Computing the collapse load and identifying the associated mechanism of block assemblage structures is an important task in historical restoration works. The research in this thesis is to contribute to the modelling and analysis of rigid block structures in dry frictional contact. The models can be applied to various structures such as masonry arches) vaults) domes, walls or piles in heritage buildings. Estimates of the collapse load are made by solving the underlying mathematical programming problems. 2-D and 3-D models are formulated and different contact assumptions (concavity and convexity) are investigated. Both associated and nonassociated flow rules are accounted for. The associative problem can be robustly solved by the bound theorems of classical plasticity theory. However, the result is often not valid. The nonassociated rule which shows up as a complementarity relationship in the governing system is the major challenge of the project. The 3-D model is more difficult to deal with as compared to the 2-D case. Proposed methods are presented to treat the nonlinear 3-D problem (Lorentz cone) by formulating it as a second-order cone problem or by piecewise linearizing the cone as a polyhedral. Various computational methods are proposed to obtain the best upper bound collapse load for the nonassociative model. One method formulates some extended complementarity problems that are able to restrict the domain of the collapse load variable to search for better solutions. The best method uses nonlinear programming) specifically mathematical programming with equilibrium constraints problems) to attempt to directly minimize the collapse load. Some enumerative schemes are also attempted to map out all the nonassociative solutions but proved to be computationally expensive. Various 2-D and 3-D examples are demonstrated for several different types of structures.

This study presents an upper bound plastic limit analysis for predicting drag anchor trajectory and load capacity. The shank and fluke of the anchor are idealized as simple plates. The failure mechanism involves the motion of the anchor about a center of rotation, the coordinates of which are systematically optimized to determine the minimum load at the shackle. For a given anchor orientation, the direction of the shackle force is varied to establish a relationship between the magnitude and direction of the shackle load. Coupling this relationship to the Neubecker-Randolph anchor line solution produces a unique solution for the magnitude and orientation of the shackle force. The anchor is then advanced a small increment about the optimum center of rotation and the process is repeated. The upper bound method (UBM) provides a practical means to determine the trajectory of the anchor and the anchor load capacity at any point in the trajectory. To better understand of the anchor behavior, extensive parameter studies were carried out varying the properties of the anchor, anchor line, and soil. The UBM show good agreement with six full-scale tests covering several different anchor types and centrifuge model tests.

[pt] O presente trabalho apresenta um estudo de problemas de estabilidade, freqüentemente,encontrados, em Engenharia Geotécnica, através da técnica da Análise Limite associada ao Método dos Elementos Finitos (MEF).Inicialmente, faz-se uma revisão das formulações da Análise Limite, via MEF,encontradas, com maior freqüência, na literatura técnica especializada.Uma formulação mista que é descrita em detalhe na tese foi escolhida para implementação. Extensões das formulações da Análise Limite de meios contínuos são propostas para contemplar características de maciços rochosos fraturados. É proposto,também, um procedimento numérico para tratar de problemas de estabilidade de meios que exibem fluxo plástico não associado.As implementações realizadas foram validadas através de problemas cujas soluções podem ser obtidas por via analítica.Finalmente, um número considerável de problemas de interesse em Engenharia Geotécnica é estudado utilizando a implementação realizada. Os resultados destes estudos sugerem a viabilidade da utilização da técnica estudada na solução de problemas práticos de Engenharia Civil. / [en] This work presents a study of stability problems often encountered in Geotechnical Engineering, through the use of Limit Analysis in conjunction with the Finite Element Method (FEM).Initially, a literature survey of the most often found formulations in Limit Analysis through the FEM is carried out.A mixed formulation of Limit Analysis was chosen for implementation and its details are fully described. Extensions of the formulation to deal with stability problems in fractured rock media are also proposed and described. A numerical procedure to take into account the effect of non associative plastic flow is proposed.The implementations carried out were validated through problems to which analytical solutions could be found.Finally, a considerable number of problems of interest to Geotechnical Engineering is studied with the implemented formulation. The results of these studies suggest that Limit Analysis can be considered as a viable tool in the solution of practical problems in Geotechnical Engineering.

This thesis provides a contribution towards a general procedure for solving robustly and efficiently limit and shakedown analyses of engineering structures within the static approach which has been chosen for its simplicity of implementation. Throughout the thesis, attempts at improving the robustness and efficiency of the computations are presented. Beginning with efforts to prevent volumetric locking, which is a severe shortcoming of traditional low order h-type displacement elements, the investigation proposes the use of the high order p-version of the finite element method. It is shown theoretically and confirmed numerically that this p-method is not only robust in preventing locking, but also provides very accurate results. However, the use of uniformly distributed high order p-elements may be computationally demanding when the size of the problem becomes large. This difficulty is tackled by two main approaches: use of a p-adaptive procedure at the elastic computation stage and use of approximate piecewise linear yield functions. The p-adaptive scheme produces a non-uniform p-distribution and helps to greatly reduce the number of degrees of freedom needed while still guaranteeing the required level of accuracy. The overall gain is that the sizes of the models are reduced significantly and hence also the computational effort. The adoption of piecewise linear yield surfaces helps to further increase the efficiency at the expense of possibly slightly less accurate, but still very acceptable, results. State-of-the-art linear programming solvers based on the very efficient interior point methodology are used. Significant gains in efficiency are achieved. A heuristic, semi-adaptive scheme to piecewise linearize the yield surfaces is then developed to further reduce the size of the underlying optimization problems. The results show additional gains in efficiency. Finally, major conclusions are summarized, and various aspects suitable for further research are highlighted.

Limit analysis based upon the theory of plasticity is one of the very useful numerical techniques to determine the failure loads of different civil and mechanical engineering structures for a material following an associated flow rule. The limiting values of the collapse loads, namely, lower and upper bounds, can be bracketed quite accurately with the application of the lower and upper bound theorems of the limit analysis. With the advancement of the finite elements and different robust optimization techniques, the numerical limit analysis approach in association with finite elements is becoming very popular to assess the stability of various complicated structures. Although two different optimization methods, namely, linear programming and nonlinear programming, have been both successfully implemented by various researchers for solving different stability problems in geomechanics, the linear programming method is employed in the present thesis due to its inherent advantage in implementation and ease in achieving the convergence. The objectives of the present thesis are (i) to improve upon the existing lower bound limit analysis method, in combination with finite elements and linear programming, with an intention of reducing the computational time and the associated memory requirement, and (ii) to apply the existing lower bound finite element limit analysis to various important planar stability problems in geotechnical engineering. With reference to the first objective of the thesis, two new methods have been introduced in this thesis to improve upon the existing computational procedure while solving the geomechanics stability problem with the usage of the limit analysis, finite elements and linear programming. In the first method, namely, the method-I, the order of the yield polygon within the chosen domain is varied, based on the proximity of the stress state to the yield, such that a higher order polygon needs not to be used everywhere in the problem domain. In the second method, the method-II, it has been intended to use only a few selected sides, but not all, of the higher order yield polygon which are being used to linearize the Mohr-Coulomb yield function. The proposed two methods have been applied to compute the ultimate bearing capacity of smooth as well as rough strip footings for various soil frictional angles. It has been noticed that both the proposed new methods reduce the CPU time and the total number of inequality constraints required as compared to the existing lower bound linear programming method used in literature. With reference to the second objective, a few important planar stability problems in geomechanics associated with interference of footings and vertical anchors have been solved in the present thesis. Footings are essentially used to transfer the compressive loads of the super structures to underlying soil media. On the other hand, vertical anchors are used for generating passive supports to retaining walls, sheet piles and bulkheads. A large number of research investigations have been reported in literature to compute the collapse load for a single isolated strip footing and a single vertical anchor. It is a common practice to estimate the bearing capacity of footings or pullout capacity of anchors without considering the effect of interference. There are, however, clear evidences from the available literature that (i) the ultimate bearing capacity of footings, and (ii) the ultimate pullout capacity of anchors, are significantly affected by their interference effect. Based on different available methods, the interference of footings, in a group of two footings as well as an infinite number of multiple footings, has been examined by different researchers in order to compute the ultimate bearing capacity considering the group effect. However, there is no research study to find the ultimate bearing capacity of interfering footings with the usage of the lower bound limit analysis. In the present thesis, the ultimate bearing capacity of two and an infinite number of multiple strip footings placed on sandy soil with horizontal ground surface, has been determined. The analysis has been performed for smooth as well as rough footings. The failure loads for interfering footings are found to be always greater than the single isolated footing. The effect of the footings' interference is expressed in terms of an efficiency factor ( ξγ); where, ξγ is defined as the ratio of the magnitude of failure load for a footing of width B in presence of the other footing to the magnitude of failure load of an isolated strip footing having the same width. The effect of the interference on the failure load (i) for rough footings becomes always greater than smooth footings, (ii) increases with an increase in soil frictional angle φ, and (iii) becomes almost negligible beyond the spacing, S > 3B. It is observed that the failure load for a footing in a group of an infinite number of multiple strip footings becomes always greater than that for two interfering footings. Attempts have been made in this thesis to investigate the group effect of two vertical anchors on their horizontal pullout resistance (PuT). The anchors are considered to be embedded at a certain clear spacing (S) along the same vertical plane. The group effect has been studied separately for anchors embedded in (i) sandy soil, and (ii) undrained clay, respectively. For anchors embedded in clays, an increase of soil cohesion with depth, in a linear fashion, has also been taken into consideration. The magnitude of PuT has been obtained in terms of a group efficiency factor, ηγ for sand and ηc for clay, with respect to the failure load for a single isolated vertical plate with the same H/B. The pullout capacity of a group of two anchors either in sand or in undrained clay becomes quite extensive as compared to a single isolated anchor. The magnitudes of ηγ and ηc become maximum corresponding to a certain critical value of S/B, which has been found to lie generally between 0.5 and 1. The value of ηγ for a given S/B has been found to become larger for greater values of H/B, φ, and δ. For greater values of H/B, the group effect becomes more significant in contributing the pullout resistance. The horizontal pullout capacity of a single isolated vertical anchor embedded in sand in the presence of pseudo static horizontal earthquake body forces has also been determined by using the lower bound finite element limit analysis. The variation of the pullout factor Fγ with changes in the embedment ratio of the smooth and rough anchor plates for different values of horizontal earthquake acceleration coefficient ( αh) has been investigated. The analysis clearly reveals that the pullout resistance decreases quite significantly with an increase in the magnitude of the earthquake acceleration coefficient. For the various problems selected in the present thesis, the failure patterns have also been exclusively drawn in order to understand the development of the plastic zones within the chosen domain for solving a given problem. The results obtained from the analysis, for the various problems taken up in this thesis, have been thoroughly compared with those reported in literature.

Les chaînes plissées d'avant-pays sont des objets géologiques qui se forment dans un contexte compressif et représentent la partie externe des orogènes.Elles sont composées de nombreuses structures plissées associées à des chevauchements généralement enracinés au sein d'un niveau de décollement peu profond situé dans la partie basale de la couverture sédimentaire. Ces objets géologiques ont été beaucoup étudiés au XXème siècle.Ils ont été modélisés cinématiquement, analogiquement et mécaniquement.S’il est indispensable de tenir compte de l’aspect mécanique en géologie structurale, les modèles mécaniques restent cependant trop peu utilisés par le géologue structuraliste.L'objectif de cette thèse est de montrer comment apporter des contraintes mécaniques à l'étude des structures géologiques, grâce à une théorie mécanique facilement utilisable en géologie structurale.Cette théorie, le calcul à la rupture (Limit Analysis), représente en effet un bon intermédiaire entre les modèles cinématiques et les modèles mécaniques en éléments finis, très complets mais relativement complexes d'utilisation.Nous étudions ici des exemples appartenant à la chaîne plissée d’avant-pays du Jura et utilisons les logiciels Optum G2 et SLAMTec.Nous procédons alors selon deux approches. La première approche, présentée en première partie de ce manuscrit consiste à étudier la déformation passée.Nous nous focalisons sur la région de Saint-Ursanne dans le Nord-Est du Jura, en Suisse et réalisons dans un premier temps une coupe géologique d'une structure que nous étendons ensuite en 3D via une série de coupes balayant l'ensemble de la zone d'étude.Pour réaliser ces coupes nous utilisons à la fois des principes d’équilibrage cinématique, des modélisations mécaniques par calcul à la rupture et des modélisations analogiques en boîte à sable.Nous montrons ainsi l’importance de l’héritage tectonique sur la mise en place des structures de cette région et modélisons mécaniquement l’influence d’un décalage du niveau de décollement par une faille normale héritée, lors d’un épisode compressif ultérieur.Nous montrons alors que ce décalage représente un générateur de failles ainsi qu’un point d’accroche qui ralenti la propagation du front de déformation, contrôlant par ailleurs la direction de la structure qui se développera.Nous montrons également que la topographie joue un rôle majeur dans la mise en place des structures géologiques.Dans la deuxième partie du manuscrit nous abordons une seconde approche qui consiste à étudier la déformation actuelle.Nous changeons cette fois-ci d’échelle pour nous intéresser à la tectonique actuelle de l’ensemble du Jura.Nos modélisations prédisent que cette chaîne plissée, formée par une tectonique de couverture, est aujourd’hui partiellement affectée par une tectonique de socle.Alors que la partie ouest du Jura serait toujours affectée par une tectonique de couverture, le socle devrait aujourd’hui être impliqué dans la déformation au niveau de la partie nord-est.Nous montrons cependant qu’une tectonique de socle n’empêche pas l’activation simultanée des niveaux de décollements situés dans la couverture (les évaporites triasiques, dans le cas du Jura).L’intérêt pratique de ces méthodes est illustré par des questions industrielles : le dernier chapitre présente une étude des coupes géologiques d’un terrain d’étude de la Nagra, situé dans l’est du Jura.Nous regardons la réponse mécanique actuelle en compression de l’interprétation structurale proposée.Les outils numériques utilisés dans l’ensemble de ce travail ont, par ailleurs, fait l’objet d’un tutoriel réalisé dans le cadre d’une mission au sein de l’entreprise Total. / Fold-and-thrust-belts form in a compressive geological context and represent the external part of orogens.They are composed of numerous folds and thrusts generally rooted in a shallow décollement level located in the basal part of the sedimentary cover.These belts have been studied extensively during the XX century. They have been modeled kinematically, analogically and mechanically.The mechanical aspect of deformation must be taken into account in structural geology, but yet, mechanical models remain underused by the structural geologist.The objective of this thesis is to show how to bring mechanical constraints to the study of geological structures with a mechanical theory easily usable in structural geology, the Limit Analysis.This theory represents a good compromise between the kinematic models and the finite element mechanical models, very complete but relatively complicated to use.We study here examples from the Jura fold-and-thrust belt and use the softwares Optum G2 and SLAMTec.We proceed with two different approaches.The first one is presented in the first part of this manuscript and consists in studying the past deformation.We focus on Saint-Ursanne area, in the North-East Switzerland Jura.We construct first a 2D geological cross section of the Mont Terri structure and then we extend this work in 3D by a series of cross sections that scan the whole Saint Ursanne area.These cross-sections are drawn using kinematical rules, mechanical modeling with Limit Analysis and analog sandbox modeling.We show the importance of the tectonic inheritance on the onset of structures in the studied region and we mechanically model the influence of a décollement offset due to an inherited normal fault during a subsequent compressive episode.We show that this offset represents a “fault generator” and a catching point that slows the propagation of the deformation front and furthermore controls the orientation of the structure that develops.We also show that the topography plays a major role in the establishment of geological structures.In the second part of the manuscript we follow a second approach that consists in studying the current deformation.We change of scale and study the current tectonics of the entire Jura.Our models predict that this belt, formed in thin-skin tectonics, is currently partially affected by thick-skin tectonics.The western Jura would be still affected by a thin-skin tectonics, but the basement should be involved in the deformation in the north-east Jura.However, we show that a thick-skin tectonics does not preclude the simultaneous activation of the shallower décollements contained by the sedimentary cover (Triassic evaporites, in the Jura).Practical interest of these methods is illustrated by industrial questions: the last chapter presents a study of geological cross-sections from a studied area of Nagra in the eastern Jura.We look at the current mechanical answer under compression of the proposed structural interpretations.The numerical tools used in this work have been the subject of a tutorial made during a mission at Total company.

[pt] É apresentada uma metodologia para a determinação da carga de colapso e uma outra para o dimensionamento ótimo em vigas de concreto armado. Para a determinação da carga de colapso a metodologia utiliza a análise limite e o modelo da Treliça Clássica de Mörsch. A formulação do problema é obtida a partir dos teoremas limites de plasticidade, que permitem escrever o problema da análise limite na forma de programação linear (PL) e o programa LINGO é utilizado para resolvê-lo. Para o dimensionamento ótimo o mesmo modelo da treliça de Mörsch é utilizado e a formulação do problema também recai num problema de programação linear que pode ser resolvido pelo programa LINGO. A solução fornece tanto as áreas das armaduras longitudinais como dos estribos. Os exemplos apresentados ilustram a validade e eficiência da formulação proposta. Os resultados obtidos são analisados e comparados com os resultados experimentais e com os obtidos com o uso de outra metodologia. Finalmente são apresentadas as conclusões e sugeridas propostas para trabalhos futuros. / [en] Methodologies for the determination of the collapse load and for the optimal design of concrete beams are presented. For the collapse load determination the methodology makes use of the limit analysis and of the classical truss model of Mörsch. The formulation of the problem is based on the limit theorems of plasticity which allow the limit analysis problem to be expressed as a linear programming problem (LP) which is solved by the program LINGO. For the optimal design, the same truss model of Mörsch is used and the formulation also leads to a linear programming problem which can be solved by the program LINGO. The solution yields both the longitudinal reinforcement and the stirrups. The examples presented illustrate the validity and efficiency of the proposed formulation. The results obtained are analysed and compared to the experimental ones and to those obtained using other methodologies. Finally, conclusions are drawn from the results and suggestions are given for future researches in the field.

A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.

A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.

A avaliação numérica das cargas limites (de colapso) de estruturas é obtida geralmente ou pela aplicação direta dos teoremas de análise limite junto com processos de otimização ou mediante análises incrementais, que levam em consideração o comportamento elasto-plástico do material. Entretanto, ambas as estratégias conduzem eventualmente a dificuldades numéricas, particularmente para cargas próximas à de colapso. Neste trabalho, emprega-se uma alternativa que consiste em simular assintoticamente o comportamento elasto-plástico mediante uma relação elástica não-linear. As vantagens deste tipo de formulação são a possibilidade de se lidar com leis de fluxo não-associadas e um custo computacional reduzido. A relação elástica não-linear é implementada no programa comercial de elementos finitos ABAQUS, através de uma sub-rotina externa ao programa escrita em linguagem FORTRAN. Diversos exemplos de estruturas cujos materiais são regidos pelos critérios de resistência de von Mises e Drucker-Prager são modelados, verificando-se que os resultados das cargas limites obtidas com essa formulação são muito próximos daqueles encontrados na literatura. Por fim, a relação elástica não-linear é empregada para a determinação do domínio de resistência de meios porosos, com diferentes níveis de porosidade. / The numerical assessment of limit loads of structures is generally achieved through the direct implementation of limit analysis theorems together with optimization processes, or through incremental analyses, which account for the elastic-plastic behavior of the material. However, both the strategies may lead to numerical difficulties, particularly when the load is close to its limit value. In this context, the alternative approach presented in this work consists in simulating asymptotically the regime of free plastic flow by means of a fictitious non-linear elastic material. One of the main advantages of this kind of formulation lies in its ability to deal with non-associated flow rules and a reduced computational cost. The non-linear elastic behavior is implemented into the finite element computational software ABAQUS, making use of an external subroutine written in FORTRAN language. Several examples of geotechnical and structural problems with materials ruled by von Mises and Drucker-Prager failure criteria are analyzed. The results obtained with this formulation prove to be very close to those obtained through analytical solutions. At last, the non-linear elastic relation is used in the determination of the resistance domain of porous media with different levels of porosity.