The features relative importance ranking.

MAPE values can be regarded as small in view that such data set contains multisource and certain noisy data.

Features importance analysis The feature importance is an important reference for feature selection and model interpretability. A trained XGBoost-based model can automatically calculate feature importance, which can be obtained through the interface feature importance criterion, i.e., gain criterion. The gain is calculated by taking each feature’s contribution for each tree in the model, indicating the relative contribution of each feature to the model. A higher value of this metric, compared to another, implies greater importance of such a feature for generating a prediction. Fig. 5 shows the five feature variables’ average of feature relative importance (%) under 5-fold CV. In the current XGBoost-based model, PS (54.3%) is the most important feature variables, followed by VES (20.3%), PL (12.5%), W (6.8%), and LL (6.1%). This result implies significant guidance for exploring the characteristics of the undrained shear strength of soft clays, and a similar result of sensitivity analyses was also presented in D’Ignazio et al. (2016).

Summary and conclusions Ensemble learning methods are very popular and powerful machine learning tools for multivariate regression and classification problems. Unlike the extensive study of the transformation model based on empirical evidence, data-driven ensemble learning method has received limited

References

323

research attention in geotechnical engineering. This chapter initiates the use of XGBoost-based and RF-based ensemble learning methods, for prediction of the undrained shear strength of soft clays. The performance of ensemble learning methods is also compared with two baseline ML algorithms (i.e., SVM, MLP) and two transformation models from previous works, in a combination database from F-CLAY/7/216 and S-CLAY/ 7/168. Based on this study, the following conclusions may be drawn. All proposed ML models are validated over the database using four performance measures under a 5-fold CV. The comparisons with baseline models show the superiority of the XGBoost-based and RF-based model in terms of predictive performance, and the overall order of prediction accuracy followed by XGBoost RF, SVM, and MLP. ML methods have obvious advantages in prediction accuracy compared with previous transformation models. The mean MAPE value of the two ensemble learning models for the testing set is 19.23% and 19.63%, respectively, and these MAPE values can be considered small for such data set containing multisource and certain noisy data. A Bayesian hyperparameter optimization method is employed to tune ML algorithms’ hyperparameters. This measure ensures that the proposed model can fully tap the predictive potential, and perform better than previous transformation models for capturing the relationships between the USS and various basic soil parameters. The proposed XGBoost-based model can automatically calculate the relative feature importance of soil parameters in USS, PS is identified as the most important feature variables, followed by VES, PL, W, and LL. This measure enhances the model’s interpretative and comprehensive and provides guidance on feature selection for Geotechnical practitioners in practical applications.

Acknowledgments The authors acknowledge the ISSMGE TC304 for compiling the database 304 dB, as well as making it open for research and further investigation. Special thanks are given to the anonymous reviewers who have helped to improve the chapter. The first author is grateful for the financial support from was supported by High-end Foreign Expert Introduction program (No. G20190022002), Chongqing Construction Science and Technology Plan Project (2019-0045) as well as Chongqing Engineering Research Center of Disaster Prevention & Control for Banks and Structures in Three Gorges Reservoir Area (Nos. SXAPGC18ZD01 and SXAPGC18YB03).

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C H A P T E R

15 Limit state analysis on deep braced excavation adjacent to an existing upper slope in mountainous terrain Wengang Zhanga, Runhong Zhangb, Li Hongc, and Anthony Goh (Teck Chee)d a

Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, People’s Republic of China, b National Joint Engineering Research Center of Geohazards Prevention in the Reservoir Areas, Chongqing University, Chongqing, People’s Republic of China, cSchool of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China, dSchool of Civil and Environmental Engineering, Nanyang Technological University, Singapore, Singapore

Introduction With rapid urbanization and infrastructure development in China, there is an increasing demand to construct high-rise apartments, shopping malls, and public transportation systems and networks. For deep excavations in cities with mountainous terrain such as Hong Kong, Chongqing, and Guiyang, one concern is that the stress relief may have an adverse influence on the stability of nearby existing slopes. However, nowadays, there have been only limited investigations into the interaction between braced excavation and the adjacent slope and the influence of such interaction on the overall stability. Li, Liu, Liu, and Han (2011) investigated the behavior of a deep braced excavation for the Shangshuijing station of the Shenzhen Metro Line 5 adjacent to a slope using FLAC3D (Itasca Consulting Group, 2002). Wang, Gu, Zhang, and Xiong (2011) examined

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00005-8

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© 2021 Elsevier Inc. All rights reserved.

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15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

the performance of retaining structures for a complex geotechnical system comprising a slope and a nearby deep excavation, based on field instrumentations. Luo (2013) used the shear strength reduction (SSR) method to investigate the effects of the excavation depth and the soil properties on the stability of a slope adjacent to the deep excavation. Varzaghani and Ghanbari (2014) presented a new analytical model to determine the seismic displacements of shallow foundations adjacent to slopes. Zhou et al. (2018) evaluated the impact of deep excavations on the stability of the adjacent slope based on the slip line theory. The results obtained by the finite element method indicate that the critical slip surface moves toward the slope surface and develops downward. Zhang, Goh, and Zhang (2019) conducted a system reliability assessment on deep braced excavation adjacent to an existing upper slope in mountainous terrain via a case study. However, still there is a lack of a detailed systematic study to investigate the critical parameters that influence both the ultimate and serviceability limit states (SLSs) of the excavation and slope system. In this study, the global factor of safety FS obtained via the SSR technique (also called c-φ reduction method) is used as the criterion for the ultimate limit state (ULS), and the calculated maximum lateral wall deflection is adopted as the SLS criterion. A detailed study was carried out to evaluate the influence of the excavation geometries, the supporting system stiffness, and the distance between the braced excavation and the existing slope on the comprehensive excavation-slope system response using the finite element program PLAXIS2D (Brinkgreve, Engin, & Swolfs, 2017). Based on these results, response surface models were developed for the ultimate and SLSs. Subsequently, the first-order reliability method (FORM) is adapted to perform a probabilistic analysis on the global factor of safety through setting the threshold maximum wall deflection as an optimization constraint to enable a rational design approach for deep braced excavations adjacent to slopes in mountainous terrain.

Methodology Shear strength reduction technique In this study, the global stability FS for the slope is assessed using the SSR technique. This technique has been used by various authors including Matsui and San (1992) and Dawson, Roth, and Drescher (1999); Dawson, Motamed, Nesarajah, and Roth (2000) and is now available in many commercial finite element (FEM) and finite difference (FDM) programs. This

Methodology

329

procedure essentially involves repeated analyses by progressively reducing the shear strength properties until collapse occurs. For a MohrCoulomb material, by reducing the shear strength by a factor F, the shear strength equation becomes: τf c tan φ ¼ + σn F F F τf F¼ 0 c + σ n tan φ0 where τf is the shear strength, σ n is the normal stress, and c0 ¼ c/F and φ0 ¼ arctan(tan φ/F) are the new Mohr-Coulomb shear strength parameters. Systematic increments of F are performed until the finite element or finite difference model does not converge to a solution (i.e., failure occurs). The critical strength reduction value that corresponds to nonconvergence is taken to be the global factor of safety FS. It should be noted that this FS indicates the global stability, not the ratio of strength to stress of any single element in the numerical model. Consequently, global FS ¼ 1 implies that total collapse has occurred. The technique has been applied to a number of underground excavation problems including rock caverns (Goh & Zhang, 2012; Hammah, Yacoub, & Curran, 2007; Zhang & Goh, 2012; Zhang & Goh, 2015) and tunnels (Wang, Zhu, & Zheng, 2014; Xia, Zhao, Zhang, & Wang, 2014).

Logarithmic regression Regression models including polynomial regression (PR) and logarithmic regression (LR) have been developed. In this chapter, the LR model is adopted to model the nonlinear relationship between the dependent and independent variables since the LR model is more accurate in predictive capacity. The basic form of the LR models is as follows: y ¼ aðX1 Þb ðX2 Þc ðX3 Þd ðX4 Þe … in which y is the dependent response, Xi is the input parameters (independent variables), and the coefficients a, b, c, d, e, etc. are determined through least-squares estimation.

Concept of probabilistic reliability assessment In many civil engineering applications, the assessment of safety is made by first establishing a relationship between the load S of the system and the resistance R. The boundary separating the safe and “failure” domains

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15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

is the limit state surface (boundary) defined by G(x) ¼ R  S ¼ 0, in which x ¼ vector of the random variables. Mathematically, R > S or G(x) > 0 would denote a “safe” domain. An unsatisfactory or “failure” domain occurs when R < S or G(x) < 0. Calculation of Pf involves the determination of the joint probability distribution of R and S and the integration of the probability density function (PDF) over the failure domain. Considering that the PDFs of the random variables are not known in most geotechnical applications, and the integration is computationally demanding when multivariables are involved, an approximate method, known as the FORM (Hasofer & Lind, 1974), is commonly used to assess Pf. The approach involves the transformation of the limit state surface into a space of standard normal uncorrelated variables, wherein the shortest distance from the transformed limit state surface to the origin of the reduced variables is the reliability index β (Cornell, 1969). For normal distributed random variables, Pf  1  Φ(β), in which Φ ¼ cumulative normal density function. Table 1 shows the relationship between β and Pf. Mathematically, Low and Tang (2004) showed that β can be computed using: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     xi  ui T 1 xi  ui β ¼ min ½R xF σi σi in which xi is the set of n random variables, ui is the set of mean values, σ i is the standard deviation, R is the correlation matrix and F is the failure region. Low (1996) showed that the Microsoft EXCEL spreadsheet can be used to perform the minimization and determine β. The reliability index β and the probability of failure Pf for both the ultimate and the SLSs can be calculated using the FORM spreadsheet method, as shown in Fig. 9. The EXCEL spreadsheet SOLVER function is used to search for the design point, also named the most probable failure point (Low & Tang, 2004). Detailed explanations of the setup of the spreadsheet for reliability calculations are described in section “Probabilistic reliability assessment”.

TABLE 1

Parameters considered and the ranges to be examined.

Parameters

Ranges

System stiffness in logarithmic scale ln(S)

3.794, 4.605, 5.187

Excavation width B (m)

20, 30, 40

Excavation depth He (m)

14, 17, 20

Wall thickness d (m)

0.6, 0.9, 1.2

Distance between wall and toe of slope B1 (m)

5, 10, 15, 20, 30, 40

Wall penetration ratio D/He

0.50, 0.76, 1.14

Finite element analysis

331

Finite element analysis Numerical setup and modeling The Plaxis2D software was utilized for the numerical simulations. The Mohr-Coulomb constitutive model was selected for the soil elements. A typical cross-section of the deep braced excavation and slope system, as well as the properties of the soil and the supporting elements, are shown in Fig. 1. The phreatic level is below the bottom boundary, so the pore water pressure distribution and changes, as well as the water percolation, are not considered. It should be noted that the adopted soil properties are typical of the ground conditions in Chongqing and that the soil and wall profiles, the excavation geometries, as well as the boundary conditions are simplified from a deep braced project in Chongqing. The analyses considered a plane strain excavation near an unreinforced slope supported by a retaining wall system and six levels of struts located below the original ground surface at depths of 1, 4, 7, 10, 13, and 16 m, respectively. The soil was modeled by 15-noded triangular elements, following the Mohr-Coulomb failure criterion. The structural elements were assumed to be linear elastic in Plaxis2D, the struts and the wall is represented by three-noded bar elements and five-noded beam elements, respectively. The soil properties are regarded as deterministic and presented in Fig. 1. This study mainly examines the influences of the retaining wall system stiffness and the excavation geometries on the excavation responses including the ULS represented by the global factor of safety of the braced excavation-upper slope system as well as the SLS denoted by the retaining wall deflection. The ranges of these key design parameters are listed in Table 1. It should be explained that ln(S) represents the natural logarithm of the system stiffness, e.g., ln(EI/γ wh4avg). The three values of 3.794, 4.605,

FIG. 1

Cross-sectional soil and wall profile.

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15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

and 5.187 are realized by changing the corresponding wall thickness of 0.6, 0.9, and 1.2 m with average vertical strut spacing havg ¼ 3 m, unit weight of water γ w ¼ 10 kN/m3 and keeping Young’s modulus of the wall constant at E ¼ 2.1  107 kN/m2. According to Poh, Wong, and Chandrasekaran (1997), Zhang, Goh, and Xuan (2015), and Goh, Zhang, Zhang, Zhang, and Liu (2017), the strut stiffness does not have a very significant influence on wall deflection when the strut is stiff. A constant strut stiffness EA of 3.0  105 kN/m is assumed. Based on Table 1, more than 160 hypothetical cases were analyzed. The construction assumption and sequence are as follows: (1) The wall is installed without any disturbance (wished into place). (2) The soil is excavated prior to installing the strut, uniformly 1 m below each strut level until the final depth He is reached. (3) Each phase of strut installation is followed by a subsequent phase in which the global safety factor is determined by the SSR method proposed by Zienkiewicz, Humpheson, & Lewis, 1975. The procedure of the SSR method involves systematically reducing the soil shear strength until failure occurs. It has been verified by Lian, Han, and Kong (2001) that the SSR FE method can be widely applied in the engineering practice. In addition, Cheng, Lansivaara, and Wei (2007), Ishii, Ota, Kuraoka, and Tsunaki (2012), Tschuchnigg, Schweiger, and Sloan (2015), Oberhollenzer, Tschuchnigg, and Fschweiger (2018), and Schneider-Muntau, Medicus, and Fellin (2018) also demonstrated that the SSR technique performs well in the many slope cases analyzed. The detailed construction sequence is listed in Table 2.

Numerical results and analysis The numerical results include the global factor of safety FS obtained via SSR and the maximum lateral wall deflection δhm. Global factor of safety Fig. 2 plots the variation of incremental displacement contours, as excavation proceeds, for cases with B ¼ 30 m, B1 ¼ 5 m, ln(S) ¼ 4.605. The corresponding FS values for excavation depths He of 0, 14, 17, and 21 m are also shown. It can be observed that as the excavation depth He increases, the zone showing the incremental displacement (in red) enlarges and the corresponding FS values decrease. A comparison shows that after the final stage of excavation, the FS is about 1.705, which is a decrease of 0.636 from the original FS of 2.341 (before any excavation). The results indicate that excavations close to an existing slope may result in a significant reduction of the FS of the slope, which may cause slope instability or even failure.

Finite element analysis

TABLE 2

333

Detailed construction procedures.

Phases

Construction procedures

Initial phase

Generate the soil initial effective stress and pore water pressure

Phase 1

Determine the initial global safety factor by the SSR method

Phase 2

Install the diaphragm wall

Phase 3

Excavate to 2 m below the ground surface inside the excavation, install the first level strut

Phase 4

Excavate to 5 m below the ground surface

Phase 5

Install the second-level strut

Phase 6

Calculated the global safety factor by the SSR method

Phase 7

Excavate to 8 m below the ground surface

Phase 8

Install the third-level strut

Phase 9

Calculated the global safety factor by the SSR method

Phase 10

Excavate to 11 m below the ground surface

Phase 11

Install the fourth level strut

Phase 12

Calculated the global safety factor by the SSR method

Phase 13

Excavate to 14 m below the ground surface

Phase 14

Install the fifth-level strut at

Phase 15

Calculated the global safety factor by the SSR method

Phase 16

Excavate to 17 m below the ground surface

Phase 17

Install the sixth-level strut

Phase 18

Calculated the global safety factor by the SSR method

Phase 19

Excavate to 20 m below the ground surface

Phase 20

Calculated the global safety factor by the SSR method

Fig. 3 presents some typical plots of the decrease of the factor of safety values ΔFS with increasing distance between the wall and the toe of the slope B1 for three different excavation widths B with He ¼ 20 m and S ¼ 5.187. As expected, ΔFS decreases with increasing distance between the wall and the existing slope B1. The ΔFS was also influenced by the width of the excavation B. The ΔFS was minimal for B1 ¼ 20 m when B ¼ 20 m. For the larger B values of 30 and 40 m, ΔFS was minimal at a greater distance of B1 ¼ 40 m.

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15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

(A)

(B)

(C)

(D)

FIG. 2 Contour of slip surface and FS for B ¼ 30 m, B1 ¼ 5 m, S ¼ 4.605 for different excavation depths He. (A) He ¼ 0 m, FS ¼ 2.341. (B) He ¼ 14 m, FS ¼ 2.214. (C) He ¼ 17 m, FS ¼ 2.095. (D) He ¼ 20 m, FS ¼ 1.705.

FIG. 3

Decrease of the factor of safety ΔFS for different B1 with He ¼ 20 m, S ¼ 4.605.

Lateral wall deflections and bending moments Figs. 4 and 5 show the variation of the wall lateral deformation profiles and the bending moments of the right wall (near the slope) for cases with different B1, respectively. The final excavation depth (He) is 20 m and supported by 1.2 m-thickness and 30 m-deep diaphragm wall with six levels of struts for B1 ranging from 5 to 40 m, respectively. All the plots show a

Finite element analysis

FIG. 4

Lateral deformation profile of the wall for different B1.

FIG. 5

Wall bending moment profiles for different B1 (mm).

335

concave wall deflection shape. As shown in Fig. 4, the maximum wall lateral deformation decreases as B1 increases. As B1 increased from 5 to 40 m, the maximum wall lateral deformation decreased by 30%. As shown in Fig. 5, the maximum bending moment occurs below the final excavation level (FEL). The change in the maximum bending moments is not as significant with a reduction of only 18% when B1 increased from 5 to 40 m.

336

FIG. 6

15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

Max. lateral wall deflection δhm for different B1 with He ¼ 20 m, S ¼ 4.605.

Fig. 6 shows the plot of the maximum lateral wall deflection δhm for different B and B1 with He ¼ 20 m, S ¼ 4.605. The trends are similar to that observed for ΔFS in Fig. 3, i.e., δhm decreases as the excavation width B decreases and the distance between the wall and the toe of the slope B1 increases. The influence of excavation width B on δhm is more significant when B1 is smaller, i.e., the slope toe is closer to excavation. For example, the difference Δδhm between B ¼ 40 and B ¼ 20 is 7.5 mm at B1 ¼ 5 m. It decreases to 3.2 mm at B1 ¼ 10 m.

Development of surrogate models for limit state functions To assess the performance in deep braced excavations adjacent to an existing slope, both the ULS and the SLS of the excavation-slope system should be satisfied. In the following subsections, the limit state functions for ULS and SLS are developed, respectively, based on the numerical results presented in the previous section.

Ultimate limit state surrogate model Based on the calculated FS results, a PR model has been developed for estimating the global factor of safety FS as a function of four input parameters B, B1, He, and S, with a coefficient of determination R2 of 0.881. The expression is shown in Eq. (1):

Development of surrogate models for limit state functions

337

FS_PR ¼ 7:35  102 B  1:57  101 B1 + 3:51  102 He + 1:02S  2:5  106 B2  2:29  104 B1 2  8:07  104 He 2  2:22  102 S2 + 2:4  104 BB1  4:08  103 BHe + 1:65  104 BS + 8:01  103 He B1  1:56  103 SB1  2:8  102 He S  1:84  102 BðD=He Þ + 4:83  102 B1 ðD=He Þ + 9:4  102 He ðD=He Þ  3:55  101 SðD=He Þ (1) Fig. 7 plots the factor of safety values determined from the finite element analyses FS_FEM against the calculated FS_PR results based on Eq. (1). Also shown in the plot are the 100% agreement line and the 10% error lines. The results indicate that Eq. (1) is fairly accurate in predicting the global factor of safety for deep braced excavations adjacent to slopes.

Serviceability limit state surrogate model Similarly, a LR model was developed for predicting the maximum lateral wall deflection δhm, with a fairly high coefficient of determination R2 ¼ 0.946. The expression is shown in Eq. (2): δhm_LR ¼ 0:1133B0:1086 B1 0:223 ðHe Þ2:1247 ðD=He Þ0:0568 S0:4448

FIG. 7

Comparison between FS_FEM and FS_PR.

(2)

338

FIG. 8

15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

Comparison between δhm_FEM and δhm_LR.

Fig. 8 plots the estimated maximum lateral wall deflections δhm_LR values against the finite element calculated results δhm_FEM. Also shown in the plot are the 100% agreement line and the 20% error lines, indicating that Eq. (2) is reasonably accurate in predicting the maximum lateral wall deflections for deep braced excavations adjacent to slopes.

Probabilistic reliability assessment The FORM (Hasofer & Lind, 1974) is adopted in this study for a probabilistic assessment of the two limit state functions. It can deal with the unknown PDFs of the random variables in most geotechnical applications and the demanding of computation of integration when multivariables are involved. FORM has been proved that the Microsoft EXCEL spreadsheet can be used to perform the minimization and determine reliability index β (Low & Tang, 2007, Zhang & Goh, 2012, Zhou, Gong, & Hong, 2017, Zou, Liu, Cai, Bheemasetti, & Puppala, 2017, Ji, Zhang, Gao, & Kodikara, 2019, Goh, Zhang, & Wong, 2019). The reliability index β and the probability of failure Pf for both the ultimate and the SLSs can be performed using FORM based on the built PR

Probabilistic reliability assessment

339

and LR models presented in the previous section. Based on the approach by Low and Tang (2007), the ULS model expressed as Eq. (1) has been incorporated into an EXCEL spreadsheet environment, from which the reliability index β can be determined. A sample spreadsheet for computing the factor of safety FS is shown in Fig. 9A, in which the values of the random input variables are the same as those used in the simulation model. According to Low and Tang (2007), the spreadsheet cells B3:B5 allows the distribution of the random variables has the selection of various types including lognormal, normal, triangular, and so on. Cells D3:E5 are average values and standard deviation which are set corresponding to the random variables in Cells B3:B5, three random variables are of the normal distribution. Cells G3:I5 represents the correlation matrix R used to define the correlations between  B, He, and S. Cells J3:J5 denote the ni vector which contains equations for xi  ui N =σ i N . The x∗i values were firstly set equal to the mean values (30, 17, 4.5) of the original random variables (B, He, S). Then by invoking the spreadsheet’s built-in optimization routine SOLVER to search algorithm by minimizing the cell L3, through changing the ni

FIG. 9 FORM spreadsheet setup for (A) ULS and (B) SLS. (A) Calculation on β and Pf for the ultimate limit state using FORM spreadsheet. (B) Calculation on β and Pf for serviceability limit state using FORM spreadsheet.

340

15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

values subject to G(x) ¼ 0, the design point (x∗i values) can be obtained. It should be stressed that in the spreadsheet environment, iterative numerical derivatives and directional search for the design point x∗i were automatically carried out via SOLVER search. The probabilistic assessment of SLS in Fig. 9B is almost identical to Fig. 9A except for the G(x) formulations. For the detailed procedures in performing the FORM spreadsheet framework to derive β and the corresponding Pf, refer to Zhang and Goh (2012).

Probabilistic assessment of the ultimate limit state For either the braced excavation or the slope, design codes or guidelines, specify the minimum factor of safety. For example, in Chongqing (China), the building authorities specify a minimum factor of safety of 1.30 for slopes in mountainous terrains. However, for the braced excavation-upper slope system, there are no guidelines or codes with regard to the critical safety factor values. Thus, the influence of the critical factor of safety FS_cr on the reliability index β and the probability of failure Pf of the ULS of the braced excavation-upper slope system is examined in this study, which can justify the choice of specific threshold factor of safety. Fig. 10 plots the influence of the various design parameters on the reliability index β and the probability of failure Pf of the ULS. Both the coefficient of variation of the system stiffness COVS and the critical factor of safety FS_cr has a significant influence on the β and Pf. A larger COV results in a smaller β. However, there is a less significant difference in β and Pf between COVS ¼ 0.05 and COVS ¼ 0.1. Besides, the reliability index β decreases with FS_cr and converges at FS_cr ¼ 2.2, similarly Pf increases with FS_cr and converges at FS_cr ¼ 1.6. Fig. 11 compares the influence of both the COVS and B1 on Pf for B ¼ 20, 30, 40 m, respectively, for He ¼ 17 m, S ¼ 4.5, and assuming the critical factor of safety FS_cr ¼ 2.0. Pf decreases with the increase of B1 while Pf increases with the increase of excavation width B. A larger COVS results in a larger Pf. However, there is no difference in Pf between COVS ¼ 0.05 and COVS ¼ 0.1. The Pf decreases with B1 and converged at B1 ¼ 15 when B ¼ 20 m, B1 ¼ 20 when B ¼ 30 m, B1 ¼ 30 when B ¼ 40 m. It suggests that COVS has little influence on Pf when B1  15 when B ¼ 20 m, B1  20 when B ¼ 30 m, B1  30 when B ¼ 40 m.

Probabilistic assessment of the serviceability limit state In this section, the SLS of the braced excavation-upper slope system is considered. Fig. 12 plots the influence of COVS and the threshold max. wall deflection δhm_cr on β and Pf for B ¼ 30 m, He ¼ 17 m, S ¼ 4.5 and

Probabilistic reliability assessment

FIG. 10 ¼ 4.5 d.

341

Influence of COVS and FS_cr on (A) β and (B) Pf for B1 ¼ 5 m, B ¼ 30 m, He ¼ 17 m, S

B1 ¼ 20, 20, 40 m, respectively. The results indicate that both COVS and δhm_cr significantly influence the β and Pf values. However, the influence of COVS on β and Pf is not as significant as that of δhm_cr, especially when COVS is higher than 0.20. Fig. 13 shows the influence of COVS on β for He ¼ 17 m, S ¼ 4.5, δhm_cr ¼ 23 mm, B ¼ 20, 30, 40 m and B1 ¼ 10, 15 m respectively. It is clear that β decreases exponentially as the COVS becomes greater. It is logical that β increases when the excavation is further away from the slope.

342

15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

FIG. 11 Influence of COVS and B1 on β and Pf for He ¼ 17 m, S ¼ 4.5, FS_cr ¼ 2.0. (A) B ¼ 20 m, (B) B ¼ 30 m, (C) B ¼ 40 m.

Probabilistic reliability assessment

FIG. 12

343

Influence of COVS and δhm_cr on β for B ¼ 30 m, He ¼ 17m, S ¼ 4.5, B1 ¼ 20,

20, 40m.

FIG. 13 Influence of COVS on β for He ¼ 17 m, S ¼ 4.5, δhm_cr ¼ 23 mm, B ¼ 20, 30, 40 m, and B1 ¼ 10, 15 m.

344

15. Limit state analysis on deep braced excavation adjacent to an existing upper slope

Summary and conclusions This study presents numerical investigations on the performance of a braced excavation adjacent to an existing slope, from the perspectives of the global factor of safety and the maximum lateral wall deflections. It also proposed a probabilistic framework for the quantitative assessment of both the ultimate and the SLSs in view of some design and construction uncertainties. The main conclusions arrived at include: (1) Based on the case study, excavations close to an existing slope may result in a significant reduction of the global FS for the braced excavation-upper slope system. The amount of reduction is closely associated with the excavation depths and widths, the penetration depth, and the distance between the excavation and the slope toe. (2) The wall deflections are also significantly influenced by the key factors listed in the conclusion point (1). (3) Surrogate models for ULS and SLS are developed and implemented into FORM for a reliability analysis. It should be noted that the proposed probabilistic framework is based on an actual case study in Chongqing as illustrated in Fig. 2 in which the slope angle and soil properties are deterministic and fixed. The effects of the inherent spatial variability of the soil properties and the slope angle on the braced excavation-upper slope system performance will be considered in a future study.

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 51608071), Chongqing Engineering Research Center of Disaster Prevention & Control for Banks and Structures in Three Gorges Reservoir Area (Nos. SXAPGC18ZD01 and SXAPGC18YB01) and Chongqing Construction Science and Technology Plan Project (20190045). The financial support is gratefully acknowledged.

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Goh, A. T. C., Zhang, F., Zhang, W. G., Zhang, Y. M., & Liu, H. L. (2017). A simple estimation model for 3D braced excavation wall deflection. Computers and Geotechnics, 83, 106–113. Goh, A. T. C., & Zhang, W. (2012). Reliability assessment of stability of underground rock caverns. International Journal of Rock Mechanics and Mining Sciences, 55, 157–163. Goh, A. T. C., Zhang, W. G., & Wong, K. S. (2019). Deterministic and reliability analysis of basal heave stability for excavation in spatial variable soils. Computers and Geotechnics, 108, 152–160. Hammah, R. E., Yacoub, T., & Curran, J. H. (2007). Serviceability-based slope factor of safety using the shear strength reduction (SSR) method. In: The Second Half Century of Rock Mechanics 11th Congress of the International Society for Rock Mechanics. Taylor & Francis, pp. 1137–1140. Hasofer, A. M., & Lind, N. (1974). An exact and invariant first-order reliability format. Journal of Engineering Mechanics, 100, 111–121. Ishii, Y., Ota, K., Kuraoka, S., & Tsunaki, R. (2012). Evaluation of slope stability by finite element method using observed displacement of landslide. Landslides, 9(3), 335–348. Itasca Consulting Group (2002). FLAC3D manual. Ji, J., Zhang, C. S., Gao, Y. F., & Kodikara, J. (2019). Reliability-based design for geotechnical engineering: An inverse FORM approach for practice. Computers and Geotechnics, 111, 22–29. Li, Y. H., Liu, P., Liu, J. Q., & Han, X. F. (2011). Stability and safety analysis of braced excavation for subway station during construction under the condition of side slope. Applied Mechanics and Materials, 99–100, 1166–1170. Lian, Z. Y., Han, G. C., & Kong, X. J. (2001). Stability analysis of excavation by strength reduction FEM. Chinese Journal of Geotechnical Engineering, 23(4), 5. Low, B. K. (1996). Practical probabilistic approach using spreadsheet. In C. D. Shackelford, P. P. Nelson, & M. J. S. Roth (Eds.), Uncertainty in the geologic environment, GSP 58 (pp. 1284– 1302). Reston: ASCE. Low, B. K., & Tang, W. H. (2004). Reliability analysis using object-oriented constrained optimization. Structural Safety, 26(1), 69–89. Low, B. K., & Tang, W. H. (2007). Efficient spreadsheet algorithm for first-order reliability method. Journal of Engineering Mechanics ASCE, 133(12), 1378–1387. Luo, P. (2013). The soil stability analysis of soft soil slope toe excavation. (Master Thesis)Hefei, China: Hefei University of Technology. (in Chinese). Matsui, T., & San, K. C. (1992). Finite element slope stability analysis by shear strength reduction technique. Soils Found, 32(1), 59–70. Oberhollenzer, S., Tschuchnigg, F., & Fschweiger, H. (2018). Finite element analyses of slope stability problems using non-associated plasticity. Journal of Rock Mechanics and Geotechnical Engineering, 10, 1091–1101. Poh, T. Y., Wong, I. H., & Chandrasekaran, B. (1997). Performance of two propped diaphragm walls in stiff residual soils. Journal of Performance of Constructed Facilities, 11(4), 190–199. Schneider-Muntau, B., Medicus, G., & Fellin, W. (2018). Strength reduction method in Barodesy. Computers and Geotechnics, 95, 57–67. Tschuchnigg, F., Schweiger, H. F., & Sloan, S. W. (2015). Slope stability analysis by means of finite element limit analysis and finite element strength reduction techniques. Part II. Back analyses of a case history. Computers and Geotechnics, 70, 178–189. Varzaghani, M. I., & Ghanbari, A. (2014). A new analytical model to determine dynamic displacement of foundations adjacent to slope. Geomechanics and Engineering, 561–575. Wang, Q. Y., Gu, D. P., Zhang, J. S., & Xiong, Z. B. (2011). Analysis of slip-risk and dynamic monitoring of a high building slope fringed a deep foundation pit. Journal of Safety and Environment, 11(2), 6. (in Chinese).

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C H A P T E R

16 Modeling of hierarchically block geomedium with application to deep openings Mikhail A. Guzeva, Vladimir Makarovb, and Vladimir Odintsevc a

Institute for Applied Mathematics FEBRAS, Vladivostok, Russia b Far-Eastern Federal University, Vladivostok, Russia c Institute for Mining IPCON, Moscow, Russia

Introduction Hierarchically block character of a constitution of geomedium is now recognized by the majority of researchers, and a search of adequate mathematical models of such medium is the actual problem demanding the prompt decision. In this direction, it is possible to emphasize two basic approaches, caused by the proposed principles of modeling. First, it is a principle of construction of a continuous medium with inhomogeneities where the hierarchically block geomedium is considered as an equivalent continuous one, whose properties are defined by averaging on volume and a selection of a comprehensible range of the inhomogeneities (Nikolaevsky, 1984; Sadovsky, Bolhovitinov, & Pisarenko, 1987). A lack of such an approach is, on the one hand, an arbitrariness in a choice of scales of averaging which results in a set of various models, and, on the other hand, at fixing any “an averaging step”— appearance of set of the models which parameters definition is a formidable problem. Besides, as basic difficulty is “extinction” of mesostructures from models because of the averaging operation which description of properties was supposed to be reached as a result of modeling. Second, it is necessary to mention attempts of direct calculation of block system motion for different sizes where blocks are presented by rigid

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00011-3

347

© 2021 Elsevier Inc. All rights reserved.

348

16. Modeling of hierarchically block geomedium

elements, and borders of blocks are presented by the deformable layerslinings replaced by rheological elements in the model. The reduction of such models to continuous ones demands the introduction of simplifications, which are analogous for similar operations in a discrete case. As a result, we obtain the quality estimations having mainly theoretical interest (Sadovskii & Sadovskaya, 2015; Vang et al., 2019). The appearance of mesomechanical representations about scales and structural levels of destruction (Panin, 1995) seemed to be has approached the decision of a problem of the geomedium adequate model construction considering as discrete as continual character of its properties (Goldin, 2005; Makarov, Smolin, Stefanov, et al., 2007). However, the difficulties linked with the effective division of scale and structural levels of such medium have not been overcome that have invoked representation about the mesomechanical approach, as a deadlock with reference to the construction of an efficient model of geomedium. Meanwhile, mesomechanics approaches to principles of the models of continuous mediums with defects construction have appeared rather effective with reference to plastic deforming and destruction of such materials, as metals and polymers (Panin, Grinjaev, Danilov, et al., 1990). The field method used here led directly to models, capable adequately to describe the mesostructures of “contrast” type, well known both at the level of rock samples, and in a rock mass (Goldin, 2005; Guzev & Makarov, 2007; Shemjakin, Fisenko, Kurlenja, Oparin, et al., 1986). The authors of the present work, leaning against results of years of own study, represent the experimental establishments and the theoretical principles leading to the necessity of introduction of non-Euclidian model of a continuous medium, combined both approaches of researches of hierarchically block rock masses, and mesomechanics achievements with reference to building of integrative a mathematical model of geomedium.

Short critical review of the modeling methods of modern geomechanics Principle of construction of a continuous medium with inhomogeneities where the hierarchically block geomedium is changed by equivalent continuous one The block constitution of a rock mass has been found since the first works on the discussion of an origin problem of tensile cracks in the conditions of all-round compression (Crosby, 1882; King, 1875). Separation of rock mechanics in independent discipline, which a subject and methods essentially differ from methods of mechanics of soils, just has been caused by a basic distinction of properties of block and continuous media (M€ uller, 1982).

Short critical review of the modeling methods of modern geomechanics

349

The development of rock mechanics has led to an understanding of a hierarchically block constitution of a rock mass that has defined the necessity of selection of hierarchy of blocks, whereas criterion of reference to this or that level ability of rocks to destruction of certain type served. The first classification has been yielded in Sadovsky’s works (Sadovsky, 1979; Sadovsky et al., 1987) which, having considered physical objects from the size of atom, to planetary, has evolved the scale multiplier 3 defining the attitude of the size of blocks of the next levels of the general hierarchy. In this case, in the continuous mechanic as a result of averaging substantially inhomogeneous medium, it is replaced with some fictitious continuum possessing thus the same reactions to external forces, as well as that medium with which it replaces (Sadovsky et al., 1987) for relations of lengths of defects: λ≪ l≪L

(1)

where λ, L—top and bottom border of a selected hierarchical level, accordingly; l—the average typical size for the selected hierarchical level of cracking defects. A similar approach is sustained in Nikolaevsky’s works who has considered the procedure of medium with cracking defects change to continuous one (Nikolaevsky, 1984). For a condition (1), the equations of mass balance, an impulse, and quantity of motion for elementary volume ΔV can be considered differential with intrinsic scale λ (the size of a crystallite of grain, a pore, etc.). In work (Oparin & Tanajno, 2009) after the introduction of “a fundamental initial series of geoblocks” (natural separatenesses), associated with a liquid core of the Earth in diameter of 2500 km, objects of an initial row of geoblocks Δi(from planetary to fine-crystalline level) are defined by a range: Δi ¼ Δ0 

pffiffiffii 2 , Δ0  2:5  106 m

(2)

where i—the integers which are taking over negative values (i ¼  1,  2,  3, …) at transferring from large representatives of geostructure to smaller; Δ0—diameter of a “liquid” core of the Earth. It is noticed that in Eq. (2) at the quantitative level, the phenomenological model of a structurally hierarchical constitution of geomedium is displayed at its partition from the big sizes to the smaller ones. Expression (2) leads to a range (Shemjakin et al., 1986): pffiffiffi pffiffiffij1 2 dj ¼ d0 2 exp ð0, 34466 jÞ, j ¼ 1, 2, …,n (3)  d0 2

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16. Modeling of hierarchically block geomedium

where dj—the representative size of natural separateness of a geomedium on j—level of hierarchy; d0—the basic (minimum) size of natural separateness (d0 ¼ 0, 001 mm, j ¼ 1, 2, …, 44). As follows from Eq. (3), at specified in Oparin and Tanajno (2009) approach 44 models for the description of hierarchically block geomedium are necessary that in engineering appendices leads to formidable problems.

Methods of direct calculation of a block system movement of all possible sizes One-dimensional mathematical models of viscoelastic deforming of block mediums are considered in work (Aleksandrova & Sher, 2004) where it is shown that representation of blocks by massive rigid bodies with elastic layers between blocks gives the chance to describe precisely enough low-frequency slow waves (a wave of pendular type), arising at shock influence. In other work (Vang et al., 2019), energy diffusion to the block medium modeled by a one-dimensional chain of masses, connected by springs and dampers is studied. The influence of parameters of block medium on an energy dissipation is investigated. The approximate analytical solution describing the total energy of block medium at a wide interval of time is received. It is necessary to mention also work (Aleksandrova, 2019) in which propagation of surface “pendular” waves at nonstationary influence on a free surface of the hole placed in block semispace is numerically investigated. A mathematical model of nonstationary viscoelastic deforming of the block medium is proposed. The model is based on the representation that the dynamic behavior of such a medium can be approximately described as the movement of rigid blocks because of the pliability of layers between them. For the description of the viscoelastic behavior of interblock layers the model of internal friction with a stuff Q-factor, as in the defining parameter is used. The medium is modeled by a threedimensional lattice of the masses connected by elastic springs and viscous dampers in axial and diagonal directions. In the framework of this model the problem about the nonstationary influence of type of “the dilating center” on a hole surface, deepen in block semispace (Fig. 1) is numerically solved. Because of numerical calculation, it is shown that the basic contribution to the wave process on a surface of block medium leads to low-frequency longitudinal and Rayleigh “pendular” waves. Depending on the size of the deepening of a hole and distance from a place of influence the amplitudes of radial and vertical speeds of blocks in a longitudinal wave can be both above, and more low, than in Rayleigh wave.

351

Short critical review of the modeling methods of modern geomechanics

A7

w, z, k

0

A5

A8 v, y, m A3 A4

(B)

m h

O

u, x, n

(A)

k

A6

l

A2 A1

(C)

FIG. 1 Schemas: (A) connections of masses by springs and dampers in three dimensional model of block medium; (B) load applies; and (C) statements of a problem (Aleksandrova, 2019).

In the considered work (Aleksandrova, 2019), the term “obvious limitation” is also underlined to the model presented in Fig. 1 for the description of real block mediums. As a whole, the specified approach allows to explain at the qualitative level the effects linked with wave anomalies, caused by the dynamic influences of mass explosions during mining processes. However, the main process of the mesocracking structures forming in block medium is not mentioned as the basic process of formation, and the problem of block hierarchy is not considered as well. Even more mathematical model is offered by the authors of work (Babeshko, Babeshko, & Evdokimova, 2009). In their opinion, now “theory of block structures is constructed” as a result of working out by authors of a new method—“a differential method of factorization” which has demanded “attraction of the big panel of modern methods of mathematics not applied together … the external analysis, methods of topology, the theory of functions of many complex variables, form-residues of Leray, the theory of group representation.” Nevertheless, it has appeared insufficient, and it was necessary to add a new and not less powerful method “an integrated method of factorization.” Such theory has allowed to create a new numerical method—a method of “a block element” which considerably differs from well-known methods of final and boundary elements. Problems of construction of mathematical models “for an intensity stressed lithosphere plates, in materials technology, building, nanotechnology” are as a result solved.

Mesomechanics approach to principles of continuous media models with defects construction The appearance of “the Physical mesomechanics” as the scientific discipline studying and modeling mesolevel of medium with defects has forced to scrap approaches to the construction of adequate mathematical

352

16. Modeling of hierarchically block geomedium

models of geomedium. Application of mesomechanics principles in geomechanics is reflected in Makarov and Goldin’s works, where the basic approaches and modeling terminology with reference to geomedium have been formulated. The authors guided by on such position of “Physical mesomechanics”: “Physical mesomechanics” considers a deformable stuff, as the organized system of structural levels of various scales. For the adequate description of processes of deformation and destruction, in accordance with the approach of physical mesomechanics, it is necessary simultaneous and inter consistent consideration of micro-, meso-, and macro-scales for which essentially different methods of modeling should be elaborated (Makarov, 1998). Therefore working out of models of geomedium is represented to them as allocation “micro-, meso-, and macro-scales” in geomedium and division into these scales of structural elements of medium-on “structural levels,” compounding hierarchy on each of scales. Let’s notice that the geomedium, since Sadovskogo’s classical works, represents a hierarchically block rock mass (Sadovsky, 1979). The block structure has, as a rule, the form of “masonry” and is formed as a result of geological processes (M€ uller, 1982), and the size of blocks varies from 0.1 mm to first hundreds kilometers. In work (Makarov et al., 2007), it is taken into consideration the concept of divisibility of rocks on structural blocks for which there is a scale multiplier “[Fcy]2” ([Fcy]2 ¼ 2618—one of Fibonacci numbers) [instead of value 3 proposed in work (Sadovsky et al., 1987)] is introduced, in which connection “block” in Makarov et al. (2007) is considered as “dissipation structure.” Such an approach forces authors to find some classes of the nonlinear models which decision in the near future is represented as hardly feasible (Makarov et al., 2007). All it creates a sensation of the deadlock for the scientific direction that has once again appeared powerless before complexity of such an object, as geomedium at the researcher. Meanwhile, to such conclusion pushes not that other, as a position of authors of the considered concept basing the conclusions, mainly, on experiments with metals.

Principles of hierarchically block geomedium modeling Selection of scale levels of hierarchically block geomedium Really, what structural elements results Goldin results in his article, naming it “Destruction of lithosphere and the physical mesomechanics” (Goldin, 2002)? Noticing that the geomedium is hierarchically blocky, he, at transferring to mesolevel, formulates: “Basic types of mesoelements:

Principles of hierarchically block geomedium modeling

353

strips of plastic flow and rather rigid domains …. The main types of deformations at mesolevel is a shear and rotation ….” However, in geomechanics, it is well known that the basic structural element of geomedium is a crack. Moreover, it is the crack of shear-tensile type (Odintsev, 1996). Crack can evolve on three scales: micro-, meso-, and macro-ones. For example, microcracks of the rock sample have intra crystalline (intra-grain) character, mesocracks evolve, as a rule, on borders of grains (diameter of grain d) and have length from d to (5–10) d (Krilov, 2015). The macrocrack is a result of integration of mesocracks, and under the conditions of compression, it is legibly possible to point out a stage of a sustainable development of a macrocrack and to find limiting length of steady macrodefect. Process of affiliation of mesocracks with macrodefect formation has the specificity: it is preceded by localization of cracking process with formation source areas where, at stability loss, macrocrack growth descends jump. The majority of models of the locus of earthquakes is based on this process (Dobrovolsky, 1991; Rice, 1980). All these stages are well investigated on samples of rocks at monaxonic and volume loading (Lockner, Kuksenko, et al., 1991). Thus, mineral grains for the rock sample can be considered as blocks. At transferring to a rock mass, we found its block constitution already at the level of the dimensions of chinks and mining openings. Blocks form a configuration, which, as it was specified above, in geomechanics, is accepted to name “masonry.” Repetition of the processes similar to the phenomena of destruction, positioned for rock samples, and unlimited growth of the macrocracks which have lost stability leads to formation in a rock mass of system of blocks. As a result, the second hierarchical level of block structure of geomedium—the first structural level of a rock mass—is formed (Figs. 3 and 4). At this level, the macrocrack of the sample acts in a role of a mesocrack of a rock mass (on borders of blocks), and mesocracks of the sample act as a microcrack (intrablock) of a rock mass. In the conditions of the big compressing stresses in rock media, on borders of a rock mass blocks, there is a formation of conditions of a rupture at compression again. Considered above for the rock sample process of mesodefects accumulation, their localization in source areas with approach of the moment of interaction is retried. That is conducting to formation of a macrocrack and the second hierarchical level of blocks of a rock mass (it already third hierarchical level of block geomedium where the level of the rock sample considered above was the first one). The macrocrack of the second hierarchical level of the rock mass blocks, in turn, is a mesocrack at a following hierarchical level of geomedium.

354

16. Modeling of hierarchically block geomedium

The considered approach essentially broadens the ideas of “Mesomechanics” with reference to hierarchically block geomedium: the dimension of steady shear-tensile macrocracks defines the minimum dimension of the block of neighbor-top level and, accordingly, limits “from below” its level in block hierarchy. Thus, the interblock sheartensile cracks of the neighbor-low hierarchical level are considered how mesocracks, and intrablock—as microcracks. Thus, at transferring to the next in hierarchy top structural level of blocks, a macrocrack of the neighbor-low level is considered as a mesocrack of this neighbor-top level, and a mesocrack of the neighbor-low level—accordingly, as neighbor-top level microcracks. As a result, it had structural hierarchy of the geoblocks, each of which had a structural level, since following the rock sample, a rock mass level (modularity of the sample is defined by the natural dimension of minerals composing it), is certain in the limiting dimensions of steady macrodefect. In addition, at each structural level of geoblocks precipitate out micro-meso-macro-the scales directly bound to scales as neighbor-low as the top next level. From this point of view, the block constitution, or the block organization, is perceived by us in the prograduated sense with reference to geomedium, since the rock sample, in comparison with traditional understanding (Oparin & Tanajno, 2009; Sadovsky et al., 1987), is introduced in Makarov et al. (2007) (Fig. 5). Other not less important element of construction of adequate model of geomedium is definition of mesovolume and on this basis—allocation of “representative” mesovolume (Makarov et al., 2007).

Selection of mesovolume from positions of physically reasonable sizing of the structural block In the considered above researches (Makarov et al., 2007), working out of models of a hierarchically block rock mass contacts with concept of mesovolume which the presence of “all significant elements of a mesoconstitution” is meant. The condition of strong compression imposes also restriction on properties of “representative” mesovolume: it is that minimum part of mesovolume is capable to reflect its properties under condition of interaction of mesodefects. We discussed above that the interaction of shear-tensile mesocracks leads to their localization with formation of source area of a macrocrack preparation. Not less important, and that “reversive” character of deformations (Makarov, Ksendzenko, Golosov, et al., 2014) is found here as shown by near-source area research.

Hierarchically block geomedium: Structural levels

355

Reversive deforming indicates local reduction of strains in the nearsource areas, while the strain continues to increase in the source area. A completely set source and near-source areas represents mesocracking structure of “contrast type” on terminology (Goldin, 2005). Therefore, all fields of a highly compressed stuff of rock or a rock mass, where there is an interaction of mesocracks, it is possible to consider as mesovolume. From these positions, it is possible to consider as representative mesovolume that quantity of blocks of the conforming structural level with the interacting mesocracks educed on their borders, which is statistically significant from modeling positions. Generally, for the rocks, the dimension corresponding to the dimensions of 30 mineral grains (Ruppenejt & Liberman, 1960) that represents the sample which dimension oversteps the bounds of concepts “small” and where normal distribution of properties of grains is prevailing was taken over. The same approach could be conserved in case of the highly compressed rock sample. For a rock mass representative, it would be possible to consider volume of 30 rock blocks of the conforming level in structural hierarchy with the mesocracks educed on their borders under condition of interaction between them. The moment of the beginning of interaction of mesocracks directly precedes a point of reversal of linear strains in near-source area of the rock sample or a rock mass. For these experimentally fixed, conditions values of the module of deformation, lateral expansion coefficient and other parametres of representative mesovolume necessary for construction of model are defined. Let us consider the realization of the offered approach on an example of the two first hierarchical levels of block geomedium: the rock sample and a rock mass round underground openings.

Hierarchically block geomedium: Structural levels As it is already considered above, in 1991, Lockner and Kuksenko conducted research of laws of formation and development of macrocracks in rock samples with application of the newest servocontrolled technics for the process of destruction (Lockner et al., 1991). After a stage of chaotic cracking, there is a localization of mesocracks and, after affiliation, they sprout in rupture. Such models of macrocracks development were considered by one of the authors in 1996 in the monography where the presence of a stage of a sustainable development of macrodefect has been proved under the specified experimental conditions of Lockner-Kuksenko (Odintsev, 1996).

356

16. Modeling of hierarchically block geomedium

Definition of limiting steady length of shear-tensile rock macrocracks On borders of mineral grains as the most relaxed fields of a geomedium, at compression, there are the shift mesodefects generating the tension fields in end section (see Fig. 2). Shear-tensile mesocracks combine with load growth, and in the sample, the rupture macrocrack educes. The direction of its growth coincides with a direction of the maximum stress at axial compression of the sample. The macrocrack is modeled by a cut with the holding apart load distributed on coast. Expression of stress intensity factor for it looks like (Odintsev, 1996):   K1 ¼ ðπlÞ1=2 γ 1 σ 01  γ 3 σ 03 (4) where l—semilength of a crack, m; σ 0 1 ,σ 0 3 —accordingly the maximum and minimum principal stresses; γ 1, γ 3—empirical coefficients. Presented in Odintsev (1996) theory of shear-tensile destruction of rocks describes all basic experimental facts known from tests of rock samples and allows to define the maximum length steady shear-tensile macrocracks. At the first hierarchical level of the block geomedium presented by rock samples (diameter of 3–5 cm, height of 6–10 cm), act as structural blocks the mineral grains having borders accurately precipitating out in experiment which set heterogeneity of the strength properties of rock of this hierarchical level. The attitude of the dimensions of such blocks

s 01

s 01

Tension

Sliding Crack Model

S

Sliding

q(x) y

Tension

a

(A)

x

s 03

s 03 Mineral Grains

b

Δ Δ

–S

Q Q q

Q

(C) FIG. 2

S

s 03

(B)

Q

s 01

x

Stages of formation shear-tensile macrocracks and its modeling.

0

Q y

c

Q

–S

357

Hierarchically block geomedium: Structural levels

600 Axial Stress (MPa)

a

d

b c

e f

400 A 200 B 0

0

2 1 Axial Stortering (nn) a

b

c

d

e

f

FIG. 3 Stages of formation of macrorupture according to АE (granite) (Lockner et al., 1991).

a

b

h

d h:d=2

(A)

(B)

FIG. 4 The first level of block hierarchy of geomedium: (A) mineral grains in the sample of a granite and (B) the schema of a cylindrical of a rock sample.

(diameter of mineral grain 0.1–0.5 mm) to the dimension of the sample on the average compounds 1:100. The dimensions of microcracks for a mineral compound size of order of the mineral grain dimensions (Krilov, 2015). The dimensions of tensile mesocracks are arising at compression of the rock sample vary in the range from 1 to 5–10 diameters of grain (Odintsev, 1996): 2lmesomin ¼ dgrain ¼ ð0.1  0.5Þ mm sam 2lmesomax ¼ ð5  10Þdgrain sam The dimensions of critical length of macrocracks at axial compression 2lmacro-max (limiting length of steady macrodefect) can be received from a ∗sam following formula (Odintsev, 1996):

358

16. Modeling of hierarchically block geomedium

Relation of the block length (Lx-y) to height (L1) 4:1

Forms I

II

III

“agregate”

III

I III II

I

“masonry”

III

II

I

I

I III II II

III

II

I

III

I II

III

II

I II

III

I II

III

45⬚ I III II

“random”

II 45⬚ V I IV

IV

“dencity”

I II

I IV

II II

I V

IV

IV

III I

III II

FIG. 5

A block constitution of a rock mass: types of blocks (Procedure of drawing up of geoblock diagrams (models) of a rock masses in the establishments of hydroconstructions, 1991).

l∗ macromax ¼ sam

h∗ E 4ð1  ν2 Þγ 1 σ c

(5)

where h∗  dmax—the dimension of the block in the laboratory sample; E— a rock elastic modulus; σ c—ultimate strength on axial compression; ν—a rock Poisson’s ratio; dmax—the maximum diameter of mineral grain. At transferring to a following top structural level in hierarchy of rock blocks of geomedium for openings in diameter of 3–5 m—that corresponds to conditions of the majority of the solitary mining openings coal and ore treating industry—the interrelation 1:100 between the dimensions of the rock sample and diameter of opening also is conserved. If it is to be believed that the principle of geometrical similarity between the block structure of the rock sample and a rock mass at hierarchically

Hierarchically block geomedium: Structural levels

359

second level of block geomedium consideration of blocks of a rock mass demands that their dimensions corresponded to the dimensions of the inferior level blocks as 1:100 (Sedov, 1977), it is fair. To them, there correspond blocks dmas ¼ (1  5) cm, and the conforming dimensions of mesodefects are defined as: 2lmesomin ¼ dmas ¼ ð1  5Þcm mas 2lmesomax ¼ ð5  10Þdmas mas

(6)

It is easy to notice that on the average 2lmaso-min ¼ dmas ¼ lmacro-max , takmas ∗sam ing into account the form of blocks of a rock mass of type of “masonry” that can be accepted as an assessment at carrying out of tests for axial compression. It allows to consider macrodefect at hierarchically first level of the rock sample as mesodefect of the minimum length at hierarchically second (top) level of geomedium—level of mining opening. In this case, the maximum length of a mesocrack of a rock mass will compound 2lmeso-max  (5  10)  2lmacro-max ¼ 10–100 cm that corresponds to majority mas ∗sam parametres of a rock mass cracking systems (Tchernyshev, 1983). We also notice that application of expression (5) is hereinafter manufactured at, in whole, little change of the attitude E/σ c  const for various levels of geomedium (Vladimir, Makarov, Guzev, Odintsev, & Ksendzenko, 2016).

Dimensions of shear-tensile cracks, faults of geomedium and the basic levels of block hierarchy Summing up to sizing of mesocracks at various levels of geomedium, it is possible to notice that as a mesocrack of next top in relation to the rock sample of hierarchical level of underground opening the steady macrocrack of the maximum testing length—the next inferior hierarchical level (Table 1) will serve. Thus, interrelation requirements 1:100 are yielded by rock mass scale in which limens conditions of geometrical similarity (Sedov, 1977) are conserved, and at conservation of mechanisms of destruction of rocks in a rock mass by rupture in the conditions of compression for openings in diameter to 3–5 m are carried out, except that, the requirement of physical similarity. Manufacturing, further, conduction of the received results on top levels of a rock mass within a crust, it is possible to receive assessments of the maximum dimensions of steady macro-ruptures. Total generalizations are resulted in Table 2. Taking linear approach of the theory into account, the settlement values resulted in Table 2, it is possible to consider satisfactory. Apparently, from Table 2, all four hierarchical levels of blocks within a crust are evolved, making it quite foreseeable to working out of the general model.

Mass

Rock block dimension dmas ¼ lmacro ∗sam

Dimension is equal to diameter of opening dop 10 dop

Meso

Macro

2lmacro-min ¼ (5  10)dmas mas

2lmeso-min ¼ dmas ¼ lmacro-max mas ∗sam

1 σ res

hmas  E ∗ 2lmacromax ¼ 2ð1ν 2 Þγ ∗mas

2lmeso-max ¼ (5  10)dmas mas

1

 σc

d ¼ 3050 mm—sample diameter h ¼ 60100 mm—sample height

Macro

h E

2lmacro-min ¼ (5  10)dmax sam

dmax—mineral grain dimension

Meso

2lmacromax ¼ 2ð1ν∗2 Þγ ∗sam

2lmeso-max ¼ (5  10)dmax sam

2lmeso-min ¼ dmax sam

Sample

Max

Min

Length of a crack

Block dimension

Scale

Hierarchical

Level

TABLE 1 Interrelation of the dimensions of defects at various hierarchical levels of block structure of rocks.

Hierarchically block geomedium: Structural levels

TABLE 2

361

Structural blocks dimensions at various hierarchical levels of geomedium.

Hierarchical level

Structural block dimension

Macrodefect steady length dimension by formula (5)

Actual range (t structural block)

Rock sample

0.5 mm

10 cm

0.1–0.5 mm

Krilov (2015) and Odintsev (1996)

Rock mass (mining opening)

5 cm

10 m

10–100 cm

Tchernyshev (1983)

Rock mass (excavation opening)

5m

1000 m

2–5 m

Lushpej, Makarov, and Laptev (1982)

Rock mass (crust)

500 m

100 km

10–230 km

Sadovsky et al. (1987) Lengths of breaks

Citations

After scaling in which frame the requirements of works (Makarov et al., 2007; Nikolaevsky, 1984; Sadovsky et al., 1987) are fulfilled on averaging of mesodefects, construction of model of a continuous medium with cracking defects at all levels of hierarchically block rock mass goes on a constructive path because the shear a component at shear-tensile destruction remains defining in processes of formation of a macrocrack. Therefore, application of models of mesomechanics (Makarov et al., 2007) to geomedium at all levels of its block hierarchy is represented and proved in accordance with certain principles (Guzev, Odintsev, & Makarov, 2018). Defining abandoning of kinematic conditions of Saint-Venant deformations compatibility ( Jaeger, Cook, & Zimmerman, 2007) on mesoscale leads to transferring from classical elastic model of medium to nonEuclidian model. It allows to formulate a principle of non-Euclidian hierarchies at construction of model of a continuous medium with types of structural defects as shear-tensile cracks. It consists that the hierarchically block rock mass is changed with system of gauge (non-Euclidian) models, each of which covers this or that structural level of block hierarchy, allowing to set the adequate description to form mesocracking structures. The principle of non-Euclidian hierarchies is supplemented with a principle of “monolithic block,” which means the consideration of intra-block defects as microcracks at each structural level. Thus, the contouring of block by interacting mesocracks answers an absence condition generally to compatibility of deformations in each

362

16. Modeling of hierarchically block geomedium

physical point presented by the block. Macrocracks at each structural level are the solitary ruptures, interaction between which is absent. In the presence of a panel of such models and observance of the specified principles, there is a possibility of definition of the basic characteristics of dissipative mesocracking structures of geomedium, such as the dimensions of the source and near-source areas of the source type mesostructures, radial width of zones in case of the zonary type mesostructures, and others on each of the considered structural levels.

Conclusion In conformity with the formulated principle of non-Euclidian hierarchy, the modeling of dissipative mesocracking structures characteristics of rock samples and rock mass should be realized by calculation with application of non-Euclidian models of the conforming hierarchical level. With that purpose, in view performance, the following sequence of actions is necessary. 1. Establishment of hierarchical structural level of block geomedium on the basis of a geological assessment of the dimensions of geoblocks. 2. For the positioned structural level of geomedium, the definition of scale level (meso-macro) defective (cracking) structures. 3. Choice of non-Euclidian models of a continuous medium of hierarchy corresponding to chosen level and to the meso-dimension defined for it. 4. Working out of methods of definition of phenomenological parametres of non-Euclidian continuous medium models. 5. Decision of the boundary value problem corresponding to problems of modeling. 6. Working out of algorithms and programs of calculation of dissipative mesocracking structures parametres of the conforming level of block geomedium (rock samples or masses). Thus, the general method of definition of characteristics of dissipative mesocracking structures of samples and rock masses is that is that the nonEuclidean model of the continuous medium corresponding to the chosen conditions is developed for excretion on the basis of a geological evaluation of the dimensions of the geoblocks of the hierarchical level of the block geomedium and certain level of mesodefects, and mesostructure characteristics are defined by calculation after the decision of the conforming boundary value problem and definition of parametres of model on the basis of the yielded experiments. In Chapter 4, the problem of modeling of hierarchically block geomedium it is considered on an example of zonary mesostructure of contrast type in a rock mass round underground openings.

References

363

Acknowledgments The research is supported by the Ministry of Science and High Education of the Russia Federation, unique identifier of the agreement RFMEFI58418X0034.

References Aleksandrova, Н. I. (2019). Three-dimensional modelling of diffusion of pendular waves at dynamic loading surfaces of underground opening. Fundamental and Applied Questions of Mining Sciences, 6(1), 32–38. (in Russian). Aleksandrova, N. I., & Sher, E. N. (2004). Modelling of process of wave propagation in block mediums. Journal of Mining Science, (6), 49–57. (in Russian). Babeshko, V. A., Babeshko, O. M., & Evdokimova, I. C. (2009). About a solution of a problem of block structures of academician. In М. А. Sadovsky (Ed.), Ecological Bulletin of the Black Sea Economic Cooperation (pp. 18–23). No. 1. (in Russian). Crosby, W. O. (1882). On the classification and origin of joint structures. Proc. Boston Society Natural History, 1882-1883. Vol. 22, pp. 72–85. Boston: Printed for the Society 1884. Dobrovolsky, I. P. (1991). Theory of preparation of tectonic earthquake. Мoscow: RAS. 217 p. (in Russian). Goldin, S. V. (2002). Destruction of lithosphere and the physical mesomechanics. Physical Mesomechanics, 5(5), 5–22. (in Russian). Goldin, S. V. (2005). Macro- and mesostructures source earthquake areas. Physical Mesomechanics, 8(1), 5–14. (in Russian). Guzev, M. A., & Makarov, V. V. (2007). Deforming and destruction of highly compressed rocks and rock masses. Vladivostok: Dalnauka. 232 p. (in Russian). Guzev, M. A., Odintsev, V. N., & Makarov, V. V. (2018). Principals of geomechanics of highly stressed rock and rock massifs. Tunnelling and Underground Space Technology, 81(November), 506–511. https://doi.org/10.1016/j.tust.2018.08.018. Jaeger, C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of rock mechanics. Blackwell Publishing. 475 p. King, W. (1875). Report on the superinduced divisional structure of rocks, called jointing, and its relation to slaty cleavage. The transactions of the Royal Irish Academy. Vol.25, pp. 605–662. Krilov, A. N. (2015). Microcracking heads of cabbage in a solid on an example of rocks. Mining Information-Analytical Bulletin (Scientific and Technical Magazine), (7), 221–225. (in Russian). Lockner, D. A., Kuksenko, V. S., et al. (1991). Quasi-static fault growth and shear fracture energy in granite. Nature, 350(6313), 39–42. Lushpej, V. P., Makarov, V. V., & Laptev, A. S. (1982). Tectonophysics an assessment of strains for conditions of the Natalkinsky deposit and means of increase of stability of detection. Kolyma, 3–4, 18–21. (in Russian). Makarov, P. V. (1998). Approach of physical mesomechanics to modelling of processes of deformation and destruction. Physical Mesomechanics, 1(1), 61–68. (in Russian). Makarov, V., Ksendzenko, L. S., Golosov, A. M., et al. (2014). Reversible deformation phenomena of a high stressed rock samples. In Rock engineering and rock mechanics (pp. 267–272). Proceedings of EUROCK. Makarov, P. V., Smolin, I. J., Stefanov, J. P., et al. (2007). In L. B. Zuev (Ed.), The nonlinear mechanics of geostuffs and geomediums. Novosibirsk: SB RAS Geo. 236 p. (in Russian). M€ uller, L. (Ed.). (1982). Rock mechanics. Wien; New York: Springer. 390 p. Nikolaevsky, V. N. (1984). Mechanic of osculiferous and fissured mediums. Moscow: Nedra. 232 p. (in Russian).

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Odintsev, V. N. (1996). Tensile destruction of a rock mass. Moscow: Ipkonran. 166 p. (in Russian). Oparin, V. N., & Tanajno, A. S. (2009). Representation of the dimensions of natural separateness of rocks in an initial scale. In Classification/physicotechnical problems of working out of minerals (pp. 40–53). No. 6. (in Russian). Panin, V. E. (Ed.). (1995). The physical mesomechanics and computer designing of stuffs. Novosibirsk: Nauka. Vol. 1. 297 p.; Vol. 2. 320 p. (in Russian). Panin, V. E., Grinjaev, J. V., Danilov, V. I., et al. (1990). Structural levels of a plastic strain and destruction. Novosibirsk: Nauka. 255 p. (in Russian). Procedure of drawing up of geoblock diagrams (models) of a rock masses in the establishments of hydroconstructions. (1991). The grant to СНиП 2.02.02-85 - L: VNIIG, 119 p. (in Russian). Rice, J. R. (1980). The mechanics of earthquake rupture. In A. M. Dziewonski, & E. Boschi (Eds.), Physics of the earth’s interior (pp. 555–649). Italian Physical Society. 216 p. Ruppenejt, K. V., & Liberman, J. M. (1960). Introduction in rocks mechanics: (pp. 95–111). 43. Мoscow: Gosgortehizdat. (in Russian). Sadovskii, V. M., & Sadovskaya, O. V. (2015). Modeling of elastic waves in a blocky medium based on equations of the Cosserat continuum. Wave Motion, 52, 138–150. Sadovsky, M. A. (1979). Natural кусковатость rock. RAS USSR, 274(4). (in Russian). Sadovsky, M. A., Bolhovitinov, L. G., & Pisarenko, V. F. (1987). Deforming of geophysical medium and seismic process. Moscow: Nauka. 100 p. (in Russian). Sedov, L. I. (1977). Methods of similarity and dimension in the mechanic. Moscow: Nauka. 440 p. (in Russian). Shemjakin, E. I., Fisenko, G. L., Kurlenja, M. V., Oparin, V. N., et al. (1986). Effect of zonary disintegration of rocks round underground openings. RAS USSR, 289(5), 1088–1094. (in Russian). Tchernyshev, S. N. (1983). Discontinuities of rocks. Moscow: Nauka. 240 p. (in Russian). Vang, К., Aleksandrova, Н. И., Sir, И., Oparin, Century Н, Dow, Л., & Chanyshev, A. I. (2019). Vlijanie of parametres of block medium on process of a dissipation of energy. Applied Mechanics and the Technical Physics, 60(5), 168–177. (in Russian). Vladimir, V., Makarov, M., Guzev, A., Odintsev, V. N., & Ksendzenko, L. S. (2016). Periodical zonal character of damage near the openings in highly-stressed rock massif conditions. Journal of Rock Mechanics and Geotechnical Engineering, 8(2), 164–169. https://doi.org/ 10.1016/j.jrmge.2015.09.010.

C H A P T E R

17 Modeling of liquefaction and large-strain response of sand in sloping ground Gabriele Chiaroa, Nalin L.I. De Silvab, and Junichi Kosekic a

Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand, bDepartment of Civil Engineering, University of Moratuwa, Moratuwa, Sri Lanka, cDepartment of Civil Engineering, University of Tokyo, Tokyo, Japan

List of symbols CRR, CSR, α d1, d2 dp0 dγ, dγ e, dγ p dεvol, dεevol, dεpvol dτ D, Dult Dr,ic D50 e, eic emax, emin Fc F1, F2 f(e), f(eic) G, Gmax Ga Gs H, Hult K

cyclic resistance ratio, cyclic stress ratio, and static stress ratio dilatancy parameters effective mean stress increment total, elastic and plastic shear strain increments total, elastic and plastic volumetric strain increments shear stress increment damage function and maximum value of D relative density at the end of consolidation mean grain diameter current void ratio and void ratio at the end of consolidation maximum and minimum void ratios soil fines content drag parameters void ratio functions current shear modulus and small-strain shear stiffness small-strain shear stiffness parameter specific gravity hardening function and maximum value of H bulk modulus

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00007-1

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© 2021 Elsevier Inc. All rights reserved.

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

Kα n Nd p0 , p0 ic pa x, xa y, ya γp γref δ α0 , β0 , κ, ξ1(0), ξ1(∞), ξ2(0), ξ2(∞) ξ1(x), ξ2(x) ν ρ1, ρ2 τ, τmax τcyclic, τstatic (τ/p0 )max, (τ/p0 )PTL

factor for normalized cyclic resistance ratio soil parameter for small-strain shear stiffness stress-dilatancy factor current and initial effective mean stresses atmospheric pressure GHE normalized plastic shear strains GHE normalized stress ratios plastic shear strain GHE reference shear strain drug function GHE parameters strain-dependent GHE fitting parameters Poisson’s ratio peak stress ratio parameters shear stress and maximum shear stress cyclic and static shear stresses peak stress ratio and stress ratio at phase transformation

Introduction It is well known that the undrained response during cyclic loading of saturated sand deposits within the sloping ground is different from that of level ground conditions because these sand deposits are subjected to the gravitational driving static shear stress on the horizontal plane or assumed failure surface. During earthquake shaking, such sand deposits are subjected to further shear stresses due to shear waves propagating vertically upward from the bedrock. The superimposition of the static shear and seismically induced cyclic shear stresses can have major effects on the undrained response of the sand, leading to sand liquefaction and extremely large ground deformation (e.g., Chiaro et al., 2015a, 2017a; Cubrinovski et al., 2011, 2017; Hamada, O’Rourke, & Yoshida, 1994). Compared with the large body of experimental data describing the undrained cyclic behavior of sands under level ground conditions, there have been very limited studies focusing on the undrained cyclic response of sands under sloping ground conditions. Such studies, mainly based on triaxial tests (Castro & Poulus, 1977; Hyodo, Murata, Yasufuku, & Fujii, 1991; Hyodo, Tanimizu, & Yasufuku, Murata, 1994; Lee & Seed, 1967; Sivathayalan & Ha, 2011; Vaid & Chern, 1983; Vaid, Stedman, & Sivathayalan, 2001; Yang & Sze, 2011, among others) and occasionally on simple shear, torsional shear or ring shear tests (Tatsuoka, Muramatsu, & Sasaki, 1982; Vaid & Finn, 1979; Yoshimi & Oh-oka, 1975, among others) have produced valuable data showing that the presence of initial static shear can have major effects on the cyclic response of sands. Nevertheless, contradictory views seem to exist with respect to such effects (e.g., Chiaro, Koseki, & Sato, 2012), i.e., the cyclic strength can either increase or decrease due to the presence of static shear stress depending

Introduction

367

on the density of the sand, magnitude of static shear, the criterion used to define the liquefaction resistance, testing conditions used, etc. Due to the scarcity of experimental data and the lack of convergence and consistency in the existing data, it has been strongly recommended that such experimental findings should not be used by nonspecialists or in routine engineering practice (Youd, Idriss, Andrus, et al., 2001). There is a need for continued in-depth research to improve our understanding of the complicated effects of initial static shear on the cyclic resistance of sandy soils within the sloping ground. In any seismic event, the development of large ground deformation represents a major hazard to many engineering structures and buried lifeline facilities. Therefore, when evaluating liquefaction, it is important to assess whether or not a given soil within the sloping ground in its in situ density and stress state has the potential for large ground deformation. In this context, numerical models are frequently used for estimating liquefaction-induced ground deformations and associate damage of soil and soil-structure systems subject to seismic loading. The accuracy of such models mainly depends on the correctness of the employed constitutive equations to approximate the pre- and post-liquefaction behaviors in the presence of initial static shear stress imposed by sloping ground or overlying structures in combination of the irregular cyclic loading imposed by the earthquake shaking (Ziotopoulou & Boulanger, 2016). Ziotopoulou (2014) examined several reputable numerical models for liquefiable soils and concluded that most of such models have notable limitations in their ability to simulate the effects of sloping ground on pore water pressure and shear strain generation for the full range of densities, initial static shear ratios, and vertical effective mean stresses important to practice. Besides, it was also found that the ability to simulate liquefaction features under sloping ground conditions may not necessarily be improved by better calibration of the model parameters, which means that such models are controlled by constitutive equations that may require key changes (Ziotopoulou, 2014). Aimed at providing new insights clarifying the effects of sloping ground on the liquefaction and cyclic resistance of sands, a two subsequent steps research activity was undertaken by the authors. First, a systematic laboratory investigation was carried out on Toyoura sand specimens subjected to various levels of combined static and cyclic shear stresses using a state-of-the-art torsional simple shear (TSS) device (Chiaro, 2020; Chiaro et al., 2012; Chiaro, Kiyota, & Koseki, 2013a; Chiaro, Umar, Kiyota, & Koseki, 2021). Then, a new state-dependent cyclic model, namely here T-sand model (Chiaro, De Silva, & Koseki, 2017b; Chiaro, Koseki, & De Silva, 2013b, 2013c; Chiaro, Koseki, De Silva, & Kiyota, 2015b; De Silva & Koseki, 2012; De Silva, Koseki, Chiaro, & Sato, 2015), was developed to support the experimental work

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and conduct detailed parametric analyses. Since the experimental findings have been exhaustively reported elsewhere, in this chapter, they are only briefly recalled for the benefit of comprehensiveness. On the other hand, the T-sand model (version 2) formulation and calibration procedure are described in detail, and the model performance is examined by simulating single-element undrained cyclic TSS tests with initial static shear carried out on Toyoura sand specimens. From the analyses of the simulation results, the effects of sloping ground conditions on the cyclic resistance of Toyoura sand are then scrutinized in terms of cyclic resistance ratio (CRR) and Kα factor for various relative density (Dr) levels ranging from 30% to 60%, and a comparison is made with independent experimental studies available in the literature.

Experimental data on the response of liquefiable soils under sloping ground conditions In view of the above background, Chiaro (2020) and Chiaro et al. (2012, 2013a, 2021) performed a number of large-strain cyclic undrained TSS tests with initial static shear on loose (Dr,ic ¼ 25%–30%) and mediumdense Toyoura sand specimens (Dr,ic ¼ 44%–48%). Toyoura sand is a fine poorly graded quartz-rich subangular sand (D50 ¼ 0.18; emax ¼0.992; emin ¼ 0.632; Gs ¼ 2.656, Fc ¼ 0.1%). The hollow cylindrical specimens (300 mm in height, 150 mm in outer diameter, and 90 mm in inner diameter) were prepared by the air pluviation method. The specimens were isotropically consolidated to an effective mean principal stress of p0 ic ¼ 100 kPa with a back pressure of 200 kPa, and then monotonically sheared by keeping drained conditions, to apply a specific value of initial static shear. Finally, undrained torsional shear loading was applied to simulate seismic conditions. Cyclic loading tests were performed over a wide range of static shear stress ratios (α ¼ τstatic/p0 ic) from 0 to 0.3. Three levels of cyclic stress ratio (CSR ¼ τcyclic/p0 ic) of 0.12, 0.16, and 0.20 were also employed to consider various combinations of initial static and cyclic shear stress reproducing stress reversal and stress nonreversal loading conditions. During the process of undrained cyclic torsional loading, the vertical displacement of the top cap was prevented to simulate as much as possible the simple shear condition that ground undergoes during horizontal seismic excitation. It was confirmed that the presence of initial static shear does play a key role on the failure modes of sand in the sloping ground (i.e., flow failure or shear failure), the way shear strain level can be developed (i.e., flow deformation or progressive deformation accumulation) and the extent of residual deformation. Moreover, under simple shear conditions, sands in the sloping ground may experience three distinct failure mechanisms during

Experimental data on the response of liquefiable soils under sloping ground conditions

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FIG. 1 Typical stress-strain relationship and effective stress path of Toyoura sand under stress reversal loading conditions in cyclic undrained TSS tests with static shear.

earthquakes, namely cyclic liquefaction, rapid (flow) liquefaction, and residual deformation failure. The most critical one is the rapid liquefaction (Fig. 1) which produces an abrupt development of large shear deformation (i.e., flow failure) without any warning. On the other hand, when the onset of initial liquefaction is not achieved, shear failure may be induced by a progressive accumulation of large residual deformation (Fig. 2). Cyclic and rapid (flow) liquefaction can only occur under stress reversal loading conditions. In contrast, residual deformation failure may occur under stress nonreversal loading conditions. It was demonstrated that the presence of initial static shear (i.e., sloping ground conditions) can be either detrimental or beneficial to cyclic resistance. In this regard, a threshold value of static stress ratio (α ¼ τstatic/pic0 ) exists after which cyclic resistance of sand tends to increase with α. The α-threshold identifies the change in failure behavior between liquefaction and residual deformation failure. The magnitude of the initial static shear (τstatic) is often expressed in terms of static shear stress ratio (α ¼ τstatic/pic0 ), which is the ratio of the

FIG. 2 Typical stress-strain relationship and effective stress path of Toyoura sand under stress nonreversal loading conditions in cyclic undrained TSS tests with static shear.

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

FIG. 3 Experimental relationships between (A) CRR and static shear stress ratio and (B) Kα factor and static shear stress ratio.

τstatic to the effective mean principal stress (pic0 ). The effects of α on the cyclic resistance ratio (CRR) of the sand is then usually expressed in terms of a Kα factor (Kα ¼ CRRα/CRRα¼0), where CRR refers to the CSR required to trigger liquefaction (according to a specific failure criterion) in a specified number of cycles; CRRα is the value of CRR for a given value of α; and CRRα¼0 correspond to the level ground condition (α ¼ 0). Experimental results on a range of sands at confining pressure less than 300 kPa showed that cyclic resistance would tend to increase with increasing α for dense sands and tend to decrease with increasing α for loose sands (Boulanger, Seed, Chan, Seed, & Sousa, 1991; Vaid & Finn, 1979). The trends reported in Figs. 3A and B show the values of CRR and Kα, respectively, from the TTS tests on Toyoura sand carried out by Chiaro (2020) and Chiaro et al. (2012, 2013a, 2021) at Dr ¼ 25%–48% with effective mean pressure pic0 ¼ 100 kPa. These trends are well in agreement with those from relevant past studies where simple shear conditions were used (Ziotopoulou & Boulanger, 2016). Therefore, the experimentally observed effects of α on CRR or Kα values, as reported in Fig. 3, provide a basis for evaluating the ability of T-sand model to simulate behaviors that have been observed for Toyoura sand in TSS tests with initial static shear under stress reversal and stress nonreversal loading conditions.

Background of the T-sand model It is widely recognized that hyperbolic equations can be used to model the highly nonlinear stress-strain behavior of soil subjected to shear loading (Cubrinovski & Ishihara, 1998a, 1998b; Duncan & Chang, 1970; Hardin & Drnevich, 1972; Konder, 1963; Tatsuoka & Shibuya, 1992).

Background of the T-sand model

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In particular, the GHE proposed by Tatsuoka and Shibuya (1992) can properly simulate stress-strain relations from very small to large strain levels for a wide range of geomaterials under general loading conditions (De Silva et al., 2015; HongNam, 2004; HongNam & Koseki, 2005; Tatsuoka, Masuda, Siddquee, & Koseki, 2003; Tatsuoka, Siddquee, Park, Sakamoto, & Abe, 1993). De Silva (2008) successfully used the GHE combined with an empirical stress-dilatancy equation valid for simple shear conditions (Nishimura & Towhata, 2004) to simulate the overall behavior of Toyoura sand undergoing drained/undrained monotonic/cyclic TSS loading conditions. However, neither the density nor the combined influence of density and stress level was considered as a variable. To be more precise, the same sand at different densities was regarded as a different material, and the effects of confining pressure were considered to be independent of the density state. As a consequence, different sets of model parameters were needed for simulating different density and stress-level conditions. Following, Chiaro (2010) and Chiaro et al. (2013b) presented an improved GHE-based model that deals with density and stress statedependency upon drained/undrained behavior of sand. Such a model was able to predict sand behavior in monotonic undrained TSS tests over a wide range of void ratios and confining pressures, including the effects of initial static shear, using a single set of soil parameters. Aimed at developing a more robust constitute model able to describe the cyclic response of saturated sandy soils under level and sloping ground conditions over a wide range of densities and stress levels, the former monotonic state-dependent model by Chiaro (2010) and Chiaro et al. (2013b) and the cyclic model by De Silva and Koseki (2012) and De Silva et al. (2015) were combined and a new cyclic state-dependent model (i.e., the T-sand model) which makes it possible to simulate the effects of initial static shear on the undrained cyclic behavior of saturated sand was obtained. The main advantage of this new model is the use of a limited number of soil parameters, which have a clear physical meaning and can be straightforwardly determined in the laboratory. The model calibration has been improved over time as more TSS test results become available (Chiaro et al., 2013b, 2015b, 2017b). The T-sand model can be considered as a suitable tool to evaluate the liquefaction potential and cyclic strain accumulation of sands under both level and sloping ground conditions, and its capability has been proven over the time by simulating experimental results of monotonic and cyclic undrained TSS tests on Toyoura sand under different density states and confining pressure levels (Chiaro et al., 2013b, 2015b; De Silva et al., 2015), and initial static shear stress loading conditions (Chiaro et al., 2017b).

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

Formulation and calibration of the T-sand model In this section, the formulation of the T-sand model (version 2) is reported in detail. It should be noted that compared to the T-sand model (version 1) described in Chiaro et al. (2017b), some modifications have been made to the model to simplify its mathematical formulation as well as improve its predictive capability, the major being: (i) modeling of the isotropic consolidation process was added; (ii) pore pressure/dilatancy ratio parameters, originally derived from drained TSS tests, have been redefined directly by means of undrained TTS tests; and (iii) peak stress ratio function has been recalibrated by using a larger number of undrained TTS tests covering a wider range of void ratio values.

Modeling the isotropic consolidation process of sand in torsional shear tests The undrained shear response of sand is influenced by the consolidation history. Therefore, in the T-sand model, the isotropic consolidation behavior of Toyoura sand is carefully modeled. Fig. 4A shows the change of the void ratio (e) during isotropic consolidation for three fully saturated Toyoura sand specimens. Irrespective of the initial void ratio (e0) at which the specimens were prepared, it is observed that the void ratio decreases with increasing effective mean stress (p0 ) and such behavior can be described by the following exponential relation:
 p0 eic ¼ e1 exp 0:0054 ic (1) pa where eic is the void ratio at the end of the isotropic consolidation process, 0 e1 is the void ratio measured at p0 ¼ 1 kPa (i.e., e1  e0), pic is the effective

FIG. 4

Isotropic consolidation behavior of hollow cylindrical Toyoura sand specimens in torsional shear apparatus.

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373

mean stress at the end of isotropic consolidation, and pa is the reference atmospheric pressure ( 100 kPa). For completeness, Fig. 4B reports the experimental and predicted trends plotted in terms of normalized void ratio versus effective mean stress ratio in a semilogarithmic plot.

Modeling the nonlinear stress–strain response of sand Elasto-plastic theory In the T-sand model, a key hypothesis is that for any given shear stress increment (dτ) both the elastic (γ e) and plastic (γ p) shear strains do always occur so that sand continuously yields from the very small strain levels and a purely elastic region does not exist. As per the classic elasto-plastic theory, the shear strain increment (dγ) is defined as the sum of the elastic (dγ e) and plastic (dγ p) shear strain increments: dγ ¼ dγ e + dγ p

(2)

e

In the above equation, the dγ is calculated using the quasielastic constitutive model proposed by HongNam and Koseki (2005): dγ e ¼

dτ Gmax

(3)

where dτ is the shear stress increment, and Gmax is the small-strain shear stiffness. Formulation of the extended GHE The highly nonlinear stress-strain behavior of sand subjected to drained/undrained shearing is modeled by using the generalized hyperbolic equation (GHE) proposed by Tatsuoka and Shibuya (1992). Typically, GHE is expressed in the form of: x y¼ (4) 1 x + ξ 1 ð xÞ ξ 2 ð xÞ where x and y are two functions representing the normalized plastic shear strain and shear stress, respectively. In the original GHE formulation, x and y are defined as:  Gmax  x ¼ γp original GHE (5) τmax  τ  original GHE (6) y¼ τmax

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where γ p is the plastic shear strain; τ and τmax are the current and the maximum shear stresses, respectively; and Gmax is the small-strain shear stiffness. Furthermore, ξ1(x) and ξ2(x) are two shear-strain-dependent fitting parameters necessary to simulate in a more realistic way the highly complex nonlinear stress-strain behavior of sand (Tatsuoka & Shibuya, 1992). In the T-sand model, to describe torsional shear loading conditions, they are formulated as follows: ξ 1 ð xÞ ¼

1 + ψα 1  ψα ξ1 ð0Þ + ξ 1 ð ∞Þ 2 2

1 + ψβ 1  ψβ ξ2 ð0Þ + ξ 2 ð ∞Þ 2 2   π ψ α ¼ cos ðα0 =xÞκ + 1   π ψ β ¼ cos κ ðβ0 =xÞ + 1

ξ2 ðxÞ ¼

(7) (8) (9) (10)

where ξ1(0), ξ1(∞), ξ2(0), ξ2(∞), α0 , β0 , κ are model parameters obtained by fitting the experimental data plotted in terms of y/x versus y relationship, as typically shown in Fig. 5. Specifically, ξ1(0) is the initial normalized plastic shear modulus and ξ2(∞) represents the normalized peak strength of sand.

FIG. 5

Evaluation of GHE model parameters for Toyoura sand.

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375

Fig. 6 reports the behavior of three Toyoura sand specimens monotonically sheared under undrained conditions in a torsional shear apparatus. The corresponding simulations obtained using the T-sand model are reported as well for comparison. The stress-strain curves (Fig. 6A) and effective stress paths (Fig. 6B) are highly nonlinear. However, when the data are replotted in terms of shear stress ratio (τ/p0 ) versus plastic shear strain (γ p), the overall sand response assumes the form of hyperbola (Fig. 6C) that can be modeled using the GHE approach. Yet, if the data are further normalized in terms of an extended x versus y relationships using the extended GHE method, the sand response becomes unique irrespective of the void ratio and mean effective stress (Fig. 6d). Consequently, modified x and y functions have been proposed to account for the void ratio and confining stress level dependence of drained/undrained stress-strain behavior of sand into the GHE: x ¼ γp

FIG. 6

Gmax =p0ic ðextended GHEÞ ðτ=p0 Þmax

(11)

Monotonic undrained torsional shear behavior of Toyoura sand: (A–D) experimental results; and (Continued)

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Shear stress, t (kPa)

100 pic' =100 kPa

eic=0.691

75 50

eic=0.820

25 eic=0.859

0 T-sand model simulations

0

(E)

10

20

Shear strain, g (%) 100

Shear stress, t (kPa)

pic' =100 kPa

75

eic=0.691

50

eic=0.820

25 eic=0.859

0

T-sand model simulations

0

50

100

150

Effective mean stress, p' (%)

(F)

FIG. 6, CONT’D (E and F) simulated response using the T-sand model.

y¼

τ=p0 ðextended GHEÞ ðτ=p0 Þmax

(12)

where γ p is the plastic shear strain; τ is the shear stress; p0 is the current effective mean stresses during shearing; p0 ic is the effective mean stresses at the end of isotropic consolidation; (τ/p0 )max is the peak shear stress ratio; and Gmax is the small-strain shear stiffness. As shown below, in Eq. (13), in the extended GHE approach, the dependence of void ratio (eic) and stress level (pic0 ) is accounted for by both Gmax and (τ/p0 )max, which are two factors with a clear physical meaning and are estimated as follows. For clean sands, a number of empirical relationships have been proposed to relate Gmax to the confining pressure and void ratio (e.g., Hardin & Richart, 1963; Iwasaki, Tatsuoka, & Takagi, 1978). The

Formulation and calibration of the T-sand model

FIG. 7

377

Evaluation of shear modulus model parameters for Toyoura sand.

experimental data reported in Fig. 7 indicate that for Toyoura sand, the following expression is valid (experimental data from De Silva, 2008; Kiyota, De Silva, Sato, & Koseki, 2006):  0 n p Gmax ¼ Ga f ðeic Þ 0 (13) pa f ðeic Þ ¼

ð2:17  eic Þ2 1 + eic

(14)

where Ga is the small-strain shear stiffness at the reference atmospheric pressure (pa  100 kPa) and n is a soil parameter to express the stress-level dependence of Gmax. Note that f(eic) is the void ratio function proposed by Hardin and Richart (1963) for sands with angular grains. Fig. 8 reports a compilation of experimental data points obtained from monotonic and cyclic undrained torsional shear tests on Toyoura sand specimens. It can be observed that there exists a clear dependency between the peak strength (τ/p0 )max and the void ratio (eic) that can be expressed by a linear equation (experimental data from Chiaro et al., 2013b; Cubrinovski & Ishihara, 1998a, 1998b; Umar, Chiaro, & Kiyota, 2017): ðτ=p0 Þmax ¼ ρ1 + ρ2 eic

(15)

where ρ1 and ρ2 are two strength parameters of sand. Modeling the initial static shear effects on stress-strain response The presence of initial static shear stress is introduced in the model by means of a monotonic drained shear loading path before the undrained one. As shown in Fig. 9, the x-y relationships from drained

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

FIG. 8

Variation of the undrained peak shear stress with void ratio.

FIG. 9

Experimentally observed and predicted two-phase skeleton curves.

to undrained (two-phase stress-strain path) display a continuity of strain development during the change of loading from drained to undrained, making it possible to model the entire two-phase backbone curve by employing Eqs. (11), (12) into Eq. (4), with single set of GHE parameters (Fig. 2). Cyclic stress-strain response from small to large shear strain levels It is recognized that the cyclic behavior of soil can be modeled by employing the well-known second Masing’s rule (Masing, 1926). As schematically shown by Fig. 10, if the backbone curve can be described by an

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379

FIG. 10

Modeling of hysteresis curve using Masing’s second rule. Adopted from De Silva, L. I. N. (2008). Deformation characteristics of sand subjected to cyclic drained and undrained torsional shear loadings and their modeling. Ph.D. Thesis. Japan: University of Tokyo.

odd function y ¼ f(x), the hysteretic unloading curve passing the initial point A having coordinates of xa, ya can be obtained by

x  x  y  ya a ¼f (16) 2 2 If the backbone curve is then simulated by the extended GHE model, Eq. (16) yields x  xa (17) y ¼ ya + 1 jx  xa j + ξ 1 ð xÞ 2 ξ 2 ð xÞ However, due to the rearrangement of soil grains, the original Masing’s rule is not sufficient to describe the soil behavior during cyclic loadings (Tatsuoka, Jardine, Lo Presti, Di Benedetto, & Kodata, 1997). The grain rearrangement effect can be taken into account by dragging the skeleton curve in opposite direction to the loading path by an amount δ while applying the Masing’s rule (Masuda, Tatsuoka, Yamada, & Sato, 1999; Tatsuoka et al., 2003). Following this concept, in the T-sand model, the drag function proposed by HongNam (2004) is used: X dx X δ¼ (18) dx 1 + F1 F2

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

where F1 is the maximum amount of drag; F2 is a fitting parameter, which is equivalent to the initial gradient of the drag function; and dx denotes the increment of normalized plastic shear strain. De Silva and Koseki (2012) pointed out that the application of the drag alone is not satisfactory for simulating the cyclic stress-strain relationship close to the peak stress state of the soil. Hence, De Silva et al. (2015) introduced two conceptual factors to take into account for the damage (D) of plastic shear modulus at large stress level and the hardening (H) of the soil during cyclic loading: D¼

Gp 1:45ð1  Dult Þ

X  ¼ p Ginitial 1 + exp jΔγ p j  0:8

(19)

p

where Dult is the minimum value of D; Σ jΔγ p jp is the plastic shear strain accumulated between the current and the previous turning points; H¼1+

Hx F2 Hx + F1 Hult  1

(20)

P where Hx is the j dxj up to current turning point; Hult is the maximum value of H after applying an infinite number of cycles; F1 and F2 are the same parameters used in the drag function. To summarize, after introducing the drag, damage, and hardening effects, in the T-sand model, the skeleton curve during cyclic loading/ unloading is modeled as follows: y ¼ ya +

ð x  δ Þ  xa 1 jðx  δÞ  xa j + 2 H ξ 2 ð xÞ D ξ 1 ð xÞ

(21)

where x and y are defined by Eqs. (12), (13).

Modeling the excess pore water pressure generation Excess pore water pressure generation for saturated undrained shear loading conditions can be computed from volume compatibility (Byrne, 1991): p

dεvol ¼ dεevol + dεvol

(22)

where dεevol is the total volumetric strain increment, dεevol is the elastic volumetric strain increment, and dεpvol is the plastic volumetric strain increment. Such volumetric strain is the result of a change of effective mean stress (dp0 ) during undrained loading that causes re-compression/ swelling of the specimen and a change of shear stress (dτ) that causes

Formulation and calibration of the T-sand model

381

the dilatation of the specimen. For simple shear conditions, the elastic volumetric strain increment can be defined as dεevol ¼

dp0 K

(23)

in which K is the bulk modulus that is a function of the shear modulus (G) and Poisson’s ratio (ν): K¼

2G ð1 + νÞ 3ð1  2νÞ

(24)

Assuming that dεvol  0 in undrained shearing conditions, Eq. (22) yields: p

dεevol ¼ dεvol

(25)

Therefore, in the T-sand model, by combining Eqs. (23), (25), the pore water pressure generation is evaluated as  p  dp0 ¼ K dεvol (26) where the plastic volumetric strain increment (dεpvol) is estimated as described below. Volume change in drained shear tests can be considered as the mirror image of pore water pressure build-up during undrained shear tests. Change of volumetric strain in different stages of shear loading can be described by the stress-dilatancy relationship, which relates the dilatancy ratio ( dεpvol/dγ p) to the shear stress ratio (τ/p0 ) (e.g., Pradhan, Tatsuoka, & Sato, 1989; Shahnazari & Towhata, 2002). Nevertheless, theoretical stress-dilatancy relations, such as Rowe’s equations (Rowe, 1962), are not directly applicable to the case of torsional shear loading. Nevertheless, the results from torsional shear tests show that unique relationships between  dεpvol/dγ p and τ/p0 exist either for loading (dγ p > 0) and unloading (dγ p < 0) conditions (De Silva, Koseki, Wahyudi, & Sato, 2014; Pradhan et al., 1989). Nishimura and Towhata (2004) recommended an empirical bi-linear stress-dilatancy relationship for sands undergoing cyclic torsional shear loading: p

dε τ ¼ Nd vol  ðτ=p0 ÞPTL p0 dγ p

(27)

In the above, Nd is a soil dilatancy parameter and (τ/p0 )PTL is the stress ratio at the phase transformation (i.e., zero dilatancy state; Ishihara, Tatsuoka, & Yasuda, 1975). Rearranging Eq. (27), the following expression for calculating the plastic volumetric strain increment is obtained:  1 τ p 0 dεvol ¼  ðτ=p ÞPTL dγ p (28) N d p0

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

FIG. 11 (A) Illustration of pore pressure/dilatancy characteristics of sand in TSS tests and (B) variation of the dilatancy parameter Nd with void ratio.

In conclusion, by substituting Eq. (27) into Eq. (26), the pore water pressure generation is evaluated as follows:  1 τ 0 p dp0 ¼ K  ð τ=p Þ (29) PTL dγ N d p0  0 n p 2ð 1 + ν Þ (30) K ¼ Ga f ðeic Þ 0 pa 3ð1  2νÞ Fig. 11A shows schematically the variation of Nd and (τ/p0 )PTL for contractive and dilative soil behaviors. As reported in Fig. 11B, Nd is a density-dependent factor, and its values can be estimated by the following liner equation: Nd ¼ d1 + d2 eic

(31)

where d1 and d2 are two parameters to express the dependence of Nd on density. On the other hand, (τ/p0 )PTL has been found to be a constant, and for the case of Toyoura sand, it is approximately 0.6. Over-consolidation effect It is well established that, during cyclic loadings, the effective mean stress (p0 ) decreases with number of cycles due to two possible mechanisms: (i) the soil is subjected to significant effects of over-consolidation until the stress state exceeds for the first time the phase transformation stress state (i.e., the first time where the volumetric behavior changes from contractive to dilative, dp0 > 0) and (ii) soil enters into the stage of cyclic mobility. In particular, the over-consolidation significantly alters the stress–dilatancy behavior of sand during the virgin loading and its effect evanishes with the subsequent cyclic loading. Oka, Yashima, Tateishi, Taguchi, and Yamashita (1999) suggested a distinct stress-dilatancy equation to reproduce the effect of over-consolidation within certain

Formulation and calibration of the T-sand model

383

boundaries, which was adopted in the original T-sand model as described by De Silva et al. (2015). " #" #3=2 p dεvol 1 τ τ=p0 τ=p0    ¼   (32) dγ p Nd p0 ln p0ic =p0 ðτ=p0 ÞPTL ln p0ic =p0 A further modification of Eq. (32), which consists of a rotation of over-consolidation (OC) boundary surface as schematically illustrated in Fig. 12A, was made to account for the combined effects of over-consolidation and initial static shear stress (τstatic) on undrained cyclic torsional shear behavior of sand. For this purpose, the following stress-dilatancy equation is proposed to define an anisotropic overconsolidation boundary surface (AOC): " #" #3=2 p dεvol 1 τ τ=p0  α τ=p0  α    ¼   (33) dγ p Nd p0 ln p0ic =p0 ðτ=p0 ÞPTL ln p0ic =p0 where α is the static shear stress ratio (α ¼ τstatic/pic0 ). Note that, whenever α ¼ 0 (i.e., τstatic ¼ 0), Eq. (33) meets Eq. (32). The proposed AOC has the same features of the one presented by Oka et al. (1999) for isotropically consolidated sands, in the sense that, it defines

FIG. 12 (A) Schematic of the four-zone effective stress path and (B–D) pore pressure/ stress dilatancy relationships used in the T-sand model.

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

the region within which the specimen behaves as less contractive while being affected by over-consolidation. As well, it takes into account the effects of anisotropic consolidation induced by the static shear stress, following the same principle of the rotation of yield surface in the stress space due to anisotropic consolidation (e.g., Taiebat & Dafalias, 2010). Four-zone effective stress path As shown in Fig. 12A, in the T-sand model, the effective stress path during undrained loading is divided into four sections namely: (i) monotonic (virgin) effective stress path—(No. 1 in Fig. 12A); (ii) stress path before PTL, but outside the AOC boundary surface— (Nos. 2 and 20 in Fig. 12A); (iii) stress path within the limits of AOC boundary surface, but before exceeding the PTL for the first time—(Nos. 3 and 30 in Fig. 12A); and (iv) stress path after exceeding the PTL for the first time—(Nos. 4 and 40 in Fig. 12A). The corresponding pore water/stress–dilatancy relationships are shown in Fig. 12B–D.

Performance of the T-sand model Model parameters The T-sand model requires a unique set of 20 parameters for simulating the monotonic/cyclic drained/undrained TSS behavior of saturated sands over a wide range of void ratios, confining pressure and static shear stress ratios. The model parameters for Toyoura sand are summarized in Table 1.

Simulations of undrained cyclic TSS tests on Toyoura sand In this section, first simulation results of undrained cyclic TSS behavior of Toyoura sand are reported. Then, the effects of sloping ground conditions on the liquefaction resistance and cyclic strain accumulation of Toyoura sand are scrutinized in terms of cyclic resistance ratio (CRR) and Kα factor for relative density (Dr,ic) ranging from 30% to 60%. Comparison is finally made with experimental results reported in Figs. 2 and 3 to evaluate the capability of T-sand model. Liquefaction and large cyclic postliquefaction deformation Fig. 13A–D shows a typical simulated behavior of medium dense Toyoura sand under undrained cyclic TSS loading (Dr,ic ¼ 45%; p00 ¼ 100 kPa, CSR ¼ 0.16) for two cases of stress reversal conditions (α ¼ 0

385

Performance of the T-sand model

TABLE 1

T-sand model parameters for Toyoura sand.

Extended GHE ξ1(0)

ξ1(∞)

ξ2(0)

ξ2(∞)

α0

β0

κ

4

0.123

0.102

1.2

0.01073

0.85012

0.2

Small-strain shear stiffness and peak shear strength Ga (kPa)

n

ρ1

ρ2

81,969

0.51

1.610

1.152

Dilatancy and bulk modulus d1

d2

(τ/p0 )PTL

K

ν

11.600

9.324

0.6

Eq. (28)

0.23

Drag, damage and hardening F1

F2

Dult

Hult

0.5

12

0.6

1.15

Four-zone effective stress path Phase

1

2 and 20

3 and 30

4 and 40

Equation

29

29

33

29

and 0.10). In these simulations, excess pore water pressure generation besides a near-zero shear strain development (γ  0) is observed until the full liquefaction state is reached at p0 ¼ 0 kPa. After this point (i.e., postliquefaction state), the rapid development of double amplitude shear strain exceeding 7.5% is clearly observed. Such simulation results are well in agreement with the experimental data reported in Fig. 1 (additional test results are reported in Chiaro, 2020; Chiaro et al., 2012, 2013a).

Rapid liquefaction-induced large shear strain On the other hand, a typical simulated behavior of medium dense Toyoura sand under undrained cyclic TSS loading (Dr,ic ¼ 45%; p00 ¼ 100 kPa, CSR ¼ 0.16) for the cases of stress reversal condition (α ¼ 0.15) are shown in Fig. 13E and F. In this case, full liquefaction (p0 ¼ 0 kPa) and the rapid development of large double amplitude shear strain exceeding 7.5% take place in less than 1 cycle of loading. Such abrupt behavior of Toyoura sand described by model simulations was also observed experimentally (Chiaro, 2020; Chiaro et al., 2012, 2013a).

386

17. Modeling of liquefaction and large-strain response of sand in sloping ground

FIG. 13 Simulated undrained cyclic TSS behavior of Toyoura sand. Simulation carried out by T-sand model.

Performance of the T-sand model

387

Progressive accumulation of residual deformation Results of a stress nonreversal TSS test (Dr,ic ¼ 45%; p00 ¼ 100 kPa, CSR ¼ 0.16, α ¼ 0.20) are shown in Fig. 13G and H. Despite the gradual pore water pressure generation, full liquefaction state is not reached (minimum value of p0  10 kPa). However, the progressive accumulation of residual (plastic) shear strain larger than 7.5% is observed. Such model simulations are clearly in agreement with the experimental results shown in Fig. 2, confirming that the proposed model is able to reproduce sand behavior also under stress nonreversal conditions.

Toyoura sand cyclic resistance Cyclic resistance ratio (CRR) and Kα factor Simulated trends of the cyclic resistance ratio (CRR ¼ CSR at 15 cycles of loading) versus α relationships are reported in Fig. 14A, for loose (Dr, ic ¼ 30%), medium dense (Dr,ic ¼ 45%), and dense (Dr,ic ¼ 60%) Toyoura sand, respectively. In Fig. 14B, the same data are also replotted in terms

FIG. 14 Simulated trends between (A) cyclic stress ratio and (B) Kα factor and static stress ratio (α) for Toyoura sand—numerical simulation performed with the T-sand model (version 2); and (C) experimental trends between Kα factor and α for simple shear tests on Ottawa sand reported by Vaid and Finn (1979) and Boulanger et al. (1991).

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17. Modeling of liquefaction and large-strain response of sand in sloping ground

of Kα factor that may provide a better indication if the presence of τstatic is detrimental (Kα < 1) or beneficial (Kα > 1) to the sand cyclic resistance. In the case of medium dense and dense Toyoura sand, the cyclic resistance (or Kα factor) first decreases (detrimental effect of τstatic) and then increases (beneficial effect of τstatic) upon the stress nonreversal loading line. On the other hand, for loose Toyoura sand, the presence of τstatic appears to be detrimental. These results and trends are not only consistent with the experimental results reported in Fig. 3A and B but also reasonably consistent with experimental data and trends available in the literature, where the simple shear conditions were employed (e.g., Boulanger et al., 1991; Vaid & Finn, 1979), that are reported in Fig. 13C for completeness.

Summary and conclusions A robust state-dependent cyclic model to describe the behavior of saturated sands subjected to undrained cyclic TSS loading with initial static shear was presented in this chapter. The model is based on an extended GHE approach and a state-dependent cyclic bi-linear stress-dilatancy relationship. This can simulate the stress-strain soil behavior over a wide range of densities and confining pressure by using a single set of 20 model soil parameters. By comparing the numerical simulations with the experimental results, it is demonstrated that the proposed model is able to describe pre- and post-liquefaction behavior of Toyoura sand, capturing the salient features of the effective stress paths and stress-strain relationships, under both stress-reversal and stress-nonreversal undrained cyclic TSS loading conditions. The model predictions also indicated that, for various levels of relative density ranging from 30% to 60%, the cyclic resistance first decreases (detrimental effect of τstatic) and then increases (beneficial effect of τstatic) upon the stress nonreversal loading line. These results and trends are reasonably consistent with experimental results reported in the literature where the simple shear conditions were employed.

References Boulanger, R. W., Seed, R. B., Chan, C. K., Seed, H. B., & Sousa, J. B. (1991). Liquefaction behavior of saturated and under uni-directional and bi-directional monotonic and cyclic simple shear loading. University of California Berkley. Byrne, P. M. (1991). A cyclic shear-volume coupling and pore pressure model for sand. In: Proceedings of the international conference on recent advances in geotechnical earthquake engineering and soil dynamics. Paper 1.

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Castro, G., & Poulus, S. J. (1977). Factors affecting liquefaction and cyclic mobility. Journal of the Geotechnical Engineering Division, ASCE, 103(GT6), 501–551. Chiaro, G. (2010). Deformation properties of sand with initial static shear in undrained cyclic torsional shear tests and their modeling: (p. 310). PhD Thesis. Japan: University of Tokyo. Chiaro, G. (2020). Cyclic resistance and large deformation characteristics of sands under sloping ground conditions: insights from large-strain torsional simple shear tests. In: SOAP Lecture, 7th International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, Bangalore, India, State-of-the-Art-and-Practice Lecture paper30. Chiaro, G., Alexander, G., Brabhaharan, P., Massey, C., Koseki, J., Yamada, S., et al. (2017a). Reconnaissance report on geotechnical and geological aspects of the 2016 Kumamoto Earthquake, Japan. Bulletin of the New Zealand Society for Earthquake Engineering, 50(3), 365–393. Chiaro, G., De Silva, L. I. N., & Koseki, J. (2017b). Modelling the effects of initials static shear on the undrained cyclic torsional simple shear behavior of liquefiable sand. Geotechnical Engineering, SEAGS, 48(4), 1–9. Chiaro, G., Kiyota, T., Pokhrel, R. M., Goda, K., Katagiri, T., & Sharma, K. (2015a). Reconnaissance report on geotechnical and structural damage caused by the 2015 Gorkha Earthquake, Nepal. Soils and Foundations, 55(5), 1030–1043. Chiaro, G., Koseki, J., & Sato, T. (2012). Effects of initial static shear on liquefaction and large deformation properties of loose saturated Toyoura sand in undrained cyclic torsional shear. Soils and Foundations, 52(3), 498–510. Chiaro, G., Kiyota, T., & Koseki, J. (2013a). Strain localization characteristics of loose saturated Toyoura sand in undrained cyclic torsional shear tests with initial static shear. Soils and Foundations, 53(1), 23–34. Chiaro, G., Koseki, J., & De Silva, L. I. N. (2013b). A density- and stress-dependent elastoplastic model for sands subjected to monotonic torsional shear loading. Geotechnical Engineering, SEAGS, 44(2), 18–26. Chiaro, G., Koseki, J., & De Silva, L. I. N. (2013c). An elasto-plastic model for liquefiable sands subjected to torsional shear loadings. Springer Series in Geomechanics and Geoengineering, 519–526. Chiaro, G., Koseki, J., De Silva, L. I. N., & Kiyota, T. (2015b). Modeling the monotonic undrained torsional shear response of loose and dense Toyoura sand. JGS Special Publication, 2(9), 407–410. Cubrinovski, M., Bray, J. D., Taylor, M., Giorgini, S., Bradley, B. A., Wotherspoon, L., et al. (2011). Soil liquefaction effects in the Central Business Districts during the February 2011 Christchurch Earthquake. Seismological Research Letters, 82(6), 893–904. Chiaro, G., Umar, M., Kiyota, T., & Koseki, J. (2021). Deformation and cyclic strength characteristics of loose and medium-dense clean sand under sloping ground conditions: Insights form cyclic undrained torsional shear tests with initials static shear. Geotechnical Engineering Journal, 52(1), 1–8. (in press). Cubrinovski, M., Bray, J. D., de la Torre, C., Olsen, M. J., Bradley, B. A., Chiaro, G., et al. (2017). Liquefaction effects and associated damages observed at the Wellington CentrePort from the 2016 Kaikoura Earthquake. Bulletin of the New Zealand Society for Earthquake Engineering, 50(2), 152–173. Cubrinovski, M., & Ishihara, K. (1998a). Modelling of sand behaviour based on state concept. Soils and Foundations, 38(3), 115–127. Cubrinovski, M., & Ishihara, K. (1998b). State concept and modified elasto-plasticity for sand modelling. Soils and Foundations, 38(4), 213–225. De Silva, L. I. N. (2008). Deformation characteristics of sand subjected to cyclic drained and undrained torsional shear loadings and their modeling. Ph.D. Thesis. Japan: University of Tokyo.

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De Silva, L. I. N., & Koseki, J. (2012). Modelling of sand behavior in drained cyclic shear. Advances in Transportation Geotechnics II, 686–691. De Silva, L. I. N., Koseki, J., Chiaro, G., & Sato, T. (2015). A stress-strain description for saturated sand under undrained cyclic torsional shear loading. Soils and Foundations, 55(3), 559–574. De Silva, L. I. N., Koseki, J., Wahyudi, S., & Sato, T. (2014). Stress-dilatancy relationships of sand in the simulation of volumetric behavior during cyclic torsional shear loadings. Soils and Foundations, 54(4), 845–858. Duncan, J. M., & Chang, C. Y. (1970). Nonlinear analysis of stress and strain of soils. Journal of Soil Mechanics and Foundation Division, ASCE, 96(SM5), 1629–1653. Hamada, M., O’Rourke, T.D. and Yoshida, N. (1994): “Liquefaction-induced large ground displacement. In: Performance of Ground and Soil Structures during Earthquakes, Proceedings of the 13th International Conference on Soil Mechanics and Foundation Engineering, 93-108 (1994). Hardin, B. O., & Drnevich, V. P. (1972). Shear modulus and damping in soils: design equations and curves. Journal of Soil Mechanics and Foundation Division, ASCE, 98(SM7), 667–692. Hardin, B. O., & Richart, F. E. (1963). Elastic wave velocities in granular soils. Journal of Soil Mechanics and Foundation Division, ASCE, 89(SM1), 33–65. HongNam, N. (2004). Locally measured quasi-elastic properties of Toyoura sand in cyclic triaxial and torsional loadings. Ph.D. Thesis. Japan: University of Tokyo. HongNam, N., & Koseki, J. (2005). Quasi-elastic deformation properties of Toyoura sand in cyclic triaxial and torsional loadings. Soils and Foundations, 45(5), 19–38. Hyodo, M., Murata, H., Yasufuku, N., & Fujii, T. (1991). Undrained cyclic shear strength and residual shear strain of saturated sand by cyclic triaxial tests. Soils and Foundations, 31(3), 60–76. Hyodo, M., Tanimizu, H., Yasufuku, N., & Murata, H. (1994). Undrained cyclic and monotonic triaxial behavior of saturated loose sand. Soils and Foundations, 34(1), 19–32. Ishihara, K., Tatsuoka, F., & Yasuda, S. (1975). Undrained deformation and liquefaction of sand under cyclic stresses. Soils and Foundations, 15(1), 29–44. Iwasaki, T., Tatsuoka, F., & Takagi, Y. (1978). Shear modulus of sand under cyclic torsional shear loading. Soils and Foundations, 18(1), 39–56. Kiyota, T., De Silva, L. I. N., Sato, T., & Koseki, J. (2006). Small strain deformation characteristics of granular materials in torsional shear and triaxial tests with local deformation measurements. In H. I. Linget al. (Ed.)Soil stress-strain behavior: Measurement, modeling and analysis (pp. 557–566). Springer Netherlands. Konder, R. L. (1963). Hyperbolic stress-strain response: cohesive soils. Journal of Soil Mechanics and Foundation Division, ASCE, 89(SM1), 115–143. Lee, K. L., & Seed, H. B. (1967). Dynamic strength of anisotropically consolidated sand. Journal of Soil Mechanics and Foundation Division, ASCE, 93(SM5), 169–190. Masing, G. (1926). Eigenspannungen und verfestigung beim messing. In: Proceedings of the 2nd International Conference on Applied Mechanics, pp. 332–335. Masuda, T., Tatsuoka, F., Yamada, S., & Sato, T. (1999). Stress-strain behavior of sand in plane strain compression, extension and cyclic loading tests. Soils and Foundations, 39(5), 31–45. Nishimura, S., & Towhata, I. (2004). A three-dimensional stress-strain model of sand undergoing cyclic rotation of principal stress axes. Soils and Foundations, 44(2), 103–116. Oka, F., Yashima, A., Tateishi, Y., Taguchi, Y., & Yamashita, S. (1999). A cyclic elasto-plastic constitutive model for sand considering a plastic-strain dependence of the shear modulus. Geotechnique, 49(5), 661–680. Pradhan, T. B. S., Tatsuoka, F., & Sato, Y. (1989). Experimental stress-dilatancy relations of sand subjected to cyclic loadings. Soils and Foundations, 29(1), 45–64.

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Rowe, P. W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proceedings of the Royal Society of London, Series A, 269, 500–527. Shahnazari, H., & Towhata, I. (2002). Torsion shear tests on cyclic stress-dilatancy relationship of sand. Soils and Foundations, 42(1), 105–119. Sivathayalan, S., & Ha, D. (2011). Effect of static shear stress on the cyclic resistance of sands in simple shear loading. Canadian Geotechnical Journal, 48(10), 1471–1484. Taiebat, M., & Dafalias, Y. F. (2010). Simple yield surface expression appropriate for soil plasticity. International Journal of Geomechanics, ASCE, 10(4), 161–169. Tatsuoka, F., Jardine, R. J., Lo Presti, D., Di Benedetto, H., & Kodata, T. (1997). Characterizing the pre-failure deformation properties of geomaterials. In: Proceeding of the 14th International Conference on Soil Mechanics and Foundations Engineering, Hamburg, Germany4, (pp. 2129–2164). Tatsuoka, F., Masuda, T., Siddquee, M. S. A., & Koseki, J. (2003). Modeling the stress-strain relations of sand in cyclic plane strain loading. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 129(6), 450–467. Tatsuoka, F., Muramatsu, M., & Sasaki, T. (1982). Cyclic undrained stress-strain behavior of dense sands by torsional simple shear stress. Soils and Foundations, 22(2), 55–70. Tatsuoka, F., & Shibuya, S. (1992). Deformation characteristics of soils and rocks from field and laboratory tests. In: Proceedings of the 9th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Bangkok, Thailand, Vol. 2, pp. 101–170. Tatsuoka, F., Siddquee, M. S. A., Park, C. S., Sakamoto, M., & Abe, F. (1993). Modelling stressstrain relations of sand. Soils and Foundations, 33(2), 60–81. Vaid, Y. P., & Chern, J. C. (1983). Effects of static shear on resistance to liquefaction. Soils and Foundations, 23, 47–60. Vaid, Y. P., & Finn, W. D. L. (1979). Static shear and liquefaction potential. Journal of the Geotechnical Engineering Division, ASCE, 105, 1233–1246. Vaid, Y. P., Stedman, J. D., & Sivathayalan, S. (2001). Confining stress and static shear effects in cyclic liquefaction. Canadian Geotechnical Journal, 38(3), 580–591. Yang, J., & Sze, H. Y. (2011). Cyclic behavior and resistance of saturated sand under nonsymmetrical loading conditions. Geotechnique, 61(1), 59–73. Yoshimi, Y., & Oh-oka, H. (1975). Influence of degree of shear stress reversal on the liquefaction potential of saturated sand. Soils and Foundations, 15(3), 27–40. Youd, T. L., Idriss, I. M., Andrus, R. D., et al. (2001). Liquefaction resistance of soils: Summary report from the 1996 NCREE and 1998 NCREE/NSF workshops on evaluation of liquefaction resistance of soils. Journal of Geotechnical Geoenvironmetal Engineering, ASCE, 127 (10), 817–833. Ziotopoulou, K., & Boulanger, R. W. (2016). Plasticity modeling of liquefaction effects under sloping ground and irregular cyclic loading conditions. Soil Dynamics and Earthquake Engineering, 84, 269–283. Ziotopoulou, K. (2014). A sand plasticity model for earthquake engineering applications. Davis: University of California. Umar, M., Chiaro, G., & Kiyota, T. (2017). Influence of density on large deformation characteristic of sand in undrained cyclic torsional shear tests with initial static shear. In: Proceedings of the 3rd international conference on performance-based design in earthquake geotechnical engineering, Vancouver, Canada (p. 8).

C H A P T E R

18 Pile behavior modeling in unsaturated expansive soils Liu Yunlonga and Sai Vanapallib a

Department of Civil Engineering, Zhengzhou University, Zhengzhou, China Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada

b

Mechanical behavior of piles in expansive soils upon infiltration Vast deposits of expansive soils are widely distributed in several countries of six of the seven continents of the world. Some of these countries include Canada and the United States from North America, Argentina from South America; Sudan and Algeria from Africa, China, India, and Israel from Asia; Spain and the United Kingdom from Europe and Australia from Australia (Al-Rawas & Qamaruddin, 1998; Chen, 1988; Rao, Reddy, & Muttharam, 2001). Expansive soils are typically referred to as problematic soils in the literature because their mechanical behavior is highly sensitive to the changes in their natural water content associated with environmental factors such as the infiltration and evaporation. Ground heave or settlement contribute to severe distress to various infrastructure constructed in expansive soil due to the changes in their natural water content and result in significant economic losses to building industry (Gourley, Newill, & Schreiner, 1993; Jaremski, 2012). Studies by Adem and Vanapalli (2016) suggest that the economic losses associated with expansive soils have been significantly increasing during the past five decades all over the world, the losses in the United States alone is estimated to be several billions of dollars. Among various choices that are available as foundations for infrastructure placed in expansive soils, pile foundations are typically preferred (Al-Rawas & Goosen, 2006). Typically, piles can be used in expansive soils as micropiles in the active zone or as group pile foundations. Micropiles reduce ground heave in the top layer of expansive soil in addition to providing support as a foundation to the infrastructure constructed in

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00003-4

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© 2021 Elsevier Inc. All rights reserved.

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18. Pile behavior modeling in unsaturated expansive soils

expansive soils (Nelson, Chao, Overton, & Nelson, 2015). Typically, small diameter steel piles (75–250 mm in diameter) are inserted in predrilled holes of larger diameter, which are then filled with compacted sand to improve the frictional resistance of micropiles (Nusier & Alawneh, 2004). Upon infiltration, heave is significantly reduced by the friction mobilized at the pile-soil interface. Micropile reinforcement technique is a rational choice to mitigate damages of lightly loaded structures on thin layer expansive soils with limited swelling potential. However, for heavy structures on thick expansive soil with high to very high swell potential, pile or group pile foundation is typically favored. Piles with high strength and stiffness extend through the active zone (depth of expansive soil layer in which moisture content changes are sensitive to environmental factors associated with infiltration and evaporation) in expansive soil and are placed on rigid bedrock or lower stable soil stratum. Such a pile foundation system not only has a significant bearing capacity but also can effectively control the nonuniform settlement, even when the mechanical behavior of shallow expansive soil layer experience significant changes under extreme conditions (heave and settlement). Two kinds of pile foundations are commonly used in engineering practice; namely, single pile (drilled pile, pushing pile) (O’Neill, 1988; Poulos & Davis, 1980) or group pile foundation (helical pile, precast pile) (Ekshtein, 1978). Pile foundation with diameters greater than 800 mm is typically cast in situ. In some scenarios, to increase the bearing capacity of the pile foundation, a belled pile that enlarges at the end is used. For enhancing the integrity of group pile foundation, grade beams that link the pile top are set to form a pile grade beam foundation system are used. Such a pile system is more reliable to prevent the nonuniform settlement and tilt of the super structure. The design of the pile foundation is conventionally based on saturated soil mechanics assuming drained condition (effective stress). However, in most cases, the soil surrounding the pile is in an unsaturated state. The in situ matric suction of expansive soils significantly influences the mechanical behavior of the piles. The load transfer mechanism of the pile foundation is sensitive to matric suction changes associated with environmental factors (i.e., infiltration and evaporation of water). As shown in Fig. 1, upon evaporation, the matric suction in the active zone increases due in comparison with the hydrostatic matric suction profile. On the contrary, matric suction in the active zone decreases upon infiltration. For a single pile installed in expansive soil, the changes in load transfer mechanism before and after infiltration are illustrated in Fig. 2. Prior to infiltration, positive friction is distributed along with the entire length of the pile and bears the upper load along with the end bearing capacity

Mechanical behavior of piles in expansive soils upon infiltration

395

FIG. 1 Variation of matric suction profile in a typical unsaturated expansive soil under the influence of environmental factors.

FIG. 2

Mechanical behavior changes of a pile in expansive soil before and after infiltration. (A) Prior to infiltration (B) after infiltration.

396

18. Pile behavior modeling in unsaturated expansive soils

(as shown in Fig. 2A). As water infiltrates into the active zone (as shown in Fig. 2B), changes mainly occur in three aspects: in the vertical direction, volume expansion of expansive soil causes ground heave. In the horizontal direction, restricted volume expansion produces lateral swelling pressure. The pile-soil interface strength properties change due to variations in the water content (matric suction) of the surrounding soil. Due to these changes, in the active zone (the depth influenced by water infiltration), uplift friction generates along with the pile as a result of displacement between the pile and adjacent soil (i.e., soil swells and moves upward relative to the pile). The magnitude of the uplift friction is determined by the increasing lateral earth pressure considering the contribution of lateral swelling pressure and the water content (matric suction)-dependent interface strength properties. A pile under a light loaded structure may get uplifted due to the uplift friction contribution. Once the pile has an upward movement, negative friction generates in the stable zone, and the pile base bearing capacity decreases significantly. The net contribution that arises from negative shaft friction, end bearing capacity, and applied load combine to balance the increased uplift shaft friction. Piles as foundation typically penetrate through the active zone and rest on bedrock or extend into soil layers with higher stiffness. In other words, the mechanical behavior of the soil under the pile end is no longer influenced by seasonal water content changes. The variations in the load transfer mechanism of pile upon infiltration are mainly associated with the variations of shaft friction in the active zone. The shaft friction in the active zone is determined by four factors including the net normal stress (lateral earth pressure), matric suction, interface shear strength properties, and the pile-soil relative displacement. In the infiltration process, mobilization of lateral swelling pressure can add an increment to the lateral earth pressure due to soil unit weight and surcharge. The pile-soil interface shear strength properties decrease with a decrease in matric suction. Ground heave also changes the pile-soil relative displacement. Considering these changes associated with the water infiltration process, the traditional load transfer curve method for the analysis of the load transfer mechanism of the pile is modified to extend their application in expansive soils. The proposed methods are verified using case studies from the published literature. The results of these studies suggest that there are reasonable comparisons between the measured and predicted results. Proposed methods are simple yet powerful tools for the estimation of mechanical behaviors of single pile in expansive soil upon water infiltration, which facilitate geotechnical engineers to provide a rational design of pile foundations in various regions of the world with expansive soils.

Mobilization of the lateral swelling pressure upon infiltration

397

Mobilization of the lateral swelling pressure upon infiltration Estimation of lateral swelling pressure with respect to matric suction reduction In this section, a theoretical method is proposed for the estimation of the lateral earth pressure considering lateral swelling pressure upon water infiltration. Passive earth pressures which are considered as the boundaries for the lateral earth pressure variations under different degrees of saturation is analyzed as well. More importantly, the influence of soil-pile interface roughness and changes in matric suction is considered in the analyses of results extending the mechanics of unsaturated soil mechanics. A superposition approach is proposed to estimate the lateral earth pressure considering lateral swelling pressure against a fixed rigid retaining structure that arises due to a matric suction reduction while the soil is still in an unsaturated state. As shown in Fig. 3, an analytical soil element behind a fixed rigid retaining structure experiences matric suction reduction from (ua
uw)a to (ua
uw)b where (ua
uw)b is an intermediate matric suction value (i.e., not equal to zero) during the infiltration process. To simplify the analysis, an analytical element around the pile is considered as a cubic soil element as well. As a consequence, lateral swelling pressure mobilizes with a matric suction reduction and adds an additional increment to the lateral earth pressure associated with soil self-weight and surcharge. To apply the superposition method, the soil element behind the retaining wall is assumed to experience a series of stress state changes

FIG. 3 Mobilization of lateral swelling pressure behind retaining structure associated with matric suction reduction ((A) analytical soil element; (B) matric suction reduction; and (C) lateral earth pressure distribution changes).

398

18. Pile behavior modeling in unsaturated expansive soils

FIG. 4 Stress states variations of the analytical soil element following different matric suction reduction paths.

following two different paths. In Path (I), soil element experiences a matric suction reduction directly from (ua
uw)a to zero. However, in Path (II), the soil element initially undergoes a matric suction reduction from (ua
uw)a to (ua
uw)b. The matric suction (ua
uw)b subsequently reduces to zero. The stress state changes in the soil element in Path (I) and Path (II) (as shown in Fig. 4) are illustrated below. (i) Following Path (I), from an initial state to State (1), there is a lateral pressure increment σ L(a
0) with matric suction reduction from (ua
uw)a to zero in the soil element. In the vertical direction, the vertical side length of the soil element increases from initial value c to b1. From State (1) to State (2), it is assumed that a vertical stress Ps(a
0) compresses the expanding soil element back to its initial volume. (ii) Following Path (II), from an initial state to State (3), the soil element experiences a matric suction reduction [(ua
uw)a
(ua
uw)b]. As a consequence, the soil element shown in State (3) gains a stress increment σ L(a
b) in the horizontal direction. In addition, the side length increases from c to b2 in the vertical direction. From State (3) to

Mobilization of the lateral swelling pressure upon infiltration

399

State (4), a vertical stress Ps(a
b) compresses the analytical element shown in State (3) back to its initial volume. From State (3) to State (5), after undergoing a matric suction reduction which is equal to [(ua
uw)a
(ua
uw)b], the soil element further experiences a matric suction reduction from (ua
uw)b to zero, which means the soil element is fully saturated. For the soil element shown in State (5), in the horizontal direction, compared to the element shown in Stage (3), there is a stress increment σ L(b
0) that arises due to matric suction reduction, while the vertical side length increases from b2 to b3. From State (5) to Stage (6), vertical stress Ps(b
0) compresses the volume of the soil element shown in State (6) back to Stage (5). The soil element further gains a stress increment σ rc, in the horizontal direction. For simplification of analysis, the soil element behind the retaining structure is considered to be isotropic, homogeneous, and elastic in nature without any plastic deformation (such as the collapse of soil structures due to overload) in the swelling process (Terzaghi, 1925, 1926, 1931). Also, for extending a conservative approach, horizontal displacement of the soil element is assumed to be strictly restricted. Constitutive relations (Eq. 1) proposed by Fredlund and Morgenstern (1976) can also be used satisfying the above assumptions for interpreting the stress state variations of the soil element shown in Fig. 4. 8  ðua
uw Þ ðσ x
ua Þ υ
> >
σ y + σ z
2ua + > εx ¼ > E E H > > > >
 > ð σ
u Þ υ ð u
uw Þ x a a > > εx ¼
σ y + σ z
2ua + > > E E H > > >
 > > > < ε ¼ σ y
ua
υ ðσ + σ
2u Þ + ðua
uw Þ y x z a E H E (1) > > τxy > > γ ¼ > > > xy G > > > τyz > > > γ yz ¼ > > G > > > > : γ ¼ τzx zx G where (σ x
ua) is net normal stress in x direction; (σ y
ua) is net normal stress in y direction; (σ z
ua) is net normal stress in z direction; (ua
uw) is matric suction; E is elastic modulus with respect to net normal stress; τxy is shear stress on the x-plane in the y-direction (i.e., τxy ¼ τyx); τyz is shear stress on the y-plane in the z-direction (i.e., τyz ¼ τzy); τzx is shear stress on the z-plane in the x-direction (i.e., τzx ¼ τxz); H is elastic modulus with respect to matric suction; G is shear modulus; υ is Poisson’s ratio. Mathematical expressions corresponding to the stress states shown in Fig. 4 are summarized as Eq. (2) for Path (I), initial state to State (1); Eq. (3)

400

18. Pile behavior modeling in unsaturated expansive soils

for Path (I), from State (1) to State (2); Eq. (4) for Path (II), from the initial state to Stage (3); Eq. (5) for Path (II), from State (3) to State (4); Eq. (6) for Path (II), from State (3) to State (5); Eq. (7) for Path (II), from State (5) to State (6), respectively. Rearranging the above equations, the lateral earth pressures corresponding to different matric suction reductions are given as Eq. (8). 8 b1
c 2υ ðua
uw Þa > > > < c ¼ Eða
0Þ σ Lða
0Þ
Hða
0Þ (2) > υ
1 ðua
uw Þa > > σ Lða
0Þ
:0 ¼ Eða
0Þ Hða
0Þ 8 Psða
0Þ c
b1 2υ > > > σ ra
< b ¼E Eða
0Þ 1 ða
0Þ (3)   > σ υ ra > > 0 ¼
+ P + σ ra sða
0Þ : Eða
0Þ Eða
0Þ where E(a
0) is the average elastic modulus over the matric suction range from (ua
uw)a to zero; H(a
0) is the average elastic modulus over the matric suction range from (ua
uw)a to zero; Ps(a
0) is the constant volume vertical swelling pressure generated from a matric suction reduction from (ua
uw)a to zero. 8   ðua
uw Þa
ðua
uw Þb > b2
c 2υ > > > < c ¼ Eða
bÞ σ Lða
bÞ
Hða
bÞ (4)   > ðua
uw Þa
ðua
uw Þb > υ
1 > >0 ¼ σ Lða
bÞ
: Eða
bÞ Hða
bÞ 8 Psða
bÞ c
b2 2υ > > > ¼ σ rb
< Eða
bÞ b2 Eða
bÞ (5)  > σ rb υ  > > + Psða
bÞ + σ rb :0 ¼
E Eða
bÞ ða
bÞ where E(a
b) is the average elastic modulus with respect to net normal stress over the matric suction range from (ua
uw)a to (ua
uw)b; H(a
b) is the average elastic modulus with respect to matric suction over the matric suction range from (ua
uw)a to (ua
uw)b; Ps(a
b) is the constant volume vertical swelling pressure generated from a matric suction reduction from (ua
uw)a to (ua
uw)b. 8 b3
b2 2υ ðua
uw Þb > > > < b2 ¼ Eðb
0Þ σ Lðb
0Þ
Hðb
0Þ (6) > υ
1 ðua
uw Þb > > σ Lðb
0Þ
:0 ¼ Eðb
0Þ Hðb
0Þ

Mobilization of the lateral swelling pressure upon infiltration

8 Psðb
0Þ b2
b3 2υ > > > ¼ σ rc
< b Eðb
0Þ Eðb
0Þ 2   > σ rc υ > > + Psðb
0Þ + σ rc :0 ¼
E Eðb
0Þ ðb
0Þ

401

(7)

where E(b
0) is the average elastic modulus with respect to net normal stress over the matric suction range from (ua
uw)b to zero; H(b
0) is the average elastic modulus with respect to matric suction over the matric suction range from (ua
uw)b to zero; Ps(b
0) is the constant volume swell pressure generated from a matric suction reduction from (ua
uw)b to zero.
 8 1
υ
2υ2 Psða
0Þ > > σ Lða
0Þ ¼ > >
 Psða
0Þ > > > 1
υ2
ð1 + υÞ 1
υ
2υ2 > > Eða
0Þ > > >
 > > > 1
υ
2υ2 Psða
bÞ < σ Lða
bÞ ¼
 Psða
bÞ (8) > 1
υ2
ð1 + υÞ 1
υ
2υ2 > > Eða
bÞ > > >
 > > 1
υ
2υ2 Psðb
0Þ > > > > > σ Lðb
0Þ ¼
 Psðb
0Þ > > : 1
υ2
ð1 + υÞ 1
υ
2υ2 Eðb
0Þ Since the soil elements following Path (I) and Path (II) experience the same matric suction reduction from (ua
uw)a to zero, under the same boundary conditions (fixed boundaries in the horizontal direction and free boundary in the vertical direction), the lateral swelling pressure in State (1) and State (5) generated due to the matric suction reduction should be the same as well (Eq. 9). The lateral swelling pressure induced by the matric suction reduction [(ua
uw)a
(ua
uw)b] can be expressed as Eq. (10). A general equation can be summarized as Eq. (11) considering the influence of lateral earth pressure due to soil self-weight and surcharge. σ Lða
0Þ ¼ σ Lða
bÞ + σ Lðb
0Þ
 1
υ
2υ2 Psða
0Þ σ Lða
bÞ ¼
 Psða
0Þ 1
υ2
ð1 + υÞ 1
υ
2υ2 E
ða
0Þ  1
υ
2υ2 Psðb
0Þ

 Psðb
0Þ 1
υ2
ð1 + υÞ 1
υ
2υ2 Eðb
0Þ
 1
υ
2υ2 Psða
0Þ σ Lða
bÞ ¼
 Psða
0Þ 1
υ2
ð1 + υÞ 1
υ
2υ2 E
ða
0Þ  1
υ
2υ2 Psðb
0Þ υ σS
+
 P 1
υ sðb
0Þ 1
υ2
ð1 + υÞ 1
υ
2υ2 Eðb
0Þ

(9)

(10)

(11)

402

18. Pile behavior modeling in unsaturated expansive soils

The Ps(a
0) and Ps(b
0), values in Eq. (11) represent constant volume vertical swelling pressure generated from initial condition to full saturation, which can be acquired from a simple laboratory test according to (ASTM, 2014). If there is no experimental data, a semiempirical prediction model (Eq. 12) proposed by Tu and Vanapalli (2016) suitable for compacted expansive soils can be used. This equation can also be extended for expansive soils behind retaining structure as they are disturbed during construction and then compacted to function as backfill material. Also, the model developed by Vanapalli and Oh (2010) (i.e., Eq. 13) is used for estimation of the modulus of elasticity of unsaturated expansive soils. In the calculation of Ea
0 and Ea
0, “E” is the average value of various Eunsat values calculated using Eq. (13) over the range of matric suction variation (ua
uw)a to zero and (ua
uw)b to zero.   Sr 2 (12) PS ¼ PS0 + βS  ψ  100 where Sr is the degree of saturation; PS0 is the intercept on the PS axis at zero suction value (PS0 is 55 kPa for compacted expansive soils); βS is a fitting parameter; βS is 23.05A32.315 (0.237IP
10.278ρdn) + 0.164; A is the activity of soils; A ¼ IP/C, IP is the index of plasticity; C is the clay content of soils; ψ is the soil suction.  ðua
uw Þ βad S Eunsat ¼ Esat 1 + αad (13) Pa =100 where Eunsat is the elastic modulus of unsaturated expansive soil; Esat is the elastic modulus of saturated expansive soil; αad and βad are fitting parameters, Adem (2015) calculated Eunsat for five different expansive soils and suggested that βad is two typically and αad varies from 0.05 to 0.15 for expansive soils. In this study, an average value, αad equals to 0.1 is used; Pa ¼ atmospheric pressure, S ¼ degree of saturation. Employing Eqs. (11)–(13), the lateral earth pressure considering lateral swelling pressure behind a fixed rigid retaining structure from an initial unsaturated state to a subsequent unsaturated state can be conveniently predicted based on matric suction profile or water content profile variations using basic soil properties including SWCC; the saturated elastic modulus, Esat; plasticity index, Ip, maximum dry density, ρd,max and the Poisson ratio, υ.

Estimation of passive earth pressure under unsaturated condition The development of lateral swelling pressure has a limiting value. As shown in the Mohr-circle below (see Fig. 5), lateral swelling pressure can be considered as an additional part to the at-rest earth pressure. The

Mobilization of the lateral swelling pressure upon infiltration

FIG. 5

403

Variation lateral earth pressure in expansive soils upon wetting and drying.

diameter of the Mohr’s circle increases upon wetting and decreases upon drying. At a certain limiting condition, the Mohr circle touches the shear strength failure envelop, which can be interpreted extending Rankine’s theory. In other words, the total lateral earth pressure acting on retaining works cannot exceed passive earth pressure or will be less than active earth pressure to avoid shear failure. However, traditional Rankine’s theory is only suitable for saturated soils against the frictionless surface of a structure. In engineering practice, there can be scenarios where the roughness of the structure surface cannot be neglected (e.g., drilled pier). In many scenarios, even after water infiltration, expansive soils may still not attain a fully saturated condition. In such situations, both the friction of the soil-structure interface and suction present within the expansive soils can significantly influence the lateral earth pressure that develops. In this section, the influence of matric suction and the roughness of soilstructure interface on the passive and active earth pressure are discussed. In the design of conventional retaining structures, Coulomb’s theory or extended Coulomb theory is used by taking account of the roughness and slope of the retaining backfill (Caquot & Kerisel, 1948; Janbu, 1957; Shields & Tolunay, 1973; Terzaghi, 1943). However, Coulomb’s theory facilitates calculating a resultant force instead of providing stress distribution curve as per Rankine’s theory. As a consequence, it cannot satisfactorily address some special problems (e.g., the calculation of pile shaft friction) in which the variation of lateral earth pressure with respect to depth is necessary. Wang, Liu, and Li (2008) extended Rankine’s earth pressure by taking account of the frictional influence between the backsurface of vertical retaining works and soils into consideration. Assuming

404

18. Pile behavior modeling in unsaturated expansive soils

the shear strength of the soil and the soil-structure interface following Coulomb’s law using Eqs. (14) and (15), respectively, Rankine’s theory was extended for the calculation of passive earth pressure (Eq. 16) and active earth pressure (Eq. 17) against rough back-surface of retaining works. τf ¼ σ nf tan ϕ0 + c0

(14)

τa ¼ σ nf tan δ0 + c0a

(15)

0

where c’ is the true cohesion of soil; ca is the effective interface cohesion; σ nf is the normal stress at failure; ϕ’ is the effective internal friction angle of soil; δ’ is the effective interface friction angle. σ hp1 ¼ σ s

1 + sin ϕ0 cos 2αp 2 cos ϕ0 cos 2αp + c0 + p0 0 1
sin ϕ cos2αp 1
sin ϕ0 cos 2αp

(16)

1 C 1 B αp ¼ arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
arctan 2 2 2 2 A A +B σ ha1 ¼ σ s

1
sin ϕ0 cos2αa 0 2 cos ϕ0 cos2αa
c + p0 1 + sin ϕ0 cos 2αa 1 + sin ϕ0 cos 2αa

(17)

1 C 1 B αa ¼ arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + arctan 2 2 2 2 A A +B

8 < A ¼ σ s sin ϕ0 + c0 cos ϕ0 B ¼ c0a sin ϕ0
σ s sin ϕ0 tan δ0
2c0 cosϕ0 tan δ0 : C ¼ σ s tan δ0 + c0a where σ s is the vertical stress due to unit weight of upper soil layers and surcharge; p0 is the pore water pressure. The concept of using two independent stress variables (i.e., net normal stress and matric suction) in the interpretation of the mechanical behavior of unsaturated soils has been widely accepted (Fredlund & Rahardjo, 1993). Eq. (18) proposed by Fredlund, Morgenstern, and Widger (1978) expressed in terms of net normal stress and matric suction is commonly used to model the peak shear strength of unsaturated soils (e.g., Escario & Saez, 1986; Gan, Fredlund, & Rahardjo, 1988; Oloo & Fredlund, 1996; Vanapalli, Fredlund, Pufahl, & Clifton, 1996).

 τf ¼ c0 + σ nf
uaf tan ϕ0 + uaf
uwf tan ϕb (18) where uaf is the pore-air pressure at failure; ϕb is the angle of friction with respect to matric suction; (σ nf – uaf) is the net normal stress at failure; and (uaf – uwf) is the matric suction at failure. However, Eq. (18) does not take into account the nonlinear increase in shear strength as the soil desaturates as a result of an increase in the matric suction. In other words, upon saturation, the friction angle ϕb may have a

Mobilization of the lateral swelling pressure upon infiltration

405

value approximately equal to ϕ’. But once the air-entry value is exceeded, ϕb tends to decrease with increasing matric suction. The nonlinear behavior of the shear strength of unsaturated soils is strongly related to the wetted contact area among air, water, and soil particles. Vanapalli et al. (1996) proposed a semiempirical equation (Eq. 19) for predicting the nonlinear increase of the shear strength of unsaturated soils with respect to matric suction by deriving the changing trend of the wetted area from the SWCC.  

 0 0 0 θ
θr τf ¼ c + σ nf
uaf tan ϕ + uaf
uwf tan ϕ (19) θ s
θr where θ is the current volumetric water content; θr is the residual volumetric water content form an SWCC; and θs is the saturated volumetric water content from an SWCC. Hamid and Miller (2009) suggested that the shear strength of the soilstructure interface, which has different roughness at different degrees of saturation, can be modeled in a similar way as Eqs. (18) and (19). Corresponding equations are given as Eqs. (20) and (21).

 τf ¼ c0a + σ nf
uaf tan δ0 + uaf
uwf tan δb (20)  

 θ
θr τf ¼ c0a + σ nf
uaf tan δ0 + uaf
uwf tan δb (21) θs
θr where δ’ is the interface friction angle with respect to net normal stress; δb is the interface friction angle with respect to matric suction. For simplicity, Eqs. (18) and (20) are used for interpreting the soil and soil-structure interface shear failure envelopes for unsaturated conditions, respectively. The modified Rankine’s theory proposed by Wang et al. (2008) is extended to include the influence of the matric suction on the soil shear strength and soil-structure interface shear strength. More discussions are available in Liu and Vanapalli (2017). From Fig. 6, it can be derived that during the desaturation process, both the passive earth pressure against the rough retaining surface and frictionless retaining surface increases. Passive earth pressure against frictionless retaining surface always has a value higher than the rough retaining surface. The passive earth pressure for saturated soil against a rough surface (σ hp1), saturated soils against a frictionless surface (σ hp2), unsaturated soil against a rough surface (σ hp3) and unsaturated soils against the frictionless surface (σ hp4) are given in Eqs. (16) and (22)–(24), respectively. σ hp2 ¼ σ hp3 ¼ σ s

σ s ð1 + sin ϕ0 Þ 2c0 cos ϕ0 + 1
sin ϕ0 1
sin ϕ0

  2cos ϕ0 cos 2αp 1 + sin ϕ0 cos2αp  0
+ c + uaf
uwf tan ϕb 0 1
sinϕ cos 2αp 1
sinϕ0 cos 2αp

(22) (23)

406

18. Pile behavior modeling in unsaturated expansive soils

FIG. 6 Development of Rankine’s passive earth pressure in unsaturated soils against frictionless and rough surface.

1 C 1 B αp ¼ arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
arctan 2 2 2 2 A A +B 8  0
  0 > A ¼ σ s sin ϕ + c + uaf
uwf tan ϕb cos ϕ0 > > 
  
  > < B ¼ c0a + uaf
uwf tan δb sinϕ0
σ s sin ϕ0 tan δ0
2 c0 + uaf
uwf tan ϕb > cos ϕ0 tan δ0 > > > 
  : C ¼ σ s tan δ0 + c0a + uaf
uwf tan δb 
  σ s ð1 + sin ϕ0 Þ 2 c0 + uaf
uwf tan ϕb cos ϕ0 σ hp4 ¼ + 1
sin ϕ0 1
sin ϕ0

(24)

Modified load transfer curve method Theoretical analysis and derivation The load transfer curve method for the analysis of single pile settlement was originally proposed by Coyle and Reese (1966). Using the curve relating the interface shear strength (shaft friction) to the pile displacement in

Modified load transfer curve method

407

different soil layers (pile-soil relative displacement), the pile head load and settlement can be calculated according to pile base resistance and settlement. Through decades of application studies, it has gained wide acceptance in practice applications (Poulos & Davis, 1980). The key factor in this method is the curve relating the interface shear strength to the pile-soil relative displacement, which is referred to as the transfer curve model in this chapter. Such transfer curve models were developed by Seed and Reese (1957), Gambin (1963) and Cambefort (1964). Several other investigators have also undertaken research studies in this area for the past half a century (Bohn, Lopes dos Santos, & Frank, 2016; Chen, Zhou, Cao, & Chen, 2007; Coyle & Reese, 1966; Liu, Xiao, Tang, & Li, 2004; Nanda & Patra, 2013; Poulos & Davis, 1980; Zhu & Chang, 2002). For example, Coyle and Reese (1966) and Coyle and Sulaiman (1967) presented the transfer curve models suitable for different situations based on either laboratory tests and/or field measurements. Several research scholars have kept improving transfer curves by taking account of the influence of several factors such as the modulus degradation, negative fiction and have also extended it for layered soils (Chen et al., 2007; Liu et al., 2004; Zhu & Chang, 2002). A detailed summary of various transfer curve models is available in Bohn et al. (2016). The pile-soil interface shear strength degradation (softening phenomenon) has been investigated from field tests by some investigators during the last few years (Zhang, Zhang, & Yu, 2011; Zhang, Zhang, Yu, & Liu, 2010; Zhang, Zhang, Zhang, & Shi, 2011; Zhao, Lu, Sun, Zhu, & Li, 2009). Extending this point of view, Zhang and Zhang (2012) proposed a simplified approach for the nonlinear analysis of the load-displacement response of a single pile considering both shaft friction degradation and base resistance hardening. This approach is based on two models; namely, the softening nonlinear transfer curve model relating the pile-soil interface shear strength to the pile displacement relative to soil in different soil layers and the bilinear model relating the pile base resistance to the pile base settlement. The shape of the softening nonlinear transfer curve model proposed by Zhang and Zhang (2012) is shown in Fig. 7A. The pile unit shaft friction shows a nonlinear increase with increasing pile head load. When the pile-soil relative displacement reaches Ssu, the unit shaft friction achieves the peak value τsu. The unit shaft friction then starts decreasing with a further increase in the pile-soil relative displacement. The mathematical expression for the curve shown in Fig. 7A is given as Eq. (25). τ s ðzÞ ¼

Ss ðzÞ½a + cSs ðzÞ ½a + bSs ðzÞ2

(25)

408

18. Pile behavior modeling in unsaturated expansive soils

FIG. 7 (A) Relationship between skin friction and relative shaft displacement at the pilesoil interface; (B) relationship between base resistance and pile-base settlement. Modified from Zhang, Q. Q., & Zhang, Z. M. (2012). A simplified nonlinear approach for single pile settlement analysis. Canadian Geotechnical Journal, 49, 1256–1266.

pffiffiffiffiffiffiffiffiffiffiffiffi 8 βs
1 + 1
βs Ssu > > > a ¼ > > 2βs τsu > > > pffiffiffiffiffiffiffiffiffiffiffiffi > < 1
1
βs 1 b¼ > τsu 2βs > > > pffiffiffiffiffiffiffiffiffiffiffiffi > > > 2
βs
2 1
βs 1 > > :c ¼ τsu 4βs βs ¼

τsr τsu

(26)

where τs(z) is the interface shear stress (shaft friction) at a given depth, z; Ss(z) is the pile–soil relative displacement at a given depth, z; Ssu is pile– soil relative displacement corresponding to peak interface shear strength; τsu is the peak interface shear strength; τsr is the residual interface shear strength; A series of field tests on bored piles under compression loading (Zhang, Zhang, & Yu, 2011; Zhang, Zhang, Yu, et al., 2010; Zhang, Zhang, Zhang, et al., 2011) demonstrated that the value of βs to be within the range of 0.83–0.97. The shape of the bilinear model relating the pile base resistance to the pile base settlement is shown in Fig. 7B and the mathematical relationship for this model is given as Eq. (27).  k w wb < Sbu τb ¼ b1 b (27) kb1 Sbu + kb2 ðwb
Sbu Þ wb  Sbu Randolph and Wroth (1978) proposed a model for the determination of kb1 shown as Eq. (28). kb1 ¼

4Gsb πr0 ð1
υb Þ

(28)

where Gsb and υb are the shear modulus and Poisson’s ratio of the soil below the pile base, respectively.

Modified load transfer curve method

409

The value of kb2 can be approximately calculated using the Eq. (29) proposed by Zhang, Zhang, and He (2010). kb2 ¼

ΔPt k ¼
t  Δwt
ΔPt L=Ep Ap 1
kt L=Ep Ap


(29)

where Δ Pt is the increased load at the pile head when the settlement at the pile base is larger than the limiting pile base settlement of the first stage of the τb versus wb curve; Δ wt is the increased settlement at the pile head induced by Δ Pt; L is the pile length; Ep is the pile elastic modulus; Ap is the cross-sectional area of the pile; and kt is the ratio of the load increment to the settlement increment at the pile head, kt ¼ Δ Pt/Δwt. Results of the field tests on seven single piles in different soils (mud, sandy silt, and clays) presented by Zhang, Zhang, Yu, et al. (2010) and Zhang, Zhang, and Yu (2011) are used to validate the reliability of the softening nonlinear transfer curve model relating the interface shear strength to the pile settlement. Similarly, the capacity of the proposed bilinear model relating the pile base resistance to the pile base settlement is verified by comparing the calculated pile base resistance settlement curve with measured results from experimental studies (Zhang, Zhang, Yu, et al., 2010; Zhang, Zhang, Zhang, et al., 2011). Since the softening nonlinear transfer curve model Eq. (25) and bilinear model Eq. (27) are efficient, relatively simple. In this study, these two models are extended for analysis of mechanical behaviors of a single pile in expansive soil introducing the necessary modifications extending the principles of unsaturated soil mechanics.

Modification of the model relating the interface shear strength to the pile-soil relative displacement The softening nonlinear transfer curve model (Eq. 25) can be extended to characterize the transfer curves in different soil layers with totally different soil properties (Zhang & Zhang, 2012). This transfer curve model contains three key parameters; namely, the peak interface shear strength (τsu), the residual interface shear strength (τsr) and the pile-soil relative displacement corresponding to the peak interface shear strength (Ssu). For pile in expansive soil, the transfer curves for the part of the pile in the stable zone keep constant during the infiltration process. For the portion of the pile embedded in the active zone, both the peak (τsu) and residual interface shear strength (τsr) can be significantly influenced by the matric suction changes in the infiltration process. Introducing Eq. (11) into Eq. (21), Eq. (30) can be obtained for the calculations of peak interface shear strength in the wetting process. From limited data collected from the literature regarding the interface direct shear

410

18. Pile behavior modeling in unsaturated expansive soils

test with matric suction control or measurement, the postpeak interface shear strength seems to be less affected by the matric suction compared to the peak interface shear strength. Hamid and Miller (2009) concluded that the possible reason for this behavior was the disruption of air-water menisci along the failure surface during continued shearing after achieving the peak interface shear strength. For this reason, the residual interface shear strength can be estimated only by considering the contribution of net normal stress as shown in Eq. (31). In Eq. (31), both the residual effective cohesion (c0 ar) and the residual interface friction angle (δ0 r) should be determined from experimental studies for specific structure surface and soil. If there is no experimental data available on the residual interface shear strength, the value of residual interface shear strength can be assumed as 0.83–0.97 times the value of peak interface shear strength based on a series of field tests on bored piles under compression loading according to the experimental studies from Zhang, Zhang, and He (2010) and Zhang, Zhang, Zhang, et al. (2011). Similar to the postpeak interface shear strength, the matric suction also poses minor influences on the critical interface shear displacement. The reason can also be given as the disruption of air-water menisci along the failure surface. While except for matric suction, the critical interface shear displacement (wcr) is influenced by various other factors like the soil type, structure surface type, stratigraphy, loading procedure, and environmental factors. Because of the complexities associated in understanding the independent contribution of each of these factors, generally, wcr is suggested to be determined experimentally or back-analysis from the field test. There are also several scholars presenting suggested values for different scenarios, although most of them are explicit and completely defined by the authors for direct use. For example, API (1993) suggested the value of wcr to be 2% of the pile diameter for piles in clay. Vijayvergiya (1977) suggested a fixed value of 7.5 mm for piles in sand while Krasi
nski (2012) suggested a fixed value of 15 mm for piles in sand. From a series of field tests (Zhang & Zhang, 2012; Zhang, Zhang, Yu, et al., 2010; Zhang, Zhang, Zhang, et al., 2011; Zhao et al., 2009) for bored piles (diameter from 0.7 to 1.1 m) in different kinds of soils (e.g., mud, clay, sandy silt, silty clay), Zhang and Zhang (2012) summarized that the value of Ssu varies within a range from 5 to 25 mm. 8 > > >
> > ð1 + υÞ 1
υ
2υ2 :1
υ 2
Eaðψmi
0Þ 0

0

9 > >  > 1
υ
2υ2 Psðψmw
0Þ υ θ
θr =
σ S + ðuar
uwr Þ +
 1
υ Psðψmw
0Þ θs
θ r > > > 1
υ2
ð1 + υÞ 1
υ
2υ2 ; Eaðψmw
0Þ


(30)

Modified load transfer curve method

411



τsr ¼ c0ar + σ nf
uaf tan δ0r

(31)

where c0 ar is the residual effective cohesion; δ0 r is the residual interface friction angle.

Modification of the model relating the pile base resistance and pile base settlement In most cases, the pile base is located in the stable zone, thus the pile base resistance can be directly calculated using Eq. (27) proposed by Zhang and Zhang (2012). However, in case that the positive friction (uplift friction) increases significantly in the infiltration process and exceeds the withholding force (pile head load and negative friction), the pile will move upward. In other words, under such a scenario, it is possible for the pile base to detach from the soil so that the pile base movement (ρt) can have a negative value. Considering this possibility, Eq. (27) proposed by Zhang and Zhang (2012) is modified as Eq. (32) for the calculation of the pile base resistance considering the possible detachment of the pile base and soil upon infiltration. 8 wb < 0 σ 2 ¼ σ 3) exhibit fundamentally different postpeak behavior compared to Class I and Class II. The new results allow proposing some new principles of dynamic instability in the Earth’s crust. It is known that the postpeak behavior of Class I and Class II (Wawersik & Fairhurst, 1970) indicates a continuous decrease in the specimen strength from peak strength to frictional (residual) strength, where the latter is considered to be the lower limit on rock shear strength. This chapter shows that in contrast with Class I and Class II, the postpeak failure of hard rocks under high σ 3, associated with the propagation of a shear rupture, has the following three stages: stage A is characterized by extreme Class II behavior during which the specimen strength decreases

Modeling in Geotechnical Engineering https://doi.org/10.1016/B978-0-12-821205-9.00013-7

473

© 2021 Elsevier Inc. All rights reserved.

474

20. Class III postpeak rock behavior and dynamic instability

sharply from the peak strength to an “abnormally” low level (up to 10 times less than the frictional strength); at stage B, the specimen strength remains “abnormally” low and constant during the further passage of the shear rupture through the specimen; at stage C, the specimen strength increases up to the level of frictional (residual) strength. The postpeak stage is characterized by the constant and “abnormally” low strength we classify as Class III. Class III behavior takes place until the rupture has crossed the specimen body. Fig. 1 illustrates the fundamental difference between the conventional Class I and Class II behavior and the new Class III behavior. Fig. 1A shows six steps of shear rupture propagation through a specimen. The rupture incorporates a process zone ‘p (spotted) representing the rupture head and a frictional zone ‘f (black) located behind the head. In front of the rupture tip, an intact zone ‘s (gray) is located. After completion of the process zone at step 1, the length of it ‘p stays constant, while the length ‘f increases and the length ‘s decreases during the rupture propagation at steps 1–4. In accordance with the conventional understanding, the specimen strength beyond the peak stress (transient strength τtr) at any failure stage

FIG. 1 Schematic illustration the fundamental difference between the conventional and the new understanding of the postpeak behavior of hard rocks tested at high σ 3. (A) Six steps of shear rupture propagation through a specimen; (B) Stress-displacement curves for Class I and Class II postpeak behavior; (C) Stress-displacement curve for Class III postpeak behavior.

Introduction

475

is determined by all three zones located along the propagating rupture (intact, process, and frictional) as described by Eq. (1): τtr ¼ τs

lp ls lf + τp + τf l l l

(1)

Here, τs is the intact material strength, τf is the frictional strength, and τp ¼ (τs + τf)/2 is the average strength of the process zone. Stress-displacement curves in Fig. 1B correspond to Class I and Class II. Points on the curve indicate six identical steps shown for the specimen in Fig. 1A. The specimen strength at each step is in accordance with Eq. (1). Here, during the failure process, the transient strength τtr decreases gradually due to the substitution of the intact material strength τs by frictional strength τf. At steps 5 and 6, the specimen strength reaches the lower limit corresponding to frictional strength τf. The postpeak rupture energy between steps 1 and 5 corresponds to the shaded area under the curve. Fig. 1C illustrates the new understanding of the postpeak properties of hard rocks at high σ 3. The postpeak curve here incorporates three stages (stages A, B, and C). It will be shown later that in this case, the failure process is governed by the recently identified fan-hinged rupture mechanism (Tarasov, 2014, 2016; Tarasov & Randolph, 2008). This mechanism provides very low shear resistance of the process zone, which can be τp  0.1τf. Furthermore, the fan mechanism represents a very powerful stress amplifier (based on an unknown before principle) providing abnormally high shear stresses at the rupture tip at low shear stresses applied. These two unique features make it possible for the shear rupture to propagate through intact rock even at very low shear stresses applied τ, which can be significantly (up to an order of magnitude) less than the frictional strength (stage B). In this case, the specimen transient strength is determined solely by the process zone strength accordingly to Eq. (2): τtrðIIIÞ ¼ τp

(2)

Fig. 1C shows that controllable failure can be provided if the testing machine is capable to unload the specimen up to the level τtr(III) ¼ τp at the moment of completion of the process zone (step 1). After that, the rupture can propagate statically through the specimen at applied stresses slightly above τtr(III) ¼ τp which represents the specimen strength between failure steps 1 and 4. This fantastic ability of a shear rupture to move through the intact hard rock at very low shear stress applied is classified as Class III behavior. The postpeak rupture energy corresponding Class III (stage B) is shown by the shaded area on the graph. This very low energy absorption implies very high brittleness of the material at the failure process. At step 5 and further, when the process zone has left the specimen, the specimen strength increases up to the level of frictional strength τf.

476

20. Class III postpeak rock behavior and dynamic instability

It should be emphasized that Class III properties of hard rocks are activated at stress conditions of seismic depths. The ability of shear ruptures to propagate through intact hard rocks at very low shear stresses allows proposing new concepts of dynamic instability in the Earth’s crust. All these questions associated with Class III behavior are discussed in this chapter.

Experimental results It is known that the level of confining pressure σ 3 affects the postpeak behavior of testing rock specimens. It is important that rising σ 3 can cause very different variations in postpeak properties depending on rock hardness. Three sets of stress-strain curves obtained at different levels of σ 3 in Fig. 2 illustrate the typical variation in postpeak properties for rocks of three types of hardness: soft, intermediate, and hard. Rock hardness here is characterized by uniaxial compressive strength UCS and increases from left to right. Levels of σ 3 are shown near each curve. The curves demonstrate how rock hardness affects the postpeak behavior at rising σ 3. Relatively, soft rocks on the left [represented by marble with UCS ¼ 130 MPa (Rummel & Fairhurst, 1970)] increase postpeak ductility with rising σ 3. All blue (gray in print version) dotted lines reflecting negative postpeak modulus M ¼ dσ/dε < 0 characterize rock behavior under all σ 3 as Class I. Rocks of intermediate hardness (represented by quartzite with UCS ¼ 180 MPa) exhibit postpeak embrittlement within a certain range of σ 3 which is expressed by a transition from Class I to Class II behavior characterized by positive postpeak modulus M ¼ dσ/dε > 0 [indicated by red (gray in print version) dotted lines]. At lower and higher σ 3, the postpeak ductility increases with rising σ 3. The typical behavior of hard rocks (represented by dolerite with UCS ¼ 300 MPa) is characterized by dramatic embrittlement at high σ 3 leading to

55 MPa 36

200

0 0

3.5

14 7

28 21

2 1.5 1 0.5 Axial deformaon (mm)

800

Class II Class I

400

0

100 75

150

Extreme Class II

1200

Class I

800 400

0 0

200 MPa

Differenal stress (MPa)

300

0

Class I

1200

Differenal stress (MPa)

Differenal stress (MPa)

Class I

Dolerite UCS = 300 MPa

Quartzite UCS = 180 MPa

Marble UCS = 130 MPa

100

Hard

Intermediate

So

30 0.01

0.02 Axial strain

0.03

UCS < 250 MPa

0

0

0

10

30

0.005

75 60

150 MPa

?

0.01 Axial strain

0.015

0.02

UCS > 250 MPa

FIG. 2 Three sets of stress-strain (deformation) curves illustrating features of postpeak behavior for rocks of different hardness.

477

Experimental results

75 MPa

600

A

400 200 0

Differential stress (MPa)

0

0.002 0.004 0.006 0.008 Axial strain

0.01

800 s3 = 60 MPa

600

A

400 200

(A)

0 0

0.002 0.004 0.006 Axial strain

0.008

Differential stress (MPa)

720

800

Differential stress (MPa)

Differential stress (MPa)

extreme Class II behavior. The transition from Class I to Class II for the dolerite occurs at σ 3 ¼ 30 MPa. At relatively low σ 3 the postpeak failure can be controlled both for Class I and Class II on stiff and servo-controlled testing machines. However, at σ 3  60 MPa, the failure process associated with the propagation of a shear rupture becomes uncontrollable and abnormally violent. With rising σ 3, rock brittleness and the violence increase. The spontaneous failure is accompanied by a shock-like sound and shudder of the massive frame of the testing machine. This indicates huge energy released during the failure process. The problem is that all existing testing machines cannot provide controllable failure of hard rocks within a certain range of high σ 3. It should be noted that the range of confining stress σ 3 at which the postpeak control is currently impossible in laboratory investigations corresponds to the seismic depths of shallow earthquakes. The lack of knowledge about the postpeak properties of the majority of the earthquake host rocks prevents us from understanding and quantifying the contribution of these rocks to natural earthquakes and human-induced dynamic events at great depths (caused, for example, by mining process and hydrofracturing). Hard rocks are represented mainly by volcanic and highly metamorphic rocks with UCS > 250 MPa. Figs. 3 and 4 introduce two unique postpeak features typical for hard rocks at high σ 3 obtained on dolerite specimens. These experiments were

0.01

ss

n

690

n * A

660 630 600 0.007

700 675

0.0075

0.008 0.0085 Axial strain

0.009

ss

650 625 600 575 550 0.0065

(B)

0.007

0.0075

0.008

0.0085

0.009

Axial strain

FIG. 3 Experimental results indicating that during spontaneous failure of hard rocks at high σ 3 (beyond point A) the postpeak modulus [red (gray in print version) dotted line] practically coincides with the unloading elastic modulus [green (gray in print version) dotted line]. Full scale (A) and fragments (B) of stress-strain curves for dolerite specimens tested at σ 3 ¼ 75 MPa and σ 3 ¼ 60 MPa.

s1

load cell

0

0.002

0.006

200

400

600

0

sfan

sf

II

0

0.008

800

1000

1200

1400

Axial strain

0.004

tr

Ex

s

as

l eC m e

s3 = 60 MPa

Differential stress (MPa) 0.01

0

200

400

600

C

0.1 Time, mS

B

0.005

0

A

Axial strain

0.01

la

0.015

II ss

0.006

sfan

sf

II

0.008

sfan

sf

s as Cl

0

200

400

600

800

1000

1200

1400

0.01

75 MPa

0.02

Axial strain

0.004

e

m

tre Ex

150 MPa

0.002

C me

0

tre

0

200

400

600

800

Ex

Differential stress (MPa)

Differential stress (MPa)

B 0.1

C

A

B

C

Time, mS

0

A

0 0.1 Time, mS

0

200

400

600

800

point A.

FIG. 4 Experimental results indicating the abnormally low level of postpeak strength for hard rocks at high σ 3 during spontaneous failure beyond

s3

0

200

400

600

Axial gauge

800

Differential stress (MPa)

800

Differential stress (MPa)

Differential stress (MPa)

Experimental results

479

conducted on a special testing machine with advanced stiffness and unloading rate which allowed obtaining very important information about postpeak behavior during the spontaneous failure process. Results in Fig. 3 allow estimating the level of postpeak modulus taking place during the spontaneous failure. Fig. 3A shows stress-displacement curves for σ 3 ¼ 75 MPa and σ 3 ¼ 60 MPa. Points A at the postpeak stage indicate the start of spontaneous uncontrollable failure. Fig. 3B shows enlarged fragments of the stress-strain curves involving the postpeak stage. Each postpeak curve of Class II here, reflecting real rock properties up to point A, was obtained in static regime due to controllable unloading and reverse deformation of the specimen provided by the servo-controlled system. Red (gray in print version) dotted lines here indicate the variation in postpeak modulus M ¼ Δ σ/Δ ε with the rupture growth. At point A, the level of postpeak modulus approaches the level of unloading elastic modulus E ¼ Δ σ/Δ ε indicated by the dotted green (gray in print version) line. The unloading modulus was determined experimentally at the peak stress (n-n line on the curve for σ 3 ¼ 75 MPa). At point A both moduli become almost the same. The postpeak behavior when M  E we call as extreme Class II. For extreme Class II, the postpeak rupture energy is vanishingly small. In this case, the controllable failure becomes impossible because all modern testing machines unable to provide sufficiently fast the specimen unloading to stop the propagation of the shear rupture (will be discussed later in more details). By analogy with the results obtained for σ 3 ¼ 60 and 75 MPa we can suppose that beyond points A for all experiments conducted at σ 3  60 MPa M  E. Another unique feature of hard rocks is the abnormally low level of strength taking place during spontaneous fracture. Fig. 4 shows stressdeformation curves involving the postpeak stage (red (gray in print version) solid lines) indicating the extreme Class II behavior, and stress-time curves ABC reflecting the abnormally low postpeak specimen strength at point B. Curves ABC were recorded by the load cell during the spontaneous failure beyond point A. The load cell was in direct contact with the specimen as shown on the schema in Fig. 4. The results were obtained for σ 3 ¼ 60, 75, and 150 MPa. After point B, the specimen strength increases to the residual strength σ f. The static residual (frictional) strength Δ σ f was determined experimentally by deforming the failed specimen at low (static) strain rates. The strength level at point B is about 10 times less than the static frictional strength σ f (point C). The recorded abnormally low specimen strength during the failure process is in conflict with the conventional understanding that the residual strength is the lower limit for rock strength.

480

20. Class III postpeak rock behavior and dynamic instability

FIG. 5 Stress-strain and stress-time curves for dolerite specimen tested at σ 3 ¼ 60 MPa, which are used for the analysis of postpeak rupture control.

Special conditions for experimental study of Class III properties of hard rocks at high s3 This section discusses the fact that postpeak characteristics recorded during spontaneous failure of hard rocks at high σ 3 depend on the unloading rate provided by the testing machine and do not reflect actual postpeak properties. This feature is illustrated in Fig. 5 representing stress-deformation and stress-time curves for the dolerite specimen tested at σ 3 ¼ 60 MPa. The actual variation in the postpeak properties of this specimen corresponds to the red (gray in print version) curve ADBCF on the stress-deformation graph. The mechanism providing these rock properties will be discussed later. Here, we demonstrate only the effect of testing machines on the recorded postpeak characteristics. As we discussed in Fig. 1C, the initial part AD of the actual postpeak curve can be obtained if the testing machine provides the sufficiently high unloading rate shown by the red (gray in print version) line AD on the stress-time graph. Then, the rest part of the actual postpeak properties DBCF (including the Class III part DB) can be obtained in full. However, if the unloading rate provided by the testing machine is slower the situation can be very different. Let us analyze the following three situations. (1) The stress-time curve ABC reflects the response provided by the testing machine used to study the dolerite specimens. This curve indicates the load (stress) applied by the machine to the specimen. Because this load during the rupture process up to point B is higher than the specimen strength (curve ADB) the rupture process should be spontaneous. However, at point B, the applied stress and the specimen

Rupture mechanism responsible for Class III behavior

481

strength become identical, after which the loading process becomes controllable. In this case, the stress-deformation curve ABCF will be recorded. (2) If the unloading rate of the machine corresponds to line AC, we have the stress-deformation curve ACF. In this case, the rupture has crossed the specimen body at the moment when the specimen strength increases up to the residual strength. This means that at unloading rates slower than AC, the specimen strength corresponding to Class III cannot be obtained experimentally. (3) If the unloading rate AE is slightly slower than AD the study of Class III postpeak properties is possible. We can conclude that to make the testing machine good enough for the study of Class III properties of hard rocks at high σ 3 the unloading rate of existing testing machines should be significantly increased. For that, it is necessary to minimize the following machine parameters: (1) The response time of the servocontrolled system. (2) The mechanical and hydraulic compliance of all load train components. (3) The inertia mass of the loading-unloading piston of the actuator. (4) The fluid volume in the loading and unloading chambers of the actuator.

Basic ideas underlying the fan-hinged rupture mechanism responsible for Class III behavior of hard rocks at high s3 To understand the failure mechanism operating in hard rocks at high σ 3, we need to analyze the evolution of failure mechanisms in hard rock specimens with rising σ 3 as shown in Fig. 6. Here, confining pressure σ 3 increases along the horizontal axis from left to right. At the origin of the horizontal axis, σ 3 ¼ 0. Acoustic emission experiments show that the failure process of brittle rocks at any level of σ 3 is accompanied by formation of tensile cracks (Lei, Kusunose, Rao, Nishizawa, & Satoh, 2000; Lockner, Byerlee, Kuksenko, Ponomarev, & Sidorin, 1991). However, the ultimate length ‘ of tensile cracks that can be developed at failure depends on the level of σ 3 because rising σ 3 suppresses the tensile crack growth. A dotted curve here shows symbolically the typical variation of ultimate length ‘ of tensile cracks versus σ 3: the higher σ 3, the shorter ‘. The length ‘ of tensile cracks in turn determines the macroscopic failure mechanism and the failure pattern shown schematically in rock specimens (i) to (iv). Within the pressure range 0  σ 3 < σ 3shear shear rupture cannot propagate in its own plane due to creation at the rupture tip of relatively long tensile cracks that prevent the shear rupture propagation (Horii & Nemat-Nasser,

482

20. Class III postpeak rock behavior and dynamic instability

FIG. 6 The evolution of failure mechanisms in hard rocks with rising confining pressure σ 3 and variable efficiency of the fan mechanism.

1985; Sahouryeh, Dyskin, & Germanovich, 2002). Two of the following failure modes typical for this pressure range may be distinguished: (i) Splitting by long tensile cracks (at low σ 3). (ii) Distributed microcracking followed by coalescence of microcracks with the formation of a macroscopic rupture (at larger σ 3). At σ 3  σ 3shear, shear rupture acquires the capability to propagate in its own plane. At these conditions, the only failure mode can take place—localized shear. Here, under the effect of high enough confining pressure σ 3, tensile cracks generated in the rupture tip become sufficiently short to assist shear rupture to propagate in its own plane. Shear ruptures are known to propagate through brittle rocks because of the creation of an echelon of tensile cracks and intercrack slabs at the rupture tip formed along with the major stress that is at angle αo  (30°  40°) to the shear rupture plane (Reches & Lockner, 1994). It was observed that at relative displacement of the rupture faces the inclined slabs are subject to rotation (King & Sammis, 1992; Peng & Johnson, 1972; Reches & Lockner, 1994). Depending on the behavior of the rotating slabs, two basic principles of the failure process may be distinguished: (iii) Frictional shear—which operates at σ 3shear  σ 3  σ 3fan(min). Here, relatively long slabs collapse at rotation leading to creation friction between the rupture faces within the rupture head (Reches & Lockner, 1994). This mechanism provides conventional Class I or

New principle of dynamic instability caused by the fan mechanism

483

Class II behavior and the failure process can be controlled on modern testing machines. (iv) Fan-hinged shear—which is active at higher confining pressures within the range σ 3fan(min) < σ 3 < σ 3fan(max) (Tarasov, 2014, 2016). Here, sufficiently short slabs can withstand rotation without collapse. Because the relative shear displacement of the rupture faces increases with distance from the rupture tip, the successively generated and rotated slabs form a fan-structure that represents the rupture head. The slabs involved in the fan-structure operate as hinges between a shearing rupture face. The completed fan-structure passes through intact rock as a wave leaving behind the shear rupture. The fan-structure has a number of fantastic features, the most important of which are (a) The fan-structure has very low resistance to shear which can be up to 10 times less than the frictional strength τfan ¼ 0.1 τf. (b) The fan-structure represents a natural mechanism of dramatic stress amplification. Low shear stresses applied to the material (specimen) can be amplified hundreds of times in the rupture tip up to levels exceeding the ultimate material strength τs. The explanation of these unique features of the fan-structure is considered in detail in Tarasov (2014, 2017) based on the physical and mathematical models. The combination of low shear resistance and high-stress concentration in the rupture tip, provided by the fan-structure, allows for the rupture propagating through intact rock at very low shear stresses applied. It should be noted, that the fan mechanism exhibits different efficiency depending on the level of σ 3. We can determine the fan-mechanism efficiency as the ratio between the frictional strength τf and the fan strength τfan: ψ ¼ τf/τfan. The variable efficiency of the fan mechanism is resulted from the fact that the length ‘ of domino-blocks further decreases with rising σ 3 in the fan-hinged pressure zone (shown in red (gray in print version) in Fig. 6). The green (gray in print version) curve in Fig. 6 illustrates graphically a possible variation of the fan-mechanism efficiency ψ versus confining stress σ 3 (see details in Tarasov, 2017; Tarasov & Randolph, 2016).

New principle of dynamic instability caused by the fan mechanism The fan mechanism provides a new principle of dynamic instability which manifests itself in laboratory experiments and in the earth crust, for example during shallow earthquakes and shear rupture rockbursts in deep mines. This question in detail can be seen in Tarasov (2017) and Tarasov and Randolph (2016). Here we demonstrate the general idea only. Fig. 7 shows the relation between shear resistance and stresses generated by

484

20. Class III postpeak rock behavior and dynamic instability

FIG. 7 Relative distribution of shear resistance (dotted curve) and amplified shear stresses (solid curve) in the rupture head caused by the fan mechanism.

the fan mechanism at low shear stresses applied τ0. A dotted curve here reflects the distribution of shear resistance along a fault involving the fan-structure. In front of the fan shear resistance corresponds to the material strength τs, behind the fan it is determined by friction τf, in the fan-zone shear resistance is significantly less than the frictional strength τfan ≪τf. A solid curve reflects the shear stress variation. A part of this curve abc corresponds to the stress distribution caused by the fan-structure. The very low shear stress applied τ0 is amplified up to the level of the material strength τs at the rupture tip. The principle of stress amplification and low shear resistance can be seen in Tarasov (2017) and Tarasov and Randolph (2016). The large difference between shear stress bc and shear resistance cd in the fan zone makes the situation self-unbalancing. The generated power in the fan zone corresponding to the shaded area on the graph can accelerate the rupture speed up to intersonic levels. It should be emphasized that despite the fact that the shear rupture here propagates through intact material the magnitude of stress drop Δτ ¼ τo  τ1 can be very low (lower than at the frictional stick-slip instability) because the rupture propagates at low shear stresses applied τ0 ≪ τf.

Class III behavior of hard rocks in laboratory specimens Fig. 8 illustrates the process of the fan-structure formation and propagation through a hard rock specimen causing the postpeak behavior of Class III. Fig. 8A shows six steps of the rupture development in the

by the fan mechanism [in red (gray in print version)], (C) Photograph of the typical dynamic shear rupture consisting of the echelon of rotating slabs (Ortlepp, 1997). Source: This was first published by the Southern African Institute of Mining and Metallurgy.

FIG. 8 (A) Illustration of the fan-structure formation and propagation through the specimen body, (B) Postpeak rock behavior of Class III generated

486

20. Class III postpeak rock behavior and dynamic instability

specimen body. Fig. 8B shows schematically a stress-displacement curve which reflects different features caused by the fan mechanism. Points on the curve indicate identical steps of the rupture propagation presented in Fig. 8A. Acoustic emission studies demonstrate that in hard rocks at high σ 3 the localized propagation of a shear rupture begins before the peak stress (Lei et al., 2000; Reches & Lockner, 1994). The shear rupture propagates due to consecutive creation of tensile cracks forming the echelon of inter crack slabs. Point 1 corresponds to this situation. The physical and mathematical models in Tarasov (2017) and Tarasov, Guzev, Sadovskii, and Cassidy (2017) show that for creation of the front half of the initial fan-structure rising shear stresses up to the material strength are required, however, during the formation of the rear half of the fan-structure its shear resistance decreases dramatically to the level τfan ¼ 0.1 τf. Thus, the peak stress on the curve (point 2 in Fig. 8B) corresponds to the completion of the front half of the fan-structure, while point 3 corresponds to the totally created fan-structure. The specimen strength after this moment corresponds to shear resistance of the fan-structure τfan ¼ 0.1 τf. The red (gray in print version) part of stress-displacement curve in Fig. 8B reflects actual postpeak properties of hard rocks caused by the fan mechanism. The transition from peak stress τs to τfan is accompanied by extremely low rupture energy. The postpeak modulus M between points 2 and 3 is very close to the elastic unloading modulus E (as discussed in Fig. 2C), representing the extreme Class II behavior. The horizontal red (gray in print version) line 3–5 indicates Class III postpeak rock behavior associated with the propagation of the completed fan-structure through the specimen body. We refer to the specimen strength at this stage of the failure process as the fan-strength τfan. This process can be controllable by the corresponding testing machine. When the fan-structure has crossed the specimen (step 6), the specimen strength returns to the conventional mode which is determined by friction between the rupture faces τf. The structure of shear rupture generated in hard rocks at high σ 3 always involves the echelon of rotated slabs. The photograph in Fig. 8C (Ortlepp, 1997) shows a typical dynamic rupture generated a severe shear rupture rockburst in a deep mine of South Africa.

Class III rock behavior as a source of shallow earthquakes In this section, we illustrate briefly features of the generation of natural dynamic events by the fan mechanism in hard rocks of Class III behavior. It is known that shallow earthquakes are normally attributed to preexisting faults (e.g. boundaries between tectonic plates). Today this feature is treated as the evidence of stick-slip nature of earthquakes associated with unstable sliding along preexisting faults. The fan mechanism allows

Class III rock behavior as a source of shallow earthquakes

487

proposing another explanation to the fact that earthquakes are generated based on preexisting faults. It was shown above that for the initial formation of the fan-structure high stresses equal to the material strength τs are required (see Fig. 8). For laboratory specimens stress equal to τs is typically applied to the whole specimen. However, at natural conditions high stresses necessary for the initial fan-structure formation can be generated locally in the vicinity of preexisting faults. Fig. 9 illustrates one of many possible models for the generation of high local stress based on preexisting faults. Fig. 9A shows a rock fragment involving a preexisting fault (black solid line) with a compressive jog. This fragment is located at great depth where the minor stress σ 3 is high enough to cause Class III behavior of intact hard rock at failure. Horizontal lines on the graph below indicate symbolically levels of the following parameters: τs is the strength of intact rock; τf is the frictional strength of the preexisting fault; τfan is the fan strength of intact rock determined by the fan mechanism; τ0 is the field shear stress applied to the rock fragment; τ1 is the field stress after the rupture propagation; Δ τ is the stress drop. The orientation of the field shear stress is shown by open arrows on the rock fragment. Because the level of field stress τ0 is significantly less than the frictional strength τf the situation on the preexisting fault is stable. However, due to deformations along the fault caused by shear stresses τ0 a high local stress can be created in the jog zone delineated by red (gray in print version) circle. If the local stress in intact rock of this zone reaches the level of rock strength τs the initial fan-structure can be formed here. After completion of the fan-structure (shown by a red (gray in print version) ellipse) shear resistance of which is τfan < τ0, it can propagate spontaneously through the intact rock mass at low shear stress τ0 and generate an earthquake. The new fault is shown by the blue (gray in print version) dotted line. Due to very low shear resistance and rupture energy provided by the fan mechanism, the failure process can be accompanied by abnormal energy release and violence. A photograph of a typical dynamic fault generated in hard rocks (taken in an ultra-deep South African mine) in Fig. 9B (Ortlepp, 1997) illustrates the fault structure consisting of the echelon of rotating slabs generated in intact hard rocks. It should be emphasized that despite the fact that the new fault is formed in intact rock the magnitude of stress drop Δ τ can be significantly less than caused by stick-slip instability along the preexisting fault. This happens because the failure process generated by the fan mechanism can occur at very low shear stresses τ0 < τf. This feature caused by the fan mechanism explains also the fact that earthquakes are normally generated at shear stresses below the frictional strength. Fig. 9A demonstrates also that each new fault (for example new fault 1) represents the source of local stress concentration based on which the new fan-structure can be generated. As a result of this, a new dynamic fault can

FIG. 9 Features of the generation of new extreme ruptures in pristine hard rock in the vicinity of preexisting faults caused by the fan mechanism at low field shear stresses τ0. (A) Schematic representation of a fragment of the rock mass involving the preexisting and two new faults; (B) A photograph of typical shear ruptures generated in intact rocks at great depths; (C) A map of faults caused many earthquakes in a New Zealand region. From (B) Ortlepp, W. D. (1997). Rock fracture and rockbursts. Johannesburg: The South African Institute of Mining and Metallurgy; (C) Temblor. (2017). http:// www.temblor.net.

Class III rock behavior as a source of shear rupture rockbursts in deep mines

489

be created (green (gray in print version) new fault 2). This explains the reason for the generation of aftershocks and also for the fact that the Earth’s crust is riddled with multiple faults. Fig. 9C (Temblor, 2017) shows a map of faults that generated many earthquakes in a New Zealand region.

Class III rock behavior as a source of shear rupture rockbursts in deep mines As hard rock mining progresses to depth, rockburst problems increase. Below a certain depth (different for different rocks) a special form of rockburst—shear rupture rockburst—comes into play. It is established (Gay & Ortlepp, 1979; McGarr, Spottiswoode, Gay, & Ortlepp, 1979) that shear rupture rock bursts are caused by the dynamic formation of new faults in pristine hard rock. These mine tremors are seismically indistinguishable from natural earthquakes and share the apparent paradox of failure under low shear stress (McGarr et al., 1979). All characteristic features of shear rupture rock bursts are great depths; generation of new faults of extreme dynamics in pristine hard rocks; fault nucleation at a point considerable distance away from the opening surface; fault propagation at low shear stresses; abnormal violence. Indeed, it is difficult to explain the entire combination of the features of this phenomenon based on the traditional understanding of rock failure mechanisms. However, all characteristic features of shear rupture rock bursts represent manifestations of the intrinsic properties of the fan mechanism. Fig. 10A shows a cross-section of the Earth’s crust involving an opening made in hard rock. The left graph illustrates the depth distribution of minor stress σ 3 in the area involving the opening. The fan mechanism can be activated below the upper cutoff corresponding to the critical level of minor stress σ 3fan(min) (see Fig. 6). The fan-mechanism efficiency ψ ¼ τf/τfan is a function of σ 3. The right green (gray in print version) graph symbolically reflects the depth distribution of the fan-mechanism efficiency. It shows that around the opening, the minor stress is below the critical level σ 3 < σ 3fan(min) at which ψ ¼ 1 (on the opening surface σ 3 ¼ 0). Hatched areas in Fig. 10A show zones where the fan mechanism cannot be generated. The zone of the fan mechanism activity is shown by the gray area. By analogy with earthquakes generated based on preexisting fault, the mine opening plays the role of stress concentration in the surrounding area. If mine induced stress locally reaches the level of intact rock strength τs somewhere in the gray area in Fig. 10A, the fan-structure will be generated in pristine rocks. This can happen within the rock space at a point some considerable distance away from the surface of an opening because around the opening σ 3 < σ 3fan(min). After formation of the fan-head, further spontaneous propagation of the new fault characterized by extremely low

490

20. Class III postpeak rock behavior and dynamic instability

FIG. 10 Features of shear rupture rockbursts in deep mines. (A) Depth distribution of σ 3

and ψ around the opening; (B) Relative position of the opening and new rupture. From (B) Ortlepp, W. D. (1997). Rock fracture and rockbursts. Johannesburg: The South African Institute of Mining and Metallurgy.

rupture energy will be accompanied by the release of very large amounts of energy. Fig. 10B from Ortlepp (1997) illustrates relative positions of the opening and new rupture, which created a shear rupture rockburst in a South African mine. The asterisk indicates the point of shear rupture nucleation.

Conclusion This chapter demonstrates that the postpeak properties of hard rocks at high confining stresses σ 3 are unexplored experimentally and incorrectly understood today. This chapter proposes new information about the

References

491

actual properties of hard rocks. It is shown that beyond the peak stress hard rocks at high σ 3 exhibit dramatic weakening and embrittlement associated with the ability of a shear rupture to propagate dynamically through intact rock at shear stresses significantly less than the frictional strength. This rock feature is classified as Class III behavior which differs fundamentally from Class I and Class II behavior. Class III rock properties are provided by a recently identified fan-hinged shear rupture mechanism operating in hard rocks at high σ 3. The unique features of this mechanism are briefly discussed. A new principle of dynamic instability based on the fan mechanism and Class III rock behavior is proposed. The role of this principle in generation of shallow earthquakes and shear rupture rockbursts in deep mines are discussed. This chapter emphasizes the vital importance of the experimental study of the unexplored Class III rocks because they represent the majority of the earthquake host rocks and also extremely dangerous in respect of shear rupture rockbursts.

Acknowledgment This work was supported by the Ministry of Science and Education of the Russian Federation (grant no. RFMEFI58418X0034).

References Gay, N. C., & Ortlepp, W. D. (1979). Anatomy of a mining-induced fault zone. Geological Society of America Bulletin, 90, 47–58. Horii, H., & Nemat-Nasser, S. (1985). Compression-induced micro-crack growth in brittle solids: Axial splitting and shear failure. Journal of Geophysical Research, 90, 3105–3125. King, G. C. P., & Sammis, C. G. (1992). The mechanisms of finite brittle strain. Pure and Applied Geophysics, 138, 611–640. Lei, X., Kusunose, K., Rao, M. V. M. S., Nishizawa, O., & Satoh, T. (2000). Quasi-static fault growth and cracking in homogeneous brittle rock under triaxial compression using acoustic emission monitoring. Journal of Geophysical Research, 105, 6127–6139. Lockner, D. A., Byerlee, J. D., Kuksenko, V., Ponomarev, A., & Sidorin, A. (1991). Quasi-static fault growth and shear fracture energy in granite. Nature, 350, 39–42. McGarr, A., Spottiswoode, S. M., Gay, N. C., & Ortlepp, W. D. (1979). Observations relevant to seismic driving stress, stress drop, and efficiency. Journal of Geophysical Research, 84, 2251–2261. Ortlepp, W. D. (1997). Rock fracture and rockbursts. Johannesburg: The South African Institute of Mining and Metallurgy. Peng, S., & Johnson, A. M. (1972). Crack growth and faulting in cylindrical specimens of Chelmsford granite. International Journal of Rock Mechanics and Mining Sciences, 9, 37–86. Reches, Z., & Lockner, D. A. (1994). Nucleation and growth of faults in brittle rocks. Journal of Geophysical Research, 99, 18,159–18,173. Rummel, F., & Fairhurst, C. (1970). Determination of the post-failure behavior of brittle rock using a servo-controlled testing machine. Rock Mechanics and Rock Engineering, 2(4), 189–204.

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Sahouryeh, E., Dyskin, A. V., & Germanovich, L. N. (2002). Crack growth under biaxial compression. Engineering Fracture Mechanics, 69(18), 2187–2198. https://doi.org/10.1016/ S0013-7944(02)00015-2. Tarasov, B. G. (2014). Hitherto unknown shear rupture mechanism as a source of instability in intact hard rocks at highly confined compression. Tectonophysics, 621, 69–84. Tarasov, B. G. (2016). Shear fractures of extreme dynamics. Rock Mechanics and Rock Engineering, 49(10), 3999–4021. Tarasov, B. G. (2017). Shear ruptures of extreme dynamics in laboratory and natural conditions. In J. Wesseloo (Ed.), Keynote addresses in eighth international conference on deep and high stress mining, Perth, Australia (pp. 1–48). ISBN: 978-0-9924810-6-3. Tarasov, B. G., Guzev, M. A., Sadovskii, V. M., & Cassidy, M. J. (2017). Modelling the mechanical structure of extreme shear ruptures with friction approaching zero generated in brittle materials. International Journal of Fracture, 207, 87–97. https://doi.org/10.1007/ s10704-017-0223-1. Tarasov, B. G., & Randolph, M. F. (2008). Frictionless shear at great depth and other paradoxes of hard rocks. International Journal of Rock Mechanics and Mining Sciences, 45(3), 316–328. Tarasov, B. G., & Randolph, M. F. (2016). Improved concept of lithospheric strength and earthquake activity at shallow depths based upon the fan-head dynamic shear rupture mechanism. Tectonophysics, 667, 124–143. Temblor. (2017). (2017). http://www.temblor.net. Wawersik, W. R., & Fairhurst, C. (1970). A study of brittle rock fracture in laboratory compression experiments. International Journal of Rock Mechanics and Mining Sciences, 7, 561–575.

Index Note: Page numbers followed by f indicate figures, t indicate tables, and b indicate boxes.

A

Adaptive neuro-fuzzy inference system (ANFIS), 39–41 Aerodynamic problems, 212–213 Aero-Servo-Hydro-Dynamic Soil StructureInteraction (ASH-DSSI), 210–212 Airy stress function method, 64–65 Amber trigger, 466 ANOVA single factor test, 50–51, 55 Artificial neural network, 38 Asymmetric multipod foundation, 245 Axi-symmetric DEM simulations, 91

B

Back analysis, uncertainty quantification, 7–9, 8f Batter piles, 37–38 angle of oblique load effect on actual vs. predicted oblique load, 53–54, 54f ANOVA single factor test, 55 Gaussian membrane functions, 54, 55f parametric analysis, 56 sensitivity analysis, 55, 56t batter angle effect on actual vs. predicted lateral load, 50, 51f ANOVA single factor test, 50–51 parametric analysis, 53 sensitivity analysis, 51–52, 52t Bayesian approach, 3 Bayesian hyperparameter optimization, 315–316 Bayesian OED, 10–11, 13–14, 17–19, 20f Beam on nonlinear Winkler foundations model, 219, 231, 231f Bearing capacity ratio (BCR), 152, 157 Bell-shaped function, 41 Biocemented soil behavior. See also Microbially induced calcite precipitation (MICP) bacterial growth and cementation solution, 108–109

biotreatment, 109, 109t cementation media concentration, 111–114 electrical conductivity, 107, 110, 111f experimental sand material, 108 Fann calcimeter pressure gauge model, 110 pH, 106–107, 110–111 SEM analysis, 114, 115f variation of calcite precipitation, 111–114, 113f Biogeochemical mechanism of MICP, 103. See also Microbially induced calcite precipitation (MICP) Bio-mediated ground improvement methods. See Microbially induced calcite precipitation (MICP) Biot’s dynamic field equations, 292 Bootstrap method, 10–11 Braced excavation finite element numerical modeling, 331–332 numerical results, analysis, 332–336 logarithmic regression, 329 probabilistic reliability assessment, 329–330 serviceability limit state, 340–343 ultimate limit state, 340 shear strength reduction technique, 328–329 surrogate models serviceability limit state, 337–338 ultimate limit state, 336–337 Buckling amplification factor, 222f Buckling instability of piles, 222t Buried continuous pipelines, 234f

C

Calcite precipitation, 105–106. See also Microbially induced calcite precipitation (MICP) Cataclastic flow, water-saturated sandstones, 188

493

494

Index

Catastrophic failure, 183, 189 Centrifuge tests, 208–209, 209f c-φ reduction method, 328 Classical theory of elasticity, 64–65 Classification and regression trees (CART), 314 Class III rock behavior. See Postpeak behavior, Class III Cohesive soil column collapse, 274–276 Column buckling, 223f Compaction localization, water-saturated sandstones, 188 Comparison models, 317 Cone penetration testing (CPTU), 309–310 Constitutive model, 262–264 Contact detection algorithms, 85–86 Contact slippage, in DEM simulation, 86–87 Coordination numbers, 90 Coulomb’s theory, 403–404 Courant-Friedrichs-Levy (CFL), 270 Current material models, 433–434 Cyclic resistance ratio (CRR), 369–370, 387–388

D

Daguangbao landslide, 277–279 Deep basements, 445–449, 448f, 449–450t, 450f Deep underground tunnels, 445, 447f Deformation bands, 187f, 191–192 in Solenhofen Limestone, 192 Degree of freedom (DOF), 435–437, 437f Delaunay triangulation, 81 Deterministic optimization techniques, 7–9 Dilatant hardening effects, 190 Discrete element method (DEM), 258 boundary types, 80–82 contact detection algorithms, 85–86 damping mechanisms, 96 data analysis, 88–89 fabric analysis, 90–91 force calculation, 86–87 macro-scale variables, 79–80, 89–90 micro-scale variables, 79–80, 90–91 numerical integration procedures, 87–88 particle types, 82 servo-control algorithm, 96 software selection and validation, 94 of soil behavior, 91–93 specimen generation, 83–84 specimen preparation and input parameters, 94–95

Distance-weighted interpolation method, 21 Drucker-Prager yield criterion, 262–263 Duxseal material, 250 Dynamic instability, 483–484, 484f

E

Earthquake-induced landslides daguangbao landslide, 277–279 tangjiashan landslide, 279–281 wangjiayan landslide, 281–283 Earthquakes, Class III rock behavior, 486–489, 488f Elastic model, 172 Elastic-perfectly plastic constitutive model, 263–264, 264f Elasto-plastic theory, 373 El-Centro earthquake, 300 Electrical conductivity, 107, 110, 111f Element tests of liquefiable soils, 232 Ensemble methods, 311 Evolutionary multi-objective optimization (EMO) algorithm, 153, 154f Excess pore pressure (EPP), 295, 300–305, 301–305f Excess pore water pressure, 380–384 four-zone effective stress path, 384 over-consolidation effect, 382–384 Existing upper slope. See Braced excavation Expansive soils, 393–396, 395f Extreme gradient boosting (XGBoost), 312–314, 319–322

F

Factor of safety (FS), 134 expressions, 125 modified Hoek vs. developed Patton’s model, 128t at peak state, 144–145, 145f at residual state, 144–145, 145f, 147–148 of rock mass, 121 of rock slope, 123 upper slope face angle effect, 127f, 129 vertical seismic coefficients, 127, 128f, 129 Failure event, 19–20 Fann calcimeter pressure gauge model, 110 FEA model, 465f Finite difference method, 172–175 Finite element, 257, 292 braced excavation construction procedures, 333t

Index

Finite element (Continued) cross-sectional soil and wall profile, 331–332, 331f lateral wall deflections, 334–335, 335f numerical results and analysis, 332–336 safety global factor, 332–333, 334f wall bending moment, 334–335, 335f laterally loaded pile (LLP), 175 beam element, 175–176 spring element, 176 u-w-p formulation, 292–294 First-order reliability method (FORM), 338–340, 339f First-order Sugeno fuzzy model, 39 Fisher information matrix (FIM), 10–11 Flexible boundaries, 81, 81f Flexural rigidity, 166, 172–173 Floating wind turbine systems, 248, 252f Front end engineering design (FEED), 433–434

G

Gaussian function, 41 Generalized hyperbolic equation (GHE), 373–377 Generalized regression neural network (GRNN), 38–39, 48, 50t Genetic algorithm (GA), 9 Genetic algorithm-based multi-objective optimization method, 153–155 Geostatistical methods, 21 Geostudio, 140 Geosynthetic reinforced soil foundation bearing capacity coefficients, 156 footing on, 155, 155f limit analysis, 152 methodology, 155–158 Nondominated Sorting Algorithm (NSGA-II), 157–158 objective functions, 158 Pareto solutions, 158–159 tensile strength of reinforcement, 158 total cost vs. BCR value, 159f of angle of internal friction, 161f for different number of layers, 160f for different tensile strength, 160–163, 162f for unit cohesion of soil, 161f for unit weight of soil, 160, 162f Geotechnical centrifuge, 208–209, 209f, 244–245 Global sensitivity analyses (GSAs), 6, 11–12 Gradient-based algorithm, 9

495

Gradient descent method, 40–41 Granular column collapse, 258–259, 273 Granular materials, 84, 90 energy dissipation, 96 stress transmission, 88–89 Gravity-based offshore platform, 453–454, 454t Grid search (GS) methods, 315–316 Ground improvement methods, 102

H

Hard rocks, 473 deep mines, 489–490, 490f fan-hinged rupture mechanism, 481–483, 482f high σ3, 474f, 480–481, 480f laboratory specimens, 484–486, 485f postpeak behavior of, 474f shallow earthquakes, 486–489, 488f shear rupture rockbursts, 489–490, 490f Hertzian fracturing, 186 Heuristic method, 66–68 Hierarchically block geomedium continuous media models, 351–352 inhomogeneities, 348–350 mathematical models, 350–351 mesovolume, 354–355 scale levels, 352–354 structural levels shear-tensile cracks, 359–362, 360–361t shear-tensile macrocracks, 356–359 Hoek’s model, 122–123, 129–130 Hybrid learning algorithm, 40–41 Hywind type floating wind turbine system failure modes, 251–252f foundation failure mechanism of, 248

I

Incompatibility function, 66 Heuristic method for, 66–68 rock mass damage, 71 Infinite element method displacements and pore pressure, 296, 298t 1-D infinite element, 295–296, 296f 2-D infinite element, 296, 297f Infinitely long pile failure mechanism, 166 Influence factors, 181, 181t Inhomogeneities, 348–350 Interaction problems, offshore wind turbines asymmetric tripod model, 214f behavior of, 210–212

496

Index

Interaction problems, offshore wind turbines (Continued) cyclic and dynamic loading conditions, 210 design steps, 215–218 geometric similarity, 215 issues, 213 loads acting on, 209–210, 211f mass similarity, 214–215 multipod foundation wind turbines, 214f scaled model tests, 216–217, 217f soil-structure interaction (SSI), 213 vibration modes, 213–215 Internal stresses, 68–69

K

Karhunen-Loe`ve expansion method, 22 Kernel function, 259–260 K-nearest neighbor (KNN), 4, 28, 30–32 Kriging, 21

L

Lagrangian particle methods, 258 Landslide daguangbao, 277–279 reactivation mechanisms pore water pressure profiles, 146f seepage force, 146 soil suction profiles, 145 suction decrease by rainfall infiltration, 145–146 tangjiashan, 279–281 wangjiayan, 281–283 Laser particle sizing technique, 192 Lateral load test, 46t batter angle effect on pile group actual vs. predicted lateral load, 50, 51f ANOVA single factor test, 50–51 parametric analysis, 53 sensitivity analysis, 51–52, 52t model pile, 44 model tank setup, 42, 43f parameter details, 46t pile caps, 44, 45f sand properties, 43t soil used for, 43 testing procedure, 44–45 training and testing data set for, 46–48, 48t Laterally loaded pile (LLP) finite difference method, 172–175 finite element method, 175 problem of, 171 simplified finite element analysis, 175

beam element, 175–176 spring element, 176 three-dimensional finite element analysis continuum element, 177–178 equivalent nodal force vector, 179 interface element, 178–179 validation, 179–181 Lateral swelling pressure matric suction reduction, 397–402 unsaturated condition, 402–406 Lateral wall deflections, 334–335, 335f Leap-Frog (LF) scheme, 270 Least square method, 40–41 Limit equilibrium method, 120–121, 123–126 Limit state functions serviceability limit state, 337–338 ultimate limit state, 336–337 Liquefaction, 291 finite element method, 292–294 infinite element displacements and pore pressure, 296, 298t 1-D infinite element, 295–296, 296f 2-D infinite element, 296, 297f numerical study, 300–305, 301–305f problem, 298–300 sloping ground conditions, 368–370 T-sand model background of, 370–371 cyclic stress-strain response, 378–380 excess pore water pressure generation, 380–384 initial static shear effects, 377–378 isotropic consolidation process, 372–373 model parameters, 384 nonlinear stress–strain, 373–380 torsional simple shear tests, 384–387 Toyoura sand cyclic resistance, 387–388 unbounded domain, 294–296 LLP. See Laterally loaded pile (LLP) Load transfer curve method field investigation case study, 417–423 model pile test, 413–417 pile base resistance and settlement, 411 pile-soil relative displacement, 409–411 theoretical analysis, 406–409 traditional load transfer curve method, 411–413 Localized elevated pore pressures, in slope stability problems, 184

Index

Local sensitivity analyses, 6 Logarithmic regression (LR), 329 Long elastic pile, 168t Long flexible pile, 171–172

M

Machine learning, 4, 25–32 accuracy of, 31–32, 32f algorithms, 310–311 K-nearest neighbor (KNN), 28, 30–32 model configuration, 29–32 support vector method, 28 Macrocrack development models, 71 Masing’s rule, 378–380 Material point method (MPM), 258 Mathematical models, 350–351 Mean absolute percentage error (MAPE), 317–319 Mean square error (MSE), 314 Mechanically stabilized soil, 152 Mechanized tunneling simulation procedure, 1–2 model adaptation framework for machine learning methods, 25–32 optimal experimental design, 10–19 reliability analysis, 19–25 uncertainty quantification, 4–9 model identification and validation, 5f Membership function, 41 Meshfree particle methods, 258 Metamodel, 7 Metropolis-Hastings algorithm, 3 Michaelis-Menten kinetics equation, 104 MICP. See Microbially induced calcite precipitation (MICP) Microbially induced calcite precipitation (MICP) bacterial growth, 104 biogeochemical mechanism of, 103 challenges, 111–114 description, 102 S. pasteurii, 102 temperature limit, 105 urea hydrolysis, 104 urease activity, 104 variation of, 111–114 Midspan collapse, of pile-supported bridges, 228–229, 228b, 229f Mohr-Coulomb shear strength criteria, 121, 123 Monopile-caisson hybrid foundation, 208 Morgenstern-Price method, 144

497

Mudline bending moment, in monopile, 247f Multilayer perceptron (MLP), 317 Multi-objective optimization algorithm, 153, 154f, 157–158

N

Natural strain localization bands in rock, 186f Navier-Stokes equations, 259 Nearest neighboring particles (NNPs) algorithm, 261, 262f Neural network (NN), 42, 48, 50t Non-cohesive soil column collapse, 271–273 Nondominated Sorting Algorithm (NSGA-II), 157–158 Non-Euclidean model classical model transition to, 64–66 of continuous medium, 63–64 and internal stresses, 68–69 rock mass damage description, 69–73 stress function of, 68–69 Nonlinear elastic contact models, 87 Norwegian soft clay, 190 Nuclear island concrete structure, 454–457, 455–457t Numerical integration procedures, 87–88 Numerical modeling, of biocemented soil behavior. See Biocemented soil behavior Numerical oscillation, SPH, 264–266

O

Oblique load test, 47t angle of oblique load effect on pile group actual vs. predicted oblique load, 53–54, 54f ANOVA single factor test, 55 Gaussian membrane functions, 54, 55f parametric analysis, 56 sensitivity analysis, 55, 56t model pile, 44 model tank setup, 42, 43f parameter details, 46t pile caps, 44, 45f sand properties, 43t soil used for, 43 testing procedure, 44–45 training and testing data set for, 48, 49t Offshore wind turbine (OWT), 457–462, 458f, 461f, 464f asymmetric tripod model, 214f behavior of, 210–212

498

Index

Offshore wind turbine (OWT) (Continued) cyclic and dynamic loading conditions, 210 design steps, 215–218 geometric similarity, 215 issues, 213 loads acting on, 209–210, 211f mass similarity, 214–215 multipod foundation wind turbines, 214f scaled model tests, 216–217, 217f soil-structure interaction (SSI), 213 vibration modes, 213–215 Optimal experimental design (OED), 2, 10 examples, 14–19 methods, 11–14 Milan Metro project, 14–17 spatial sensitivity method, 17–19 Optimization techniques, 7–9 Oso landslide description, 193–195 features of, 194f strain localization effect on, 195–198 Over-consolidation effect, 382–384 Overturning moment, in foundation, 246

P

Particle swarm optimization (PSO), 9 Pattern recognition methods, 3–4, 26, 27f Patton’s shear strength graph, 122f Peak shear strength (PSS), 134, 144 Periodic boundaries, 81–82, 81f Periscopic method, 62 pH, 106–107, 110–111 Physical mesomechanics, 351–352 Physical modeling, 205–206, 254 advantages, 206–208 failure/collapse mechanisms, 206, 207f geotechnical and complex interaction problems, 208–209 limitations, 240–242 of offshore wind turbine asymmetric tripod model, 214f behavior of, 210–212 cyclic and dynamic loading conditions, 210 design steps, 215–218 foundations, 242–248 geometric similarity, 215 issues, 213 loads acting on, 209–210, 211f mass similarity, 214–215 multipod foundation wind turbines, 214f

scaled model tests, 216–217, 217f soil-structure interaction (SSI), 213 vibration modes, 213–215 of pile-supported structure collapse, 218–232 of pipeline crossing seismic faults, 232–242 similitude (similarity) rules, 238–239 of soil-pipe interaction problems, 235–236 testing procedure, 212–213, 212f for verification of pile buckling, 223–230 Physical phenomenon, 291 Pile behavior modeling, 166–168 See also specific Pile expansive soils, 393–396 lateral swelling pressure matric suction reduction, 397–402 unsaturated condition, 402–406 load transfer curve method field investigation case study, 417–423 model pile test, 413–417 pile base settlement, 411 pile-soil relative displacement, 409–411 theoretical analysis, 406–409 traditional load transfer curve method, 411–413 unsaturated soil, 404–406, 406f Pile buckling, 223–230, 223f Pile flexibility factor, 181, 181t Pile foundations, 165–166, 394 Pile-soil relative displacement, 409–411 Pile-supported bridges, midspan collapse of, 228–229, 228b, 229f Pile-supported structure collapse, in liquefiable soils failure modes, 219 nondimensional groups, derivation of, 220 pile loading stages, 219–220, 220f postearthquake survey, 218–220 Pipe bending zone, 242 Pipe diameter to wall thickness (D/t), 241 Pipe end conditions, 241 Plane rock slope failure, 120–121, 121f, 126 Pluviation technique, 240f Polyhedra, 82, 83f Polynomial regression (PR) assessment, 329–330, 330t first-order reliability method, 338–340, 339f reliability index, 338–340

Index

Polynomial regression (PR) assessment (Continued) serviceability limit state, 340–343 ultimate limit state, 340 Population-based evolutionary multiobjective optimization, 153 Pore pressure and strain localization, 191–193 Pore water pressure build-up, 183–184 Porous medium, 292 Postpeak behavior, Class III deep mines, 489–490, 490f dynamic instability, fan mechanism, 483–484, 484f earth crust, 483–484 experimental results, 476–479, 476–478f fan-hinged rupture mechanism, 481–483, 482f hard rocks, high σ3, 474f, 480–481, 480f laboratory specimens, 484–486, 485f shallow earthquakes, 486–489, 488f shear rupture rockbursts, 489–490, 490f stress-displacement curves, 474f, 475 stress-strain curves, 476, 476f Probability density function (PDF), 329–330 p-y concept, 165–166 p-y curves, 231–232

R

Raft foundations, 449–452, 451–453f, 453t Random field concept, 2–3 Random field theory, 21 Random forest (RF), 311, 314–315 Random search (RS) methods, 315–316 Rankine’s theory, 403–404 Red trigger, 466 Reductionist approach, 253–254 Regression analyses, 21 Reinforced soil. See Geosynthetic reinforced soil foundation Residual shear strength (RSS), 134, 136 matric suction contribution, 146–147 of broken rock and bedrock, 144 of saturated soils, 134 of unsaturated soils, 134–135, 144 Reverse fault, 1-g physical model of, 235f Rigid boundaries, 81, 81f Rock mass damage failure criterion, 71–72 nonclassical boundary conditions, 70–71 stress field, 69–70 Rock material, 185–188

499

Rock mechanics, 348–350 Rock slope failure type, 120 geometry of, 123, 124f plane failure of, 120–121, 121f, 126 stability analysis factor of safety, 120 probabilistic method, 120 tension crack depth of water effect, 127, 127f, 129 at slope face, 125 in upper slope face, 125 Rock-soil mixtures, 188–191 Root mean square error (RMSE), 317–319 Runge-Kutta (RK) schemes, 270

S

Saint-Venant compatibility condition, 64–65 Sand biogeochemical properties bio-cementation effects on, 106 electrical conductivity, 107 pH, 106–107 grain size distribution curve, 108f Sandstone, triaxial compression tests on, 192 Saturated Bentheim sandstone samples, 188 Scaled model tests at 1-g (single gravity), 208 for pipelines crossing seismic faults, 233–235 Scaling laws, 253 for centrifuge tests, 209, 210t for soil-pipe interaction under faulting, 239t Seismic coefficients, 127, 128f, 129 Seismic soil-structure interaction tests flexible laminar box containers, 250 flexible soil containers, 249 requirements, 249 SEM images, of biocemented samples, 114, 115f Semivariograms for ground settlement sensors, 31–32, 31f Sensitivity analysis, uncertainty quantification, 4–7 Sensitivity index, 11–12 Serviceability limit state (SLS), 451–452 polynomial assessment, 340–343 surrogate models, 337–338 Servo-control algorithm, 96 Shallow earthquakes, 486–489, 488f Shallow foundation design criteria, 152

500

Index

Shallow twin tunnel excavation, 23–25 Shape functions, 177 of continuum element, 177 of interface element, 178 Shear bands, 187–188 Shear strength reduction (SSR) technique, 328–329 Shear strength, subglacial tills, 189–190 Shear-tensile macrocracks, 356–359, 356f Shear testing of discontinuities, 121, 122f of Saw-tooth specimen, 121–122, 122f Short rigid piles, 166–167, 168t soil reactions and bending moments for in cohesionless soil, 169, 170–171f in cohesive soil, 168, 169–170f ultimate lateral resistance of, 168–171, 170f Sigma-point method, 10–11 Similitude (similarity) rules, 238–239 Single degree of freedom (SDOF), 434, 455f, 457t Slip planes, 187f, 191 Slope stability analysis, 141 SLS. See Serviceability limit state (SLS) Small-scale models, 242 Smoothed particle hydrodynamics (SPH) advantages, 283–285 boundary condition, 268–270 constitutive model, 262–264 earthquake-induced landslides daguangbao landslide, 277–279 tangjiashan landslide, 279–281 wangjiayan landslide, 281–283 finite element method, 257 flowchart, 271, 272f gradient approximation formulations, 259–260 gradient operator, 260–261 limitations of, 283–285 nearest neighboring particles, 261, 262f soil column collapse artificial viscosity effect, 273–274 cohesive soil column collapse, 274–276 non-cohesive, 271–273 stability-related corrections numerical oscillation and artificial viscosity, 264–266 tensile instability and artificial stress, 266–268 time integration, 270–271 Soft sensitive clays, 309 Soil behavior, DEM, 91–93

Soil column collapse artificial viscosity effect, 273–274 cohesive soil column collapse, 274–276 non-cohesive, 271–273 Soil constitutive models, 432–433 Soil liquefaction effects, 224 Soil reinforcement, 152 Soils challenges associated with, 102 stabilization procedure, 103 stress dependency and nonlinearity, 240 unsaturated physical properties of, 139t shear strength envelopes of, 138–139, 139f Soil-structure interaction (SSI), 434 amber trigger level, 466 analysis, 466–467 background, 431–432 current material models, 433–434 deep basements, 445–449, 448f, 449–450t, 450f deep underground tunnels, 445, 447f gravity-based offshore platform, 453–454, 454t monitoring frequency, 464 nuclear island concrete structure, 454–457, 455–457t offshore wind turbines, 457–462, 458f, 461f, 464f raft foundations, 449–452, 451–453f, 453t red trigger level, 466 retaining walls, 429, 466 risk-based approach, 462–464 simple hand calculations, 434–435 soil constitutive models, 432–433 subsurface underground tunnels, 440–445, 443–444f, 444–446t tabletop steam turbine foundation, 435–440 Soil support element stiffness matrix, 176 Soil-water characteristic curve (SWCC), 134–135, 405 features of, 135, 135f parameters of, 143t Specimen generation approach, 83–84 SPH. See Smoothed particle hydrodynamics (SPH) Spheres and clumps, 82, 83f Sphero-simplices, 82, 83f Sporosarcina pasteurii, 102–103 SSI. See Soil-structure interaction (SSI)

Index

Stability-related corrections artificial stress, 266–268 artificial viscosity, 264–266 numerical oscillation, 264–266 tensile instability, 266–268 State-dependent model, 371 Steady-state seepage analysis, 141 Steam turbine foundation, 436f degree of freedom, 435–437, 437f impedance matrices structural model, 437–440, 439f, 440–442t piles dynamic impedances, 437, 438–439f Stiffness factors, pile behavior, 166 Stiff over-consolidated clay, 167t Stochastic optimization techniques, 7–9 Strain localization, 183 in deformation bands, 192 on Oso landslide activity, 195–198 and pore pressure, 191–193 in rock material, 185–188 in rock-soil mixtures, 188–191 Strain-softening behavior, 134 Strength reduction method, 120 Stress-dilatancy equation, 371, 382–383 Subglacial tills, deformation, 189–190 Subgrade reaction model, of soil behavior, 172 Submarine landslides, 192–193 Subsoil parameter uncertainty, 2–3 Subsurface underground tunnels, 440–445, 443–444f, 444–446t Superquadric equations, 82, 83f Support vector machine (SVM), 4, 28 Surrogate models, 7 serviceability limit state, 337–338 ultimate limit state, 336–337

T

Tangjiashan landslide, 279–281 Technology readiness level (TRL), 242, 243t Tensile instability, SPH cohesive soil column collapse, 274–276 stability-related corrections, 266–268 Terzaghi equation, 155 Testing methods, for offshore wind turbines, 244–245 Three-dimensional finite element analysis of laterally loaded pile (LLP) continuum element, 177–178 equivalent nodal force vector, 179 interface element, 178–179 validation, 179–181

501

Time series analysis methodology, 22 Time Series FeatuRe Extraction based on Scalable Hypothesis algorithm (TSFRESH), 29–30 Torsional shear tests, 372–373 Torsional simple shear (TSS) device, 367–368, 384–387 Toyoura sand cyclic resistance ratio, 387–388 torsional simple shear, 384–387 T-sand model parameters, 385t Transient seepage analysis, 141 Triangular function, 41 T-sand model background of, 370–371 cyclic stress-strain response, 378–380 excess pore water pressure, 380–384 initial static shear effects, 377–378 isotropic consolidation process, 372–373 model parameters, 384 nonlinear stress–strain, 373–380 state-dependent model, 371 torsional simple shear tests, 384–387 Toyoura sand cyclic resistance, 387–388 Tunnel boring machine (TBM) advancement, 2, 14–17 Tylor series, 260

U

Ultimate limit state (ULS) polynomial assessment, 340 surrogate models, 336–337 Uncertainty quantification back analysis, 7–9, 8f sensitivity analysis, 4–7 Undrained shear strength Bayesian hyperparameter optimization, 315–316 comparison models, 317 database, 316 extreme gradient boosting, 312–314, 319–322 performance measures, 317–318 random forest, 314–315 root mean square error, 318–319 Undrained torsional shear, 368, 377 Unsaturated soils, 404–406, 406f numerical modeling assumptions, 140 boundary conditions, 141–142 coefficient of permeability functions, 143f

502

Index

Unsaturated soils (Continued) material properties, 142–144 shear strength parameters, 142–143, 143t slope stability analysis, 141 soil-water characteristic curves, 143t steady-state seepage analysis, 141 transient seepage analysis, 141 site investigation studies electron spin resonance (ESR) tests, 136 material properties, 138–140 Outang landslide description, 136–137, 137f rainfall and river water level data, 137–138 study area, 136 Urea hydrolysis, 104 Urease activity, 104 Ureolysis, 102–103, 107

V

Variance-based sensitivity analysis method, 6–7

Vertical piles, 37 internal resistance, 166 Voronoi tessellation approach, 81

W

Wangjiayan landslide, 281–283 Wave tank testing, 244–245 Wind tunnel tests, 244–245 Wind turbine generators (WTG), 457–458, 458f Wind turbine structure, tilting of, 248f

X

XGBoost. See Extreme gradient boosting (XGBoost)

Z

Zeroth-order Bessel and Neumann functions, 70 Zeroth-order MacDonald function, 70 Zonal disintegration of rocks, 61–63 model parameter values, 72, 72t at Nikolaevsky mine, 72, 73t