The distinct element method (DEM) is a useful tool in rock engineering to model jointed rock masses. To simulate a jointed rock mass realistically, the main challenge is to be able to capture its complex geometry which consists of blocks with various shapes and sizes, and to model the interactions between these blocks. The main contribution of this thesis is the development of novel algorithms in the DEM to model jointed rock masses, namely rock slicing procedures for block generation, and algorithms for contact detection between polygonal blocks in 2-D or polyhedral blocks in 3-D. These algorithms make use of convex optimisation techniques, for which there exist efficient solution procedures. They do not rely on conventional vertex-edge-face hierarchical data structures and tedious housekeeping algorithms. The algorithms have been verified against analytical and numerical solutions, as well as validated against experimental results published in the literature. Among those, the results of DEM simulations were compared against the experimental model tests and numerical simulations of jointed beams carried out by Talesnick et al. (2007) and Tsesarsky & Talesnick (2007) respectively. Emphasis was placed on modelling the stiffness of the block interfaces accurately, and this was accomplished by reinterpreting the laboratory data published by the investigators. The capabilities of the numerical tools are also examined and demonstrated in areas for which the DEM has found practical application. A substantial fraction of this thesis is devoted to illustrating how these tools can assist the engineer in designing support systems; for example, designing the length and spacing of rock bolts and the lining thickness for a tunnel. Algorithms to model rock bolt and lining support were implemented for this purpose. Interesting comparisons with elastic solutions for supported openings were obtained. Further, it is shown that the relative benefit of introducing more rock bolts or thicker lining can be evaluated using the numerical tools with the aid of an interaction diagram. In the final part of this thesis, the case history of the 1963 Vaiont rock slide in Italy is studied. The 2-D analyses led to useful insights concerning the influence of the reservoir water level, the rock mass strength and deformability, and the slide surface shear stiffness. 3-D analyses were undertaken to investigate the influence of the eastern boundary of the slope, and interesting insights were obtained concerning the slope kinematics. Overall, the case study shows that the tools are capable of modelling problems with specific physical and geometrical detail in both 2-D and 3-D.

En tribologie, la modélisation numérique est aujourd'hui un outil indispensable pour étudier un contact afin de pallier les limites expérimentales. Pour comprendre de mieux en mieux les phénomènes mis en jeu, les modèles ne se situent plus à une seule échelle, mais en font intervenir plusieurs, rendant plus que jamais le concept de triplet tribologique incontournable. Travaillant avec cette philosophie et en se basant sur l'approche Non Smooth Contact Dynamics, dont nous rappelons les grandes lignes, nous proposons de franchir deux cas: proposer des modèles offrant des résultats quantitatifs et mettre en place les premières pièces d'une homogénéisation au niveau du contact (VER). Dans le premier cas, l'étude du couplage éléments finis/éléments discrets au sein d'une même simulation a pour but de proposer des modèles plus "réalistes". Même si l’interface utilisée est déjà présente au coeur du contact et ne va pas évoluer, elle permet de mettre en évidence l’utilisation d’outil de mesure permettant de lier le mouvement des particules aux instabilités dynamiques et permet d’avoir des résultats qualitatifs mais aussi quantitatifs puisque la comparaison avec les taux de contraintes expérimentaux sont en très bonne adéquation. Dans le second cas, le VER sous sollicitations tribologiques est étudié afin d'étendre les techniques d'homogénéisation aux problèmes de contact afin de s'affranchir de la description des interfaces aux grandes échelles en trouvant un moyen d'homogénéiser le comportement hétérogène de l'interface et de le faire dialoguer avec le comportement continu des corps en contact en faisant remonter, dans un sens, des grandeurs moyennées à l'échelle microscopique à l'échelle macroscopique des premiers corps et dans l'autre sens, se servir des données locales à l'échelle macroscopique comme conditions limites à l'échelle microscopique. / In tribology, the numerical modeling has become an indispensable tool for studying a contact to overcome the experimental limitations. To have a better understanding of the phenomena involved, the models are no longer located at a single scale, but involve several ones, more than ever, making the concept of tribological triplet as a unavoidable concept. Working with this philosophy and approach based on the Non Smooth Contact Dynamics framework, which we remind some outlines, we propose to cross two steps~: model that can offer quantitative results and that implement the first ingredient to perform a homogenization at a contact level. In the first case, the study of coupling finite elements/discrete elements within the same simulation aims to propose models that are more "realistic". Even if the interface is already present in the contact and not going to evolves, it can highlight the use of measurement tool of spot particles via dynamic instabilities and allows to have not only qualitative results but also quantitative ones since the comparison with the experimental strain rates are in very good agreement. In the second case, the study of VER in tribological charges is performed to extend the homogenization techniques to contact problems in order to overcome the interface description on large scales by finding a way to homogenize the heterogeneous behavior of the interface and make a dialogue with the continue behavior of bodies in contact by send up, in a sense, average values of the microscopic scale to the macroscopic scale and in the other sense, use local data of the macroscopic scale as boundary conditions at the microscopic scale.

In geotechnical engineering, the uncertainties such as the variability and uncertainty inherent in the geotechnical properties have caught more and more attentions from researchers and engineers. They have found that a single “Factor of Safety” calculated by traditional deterministic analyses methods can not represent the slope stability exactly. Recently in order to provide a more rational mathematical framework to incorporate different types of uncertainties in the slope stability estimation, reliability analyses and non-deterministic methods, which include probabilistic and non probabilistic (imprecise methods) methods, have been applied widely. In short, the slope non-deterministic analysis is to combine the probabilistic analysis or non probabilistic analysis with the deterministic slope stability analysis. It cannot be regarded as a completely new slope stability analysis method, but just an extension of the slope deterministic analysis. The slope failure probability calculated by slope non-deterministic analysis is a kind of complement of safety factor. Therefore, the accuracy of non deterministic analysis is not only depended on a suitable probabilistic or non probabilistic analysis method selected, but also on a more rigorous deterministic analysis method or geological model adopted. In this thesis, reliability concepts have been reviewed first, and some typical non-deterministic methods, including Monte Carlo Simulation (MCS), First Order Reliability Method (FORM), Point Estimate Method (PEM) and Random Set Theory (RSM), have been described and successfully applied to the slope stability analysis based on a numerical simulation method-Strength Reduction Method (SRM). All of the processes have been performed in a commercial finite difference code FLAC and a distinct element code UDEC. First of all, as the fundamental of slope reliability analysis, the deterministic numerical simulation method has been improved. This method has a higher accuracy than the conventional limit equilibrium methods, because of the reason that the constitutive relationship of soil is considered, and fewer assumptions on boundary conditions of slope model are necessary. However, the construction of slope numerical models, particularly for the large and complicated models has always been very difficult and it has become an obstacle for application of numerical simulation method. In this study, the excellent spatial analysis function of Geographic Information System (GIS) technique has been introduced to help numerical modeling of the slope. In the process of modeling, the topographic map of slope has been gridded using GIS software, and then the GIS data was transformed into FLAC smoothly through the program built-in language FISH. At last, the feasibility and high efficiency of this technique has been illustrated through a case study-Xuecheng slope, and both 2D and 3D models have been investigated. Subsequently, three most widely used probabilistic analyses methods, Monte Carlo Simulation, First Order Reliability Method and Point Estimate Method applied with Strength Reduction Method have been studied. Monte Carlo Simulation which needs to repeat thousands of deterministic analysis is the most accurate probabilistic method. However it is too time consuming for practical applications, especially when it is combined with numerical simulation method. For reducing the computation effort, a simplified Monte Carlo Simulation-Strength Reduction Method (MCS-SRM) has been developed in this study. This method has estimated the probable failure of slope and calculated the mean value of safety factor by means of soil parameters first, and then calculated the variance of safety factor and reliability of slope according to the assumed probability density function of safety factor. Case studies have confirmed that this method can reduce about 4/5 of time compared with traditional MCS-SRM, and maintain almost the same accuracy. First Order Reliability Method is an approximate method which is based on the Taylor\'s series expansion of performance function. The closed form solution of the partial derivatives of the performance function is necessary to calculate the mean and standard deviation of safety factor. However, there is no explicit performance function in numerical simulation method, so the derivative expressions have been replaced with equivalent difference quotients to solve the differential quotients approximately in this study. Point Estimate Method is also an approximate method involved even fewer calculations than FORM. In the present study, it has been integrated with Strength Reduction Method directly. Another important observation referred to the correlation between the soil parameters cohesion and friction angle. Some authors have found a negative correlation between cohesion and friction angle of soil on the basis of experimental data. However, few slope probabilistic studies are found to consider this negative correlation between soil parameters in literatures. In this thesis, the influence of this correlation on slope probability of failure has been investigated based on numerical simulation method. It was found that a negative correlation considered in the cohesion and friction angle of soil can reduce the variability of safety factor and failure probability of slope, thus increasing the reliability of results. Besides inter-correlation of soil parameters, these are always auto-correlated in space, which is described as spatial variability. For the reason that knowledge on this character is rather limited in literature, it is ignored in geotechnical engineering by most researchers and engineers. In this thesis, the random field method has been introduced in slope numerical simulation to simulate the spatial variability structure, and a numerical procedure for a probabilistic slope stability analysis based on Monte Carlo simulation was presented. The soil properties such as cohesion and friction angle were discretized to continuous random fields based on local averaging method. In the case study, both stationary and non-stationary random fields have been investigated, and the influence of spatial variability and averaging domain on the convergence of numerical simulation and probability of failure was studied. In rock medium, the structure faces have very important influence on the slope stability, and the rock material can be modeled as the combination of rigid or deformable blocks with joints in distinct element method. Therefore, much more input parameters like strength of joints are required to input the rock slope model, which increase the uncertainty of the results of numerical model. Furthermore, because of the limitations of the current laboratory and in-site testes, there is always lack of exact values of geotechnical parameters from rock material, even the probability distribution of these variables. Most of time, engineers can only estimate the interval of these variables from the limit testes or the expertise’s experience. In this study, to assess the reliability of the rock slope, a Random Set Distinct Element Method (RS-DEM) has been developed through coupling of Random Set Theory and Distinct Element Method, and applied in a rock slope in Sichuan province China.

In geotechnical engineering, the uncertainties such as the variability and uncertainty inherent in the geotechnical properties have caught more and more attentions from researchers and engineers. They have found that a single “Factor of Safety” calculated by traditional deterministic analyses methods can not represent the slope stability exactly. Recently in order to provide a more rational mathematical framework to incorporate different types of uncertainties in the slope stability estimation, reliability analyses and non-deterministic methods, which include probabilistic and non probabilistic (imprecise methods) methods, have been applied widely. In short, the slope non-deterministic analysis is to combine the probabilistic analysis or non probabilistic analysis with the deterministic slope stability analysis. It cannot be regarded as a completely new slope stability analysis method, but just an extension of the slope deterministic analysis. The slope failure probability calculated by slope non-deterministic analysis is a kind of complement of safety factor. Therefore, the accuracy of non deterministic analysis is not only depended on a suitable probabilistic or non probabilistic analysis method selected, but also on a more rigorous deterministic analysis method or geological model adopted. In this thesis, reliability concepts have been reviewed first, and some typical non-deterministic methods, including Monte Carlo Simulation (MCS), First Order Reliability Method (FORM), Point Estimate Method (PEM) and Random Set Theory (RSM), have been described and successfully applied to the slope stability analysis based on a numerical simulation method-Strength Reduction Method (SRM). All of the processes have been performed in a commercial finite difference code FLAC and a distinct element code UDEC. First of all, as the fundamental of slope reliability analysis, the deterministic numerical simulation method has been improved. This method has a higher accuracy than the conventional limit equilibrium methods, because of the reason that the constitutive relationship of soil is considered, and fewer assumptions on boundary conditions of slope model are necessary. However, the construction of slope numerical models, particularly for the large and complicated models has always been very difficult and it has become an obstacle for application of numerical simulation method. In this study, the excellent spatial analysis function of Geographic Information System (GIS) technique has been introduced to help numerical modeling of the slope. In the process of modeling, the topographic map of slope has been gridded using GIS software, and then the GIS data was transformed into FLAC smoothly through the program built-in language FISH. At last, the feasibility and high efficiency of this technique has been illustrated through a case study-Xuecheng slope, and both 2D and 3D models have been investigated. Subsequently, three most widely used probabilistic analyses methods, Monte Carlo Simulation, First Order Reliability Method and Point Estimate Method applied with Strength Reduction Method have been studied. Monte Carlo Simulation which needs to repeat thousands of deterministic analysis is the most accurate probabilistic method. However it is too time consuming for practical applications, especially when it is combined with numerical simulation method. For reducing the computation effort, a simplified Monte Carlo Simulation-Strength Reduction Method (MCS-SRM) has been developed in this study. This method has estimated the probable failure of slope and calculated the mean value of safety factor by means of soil parameters first, and then calculated the variance of safety factor and reliability of slope according to the assumed probability density function of safety factor. Case studies have confirmed that this method can reduce about 4/5 of time compared with traditional MCS-SRM, and maintain almost the same accuracy. First Order Reliability Method is an approximate method which is based on the Taylor\'s series expansion of performance function. The closed form solution of the partial derivatives of the performance function is necessary to calculate the mean and standard deviation of safety factor. However, there is no explicit performance function in numerical simulation method, so the derivative expressions have been replaced with equivalent difference quotients to solve the differential quotients approximately in this study. Point Estimate Method is also an approximate method involved even fewer calculations than FORM. In the present study, it has been integrated with Strength Reduction Method directly. Another important observation referred to the correlation between the soil parameters cohesion and friction angle. Some authors have found a negative correlation between cohesion and friction angle of soil on the basis of experimental data. However, few slope probabilistic studies are found to consider this negative correlation between soil parameters in literatures. In this thesis, the influence of this correlation on slope probability of failure has been investigated based on numerical simulation method. It was found that a negative correlation considered in the cohesion and friction angle of soil can reduce the variability of safety factor and failure probability of slope, thus increasing the reliability of results. Besides inter-correlation of soil parameters, these are always auto-correlated in space, which is described as spatial variability. For the reason that knowledge on this character is rather limited in literature, it is ignored in geotechnical engineering by most researchers and engineers. In this thesis, the random field method has been introduced in slope numerical simulation to simulate the spatial variability structure, and a numerical procedure for a probabilistic slope stability analysis based on Monte Carlo simulation was presented. The soil properties such as cohesion and friction angle were discretized to continuous random fields based on local averaging method. In the case study, both stationary and non-stationary random fields have been investigated, and the influence of spatial variability and averaging domain on the convergence of numerical simulation and probability of failure was studied. In rock medium, the structure faces have very important influence on the slope stability, and the rock material can be modeled as the combination of rigid or deformable blocks with joints in distinct element method. Therefore, much more input parameters like strength of joints are required to input the rock slope model, which increase the uncertainty of the results of numerical model. Furthermore, because of the limitations of the current laboratory and in-site testes, there is always lack of exact values of geotechnical parameters from rock material, even the probability distribution of these variables. Most of time, engineers can only estimate the interval of these variables from the limit testes or the expertise’s experience. In this study, to assess the reliability of the rock slope, a Random Set Distinct Element Method (RS-DEM) has been developed through coupling of Random Set Theory and Distinct Element Method, and applied in a rock slope in Sichuan province China.

Η παρούσα διατριβή επανεξετάζει την οριακή ανάλυση ευστάθειας των λίθινων αψίδων. Η οριακή ανάλυση ευστάθειας χρησιμοποιείται σήμερα ως το βασικό εργαλείο αποτίμησης της ευστάθειας τόξων και θολωτών κατασκευών από τοιχοποιία, όπως ακριβώς συνέβαινε και τους τελευταίους τέσσερις αιώνες. Παρά την τόσο μακρόχρονη ιστορία της μεθόδου, δεν έχουν πλήρως διασαφηνιστεί στην επιστημονική κοινότητα θεμελιώδης έννοιες και δεν έχουν σαφώς απαντηθεί ερωτήματα όπως: Ποιες είναι οι φυσικά πραγματοποιήσιμες γραμμές ώθησης και ποιες όχι; Ποια είναι η επίδραση της στερεοτομίας ενός τόξου στην οριακή του ευστάθεια; Ποιος είναι ο ρόλος της αλυσοειδούς καμπύλης και κατά πόσο αυτή είναι μία φυσικά αποδεκτή γραμμή ώθησης; Τι σχέση υπάρχει ανάμεσα στην κλίση της συνισταμένης θλιπτικής δύναμης και στην κλίση της γραμμής ώθησης στο σημείο εφαρμογής της; Η παρούσα διατριβή αναζητά απαντήσεις στα ερωτήματα αυτά, και έχει ως στόχο τη βαθύτερη κατανόηση της οριακής ανάλυσης ευστάθειας των τόξων, με παράλληλη ανάδειξη νέων υπολογιστικών διαδικασιών. Η δομή της παρουσιάζεται συνοπτικά παρακάτω. Στο πρώτο κεφάλαιο γίνεται ιστορική ανάλυση της μεθόδου μέσα από παρουσίαση και σχολιασμό των εργασιών με τη σημαντικότερη συμβολή, από τα μέσα του 17ου αιώνα μέχρι σήμερα. Στο δεύτερο κεφάλαιο επανεξετάζεται ένα από τα πιο κλασικά προβλήματα της μηχανικής: ποιο είναι το ελάχιστο επιτρεπτό πάχος ενός ημικυκλικού τόξου υπό τη δράση του ιδίου βάρους του για να είναι ευσταθές. Παράλληλα απαντώνται τα ερωτήματα που τέθηκαν παραπάνω αναπτύσσοντας νέες κλειστές μαθηματικές εκφράσεις των γραμμών ώθησης μέσω γεωμετρικής προσέγγισης, αλλά και μέσω του λογισμού των μεταβολών. Στο τρίτο κεφάλαιο χρησιμοποιείται παρόμοια διαδικασία για την ανάλυση της γενικής περίπτωσης των ελλειπτικών τόξων, οποιουδήποτε γεωμετρικού λόγου ύψος προς βάση, καθώς δεν είναι διαθέσιμα αναλυτικά αποτελέσματα στη διεθνή βιβλιογραφία, όπως συμβαίνει για τα κυκλικά τόξα. Στο τέταρτο κεφάλαιο εξετάζεται η οριακή ευστάθεια κυκλικών τόξων οποιασδήποτε γωνίας εναγκαλισμού, υπό την ταυτόχρονη δράση του ιδίου βάρους τους και σταθερής οριζόντιας εδαφικής επιτάχυνσης, ενώ υπολογίζεται με ακρίβεια η μορφή που θα έχει ο επικείμενος μηχανισμός κατάρρευσης μαζί με το οριακό πάχος, συναρτήσει της σεισμικής φόρτισης. Τα αποτελέσματα της μαθηματικής ανάλυσης (Κεφ. 2-4) επιβεβαιώνουν την ακρίβεια του λογισμικού που αναπτύχθηκε για τις ανάγκες της διατριβής, καθώς και τα αποτελέσματα που προκύπτουν από εμπορικό λογισμικό της μεθόδου των διακριτών στοιχείων. Στο πέμπτο κεφάλαιο γίνεται εφαρμογή και σύγκριση των πιο αντιπροσωπευτικών υπολογιστικών μεθόδων που απαντώνται σήμερα στη βιβλιογραφία για την αποτίμηση της ευστάθειας και φέρουσας ικανότητας της υπόγειας Θολωτής Διόδου του Σταδίου της Αρχαίας Νεμέας, ενώ η οριακή ανάλυση ευστάθειας αναδεικνύεται ως ένα μοναδικό εργαλείο για την κατανόηση της αλληλεπίδρασης της κατασκευής με το περιβάλλον έδαφος. Επιπλέων των συμπερασμάτων στο τέλος κάθε κεφαλαίου (Κεφ. 2 έως 5), στο έκτο κεφάλαιο παρουσιάζονται τα πιο σημαντικά συμπεράσματα και η συνεισφορά της παρούσας διατριβής. / This doctoral thesis revisits the limit equilibrium analysis of masonry arches. Limit equilibrium analysis is used today as the main analysis method for the assessment of the stability of masonry arches and vaulted structures, and is the outcome of important contributions that happened during the last four centuries. Although this method has a long history and a rich literature, there are still fundamental concepts that have not been thoroughly clarified, such as: What are the physically admissible thrust lines of an arch? How the stereotomy of an arch affects its limit stability? What is the role of the catenary curve (the alysoid)? Is the catenary curve a physically admissible thrust line? What is the relation between the direction of the thrust force and the slope of the thrust line at the point of application of the force? This thesis investigates these questions and aims to a better understanding of the limit equilibrium analysis of masonry arches, and at the same time, to present innovative methodologies and new analysis tools. Chapter 1 presents the work of other authors that have contributed the most to the stability analysis of masonry arches and vaulted structures over the last centuries. Chapter 2 revisits one of the most classical problems of Mechanics—what is the minimum thickness of a semicircular masonry arch subjected to its own weight. At the same time, the analysis presented in this chapter answers to the aforementioned questions through the development of closed-form expressions of the thrust line and the application of calculus of variation. Chapter 3 is focused on the limit equilibrium state of elliptical masonry arches, using the same approaches that were used in Chapter 2. This analysis was motivated from the fact that numerical results have been available in literature only for circular and not for elliptical masonry arches. Chapter 4 computes the location of the imminent hinges and the minimum thickness of circular masonry arches, for every given embrace angle, which can just sustain their own weight, together with a given level of horizontal ground acceleration. The numerical results presented in Chapters 2 to 4 confirm the accuracy of the in-house software that was developed for the needs of this thesis and the results obtained with a representative, commercially available software of the distinct element method. Chapter 5 present a comprehensive structural analysis of the Tunnel-Entrance to the Stadium of Ancient Nemea which ranges from the thrust line limit analysis and the discrete element method, to a 3-dimensional finite-element analysis. Limit equilibrium analysis emerges as a unique analysis method for the assessment of the stability of the structure and its interaction with the surrounding soil. While at the end of every chapter (Chapters 2 to 5) are presented detailed comments and conclusions, Chapter 6 is focused on outlining the most important conclusions and the main contribution of this thesis.