Dans ce manuscrit nous développons l'approche cinématique 2D du calcul à la rupture pour examinerles effondrements en extension (ou failles normales) de la croûte supérieure cassante qui résultent desurpressions de fluides. Les sujets d'intérêt liés à la déformation en extension sont (1) les roles de lapression des fluides, des processus de surface, et des propriétés des matériaux et des failles sur lastabilité des structures d'extension; (2) la formation de failles normales à faible pendage et de failleslistriques; (3) la distribution de la déformation au dessus d'un glissement à faible pendage; et (4)l'influence de l'adoucissement mécanique des failles et des processus de sédimentation sur cettedistribution.Cette approche mécanique est vérifiée par la théorie du prisme critique de Coulomb, et la généralise pour étudier la topographie complexe de la péninsule de Mejillones dans le Nord du Chili. Cetteapproche est aussi appliquée à l'instabilité gravitaire dans le delta du Niger en reliant les structurescompressives en bas de pente aux structure extensives en amont par un détachement profond. Nousprédisons des surpressions de fluides beaucoup plus élevée que celles obtenues par application duprisme de Coulomb. Enfin, cette méthodologie est appliquée à l'étude de la forme de failles normalesreliant un détachement profond à la surface. Dans le cas du delta du Niger, nous montrons que lesfailles à faible pendage et les failles listriques impliquent que la profondeur de rétention des fluides estfaible. La version séquentielle de l'analyse limite ouvre de nouvelles voies pour suivre l'évolutionstructurale dans le temps du jeu sur les failles normales. Les simulations montrent en particulier qu'unefaille normale tourne vers des pendage plus faibles au fur et à mesure de la dénudation du mur, formantune région qui passe du mur au toit de la faille active en rotation. La prédiction de cette région estillustrée par des expériences analogiques et des exemples de terrain. / This manuscript develops a 2D kinematic approach of Limit Analysis to examine the extensionalfailures in the brittle, upper crust resulting from fluid overpressures and normal faulting. There aremany interesting topics related to the extensional deformation such as (1) the roles of fluid pressure,topographic process, material and fault properties on the stability of extensional structures; (2) theformation of low-angle and listric normal fault; (3) the deformation pattern due to slip on a low-anglefault; and (4) the influence of fault softening and sedimentation processes on this deformation pattern.This mechanical approach applied to wedge prototypes is validated by the critical Coulomb wedge(CCW) theory, and it generalizes the CCW theory to investigate the complex topography on theMejillones peninsula, Northern Chile. Additionally, this approach is also applied to investigate gravityinstability of Niger Delta by linking down-slope compressional to up-slope extensional failures througha deep detachment. We predict much higher fluid overpressures than that of the CCW theory. Finally,this Limit Analysis methodology is applied to investigate the shape of normal fault linking a lowdetachment to the surface. The application to Niger Delta implies that the formation of very low-angleand strongly listric faults results from a shallow fluid-retention depth. The sequential version of LimitAnalysis opens new ways to envision the structural evolution through time resulting from normalfaulting. The simulations show that the normal fault rotates during extension, forming a region of Footto-Hanging Wall (FHW) where the material in the footwall is sheared upon entering the hanging wall.The creation of the FHW region is illustrated by sandbox experiments and field examples.

[pt] O presente trabalho tem como objetivo estudar o comportamento de estruturas geotécnicas mediante o uso de Análise Limite Numérica. Para isto foi desenvolvido o programa GEOLIMA (GEOtechnical LIMit Analysis) com base na teoria de Análise Limite Numérica utilizando o Método de Elementos Finitos (MEF), considerando problemas bidimensionais e tridimensionais. Devido ao fato das propriedades do solo serem variáveis aleatórias, a Análise Não Determinística também foi considerada mediante o uso do Método Estatístico Linear e do Método de Monte Carlo. Inicialmente, são apresentados os fundamentos da teoria de Análise Limite Determinística e sua formulação mista pelo Método de Elementos Finitos. A seguir são apresentados os fundamentos de Análise Não Determinística, onde os métodos Estatístico Linear e Monte Carlo são descritos. As fases de desenvolvimento do GEOLIMA são descritas de forma resumida e a validação é feita mediante a comparação de resultados obtidos com soluções analíticas ou outras soluções. A seguir, uma aplicação em 2D é apresentada com a finalidade de ilustrar a Análise Limite Determinística e Não Determinística mediante o método Estatístico Linear e o método de Monte Carlo. Finalmente, duas aplicações em 3D são apresentadas: um problema relativo à frente de escavação de um túnel e um estudo de painéis de mineração. Os resultados deste trabalho indicam a viabilidade de usar Análise Limite Determinística e Não Determinística no estudo de problemas geotécnicos. / [en] The present work has the purpose of studying the behavior of geotechnical structures by means of numerical analysis. For this, program GEOLIMA (GEOtechnical LIMit Analysis) was developed based on the theory of Numerical Limit Analysis using the Finite Element Method (FEM), considering bidimensional and three-dimensional problems. Due to the fact that the properties of the ground are generally random variables, Non Deterministic Analysis was also considered by means of the Linear Statistical and the Monte Carlo Methods. Initially, the fundamentals of Deterministic Limit Analysis and its mixed formulation are presented. Then, the fundamentals of Non Deterministic Theory are presented, and the Linear Statistic and the Monte Carlo Methods are described. The development phases of GEOLIMA are briefly described. Its validation is made by comparing the results obtained with analytical solutions or other solutions. Following, a 2D application is made with the purpose of illustrating Deterministic and Non Deterministic Limit Analysis. Finally, two 3D applications are presented: a problem related to the excavation of a tunnel front and a problem related to mining panels. The results of this work indicate the viability of using Deterministic and Non Deterministic Limit Analysis in the study of geotechnical problems.

Im Rahmen dieser Arbeit wurde das Umformverhalten der gieß- und warmgewalzten Magnesiumlegierung des Mg-Al-Zn-Systems (AZ31) unter einachsiger Zugbeanspruchung in Abhängigkeit von Temperatur, Dehnrate und Entnahmerichtung untersucht. Weiterhin wurde das richtungsabhängige Grenzformänderungsverhalten im statischen sowie erstmalig im dynamischen Lastfall bei verschiedenen Temperaturen bewertet. Für die dynamischen Versuche wurde ein eigens konstruiertes Prüfmodul entwickelt und eingesetzt. Die Ergebnisse zeigen, dass sich bei einer Variation der Beanspruchungsbedingungen die Ausprägung der Versetzungsbewegung, der mechanischen Zwillingsbildung sowie der Hochtemperaturmechanismen signifikant verändert. Die Verknüpfung der sich dadurch ändernden mechanischen Eigenschaften mit der Mikrostrukturentwicklung konnte mit der Licht- und Rasterelektronenmikroskopie sowie der Texturbestimmung mit der Röntgenbeugung (XRD) nachgewiesen werden.

Pour l’évaluation de la résistance ultime des ouvrages l’ingénieur de génie civil fait appel à différentes méthodes plus ou moins empiriques, dont de nombreuses manuelles, du fait de la lourdeur excessive des méthodes par éléments finis non-linéaires mises en œuvre dans les logiciels de calcul à sa disposition. Le calcul à la rupture, théorisé par J. Salençon, indique la voie de méthodes rigoureuses, tout à fait adaptées à cette problématique, mais dont la mise en œuvre systématique dans un logiciel a longtemps buté sur l’absence de méthodes numériques efficaces. Ce verrou de mathématique numérique a été levé récemment (Algorithme de point intérieur).Dans ce contexte l’objectif de la présente thèse est de mettre au point les méthodes permettant d’analyser, au moyen du calcul à la rupture, la capacité ultime d’éléments en béton armé tridimensionnels. Les deux approches du calcul à la rupture, que sont les approches statique et cinématiques, seront mises en œuvre numériquement sous la forme d’un problème d’optimisation résolu à l’aide d’un solveur mathématique dans le cadre de la programmation semi définie positive (SDP).Une large partie du travail sera consacré à la modélisation des différents matériaux constituant le béton armé. Le choix du critère pour modéliser la résistance du béton sera discuté, tout comme la méthode pour prendre en compte le renforcement. La méthode d’homogénéisation sera utilisée dans le cas de renforcement périodique et une adaptation de cette méthode sera utilisée dans le cas de renforts isolés. Enfin, les capacités et le potentiel de l’outil développé et mis en œuvre au cours de cette thèse seront exposés au travers d’exemples d’application sur des structures massives / To evaluate the load bearing capacity of structures, civil engineers often make use of empirical methods, which are often manuals, instead of nonlinear finite element methods available in existing civil engineering softwares, which are long to process and difficult to handle. Yield design (or limit analysis) approach, formalized by J. Salençon, is a rigorous method to evaluate the capacity of structures and can be used to answer the question of structural failure. It was, yet, not possible to take advantage of these theoretical methods due to the lack of efficient numerical methods. Recent progress in this field and notably in interior point algorithms allows one to rethink this opportunity. Therefore, the main objective of this thesis is to develop a numerical model, based on the yield design approach, to evaluate the ultimate capacity of massive (3D) reinforced concrete structural elements. Both static and kinematic approaches are implemented and expressed as an optimization problem that can be solved by a mathematical optimization solver in the framework of Semi-Definite Programming (SDP).A large part of this work is on modelling the resistance of the different components of the reinforced concrete composite material. The modelling assumptions taken to model the resistance of concrete are discussed. And the method used to model reinforcement is also questioned. The homogenization method is used to model periodic reinforcement and an adaptation of this technique is developed for isolated rebars. To conclude this work, a last part is dedicated to illustrate the power and potentialities of the numerical tool developed during this PhD thesis through various examples of massive structures

Résumé Nous présentons en premier lieu dans ce travail les principales notions de la théorie de l'Analyse Limite (AL) — ou théorie des charges limites — en mécanique. Puis nous proposons une méthode de point intérieur destinée à résoudre des problèmes de programmation convexe posés par la méthode statique de l'AL, en vue d'obtenir des bornes inférieures de la charge limite (ou de ruine) d'un système mécanique. Les principales caractéristiques de cette méthode de point intérieur sont exposées en détail, et particulièrement son itération type. En second lieu, nous exposons l'application de cet algorithme sur un problème concret d'analyse limite, sur une large gamme de tailles numériques, et nous comparons pour validation les résultats obtenus avec ceux déjà existants ainsi qu'avec ceux calculés à partir de versions linéarisées du problème statique. Nous analysons également les résultats obtenus pour des problèmes classiques avec matériaux de Gurson, pour lesquels la linéarisation ou la programmation conique ne s'applique pas. La deuxième partie de cet ouvrage a trait à la méthode cinématique de l'analyse limite, qui, elle, s'occupe de fournir des bornes supérieures des charges limites. En premier lieu, nous traitons de l'équivalence entre la méthode cinématique classique et la méthode cinématique mixe, en partant d'une l'approche variationnelle fournie précédemment par Radenkovic et Nguyen. Ensuite, prenant en compte les exigences particulières aux formulations numériques, nous présentons une méthode mixte originale, parfaitement cinématique, utilisant aussi bien des champs de vitesses linéaires que quadratiques, continus ou discontinus. Son modus operandi pratique est tiré de l'analyse des conditions d'optimalité de Karush, Kuhn et Tucker, fournissant par là un exemple significatif d'interaction fructueuse entre la mécanique et la programmation mathématique. La méthode est testée sur des problèmes classiques avec les critères de plasticité de von Mises/Tresca et Gurson. Ces test démontrent l'efficacité remarquable de cette méthode mixte — qui par ailleurs n'utilise que le critère de plasticité comme information sur le matériau — et sa robustesse, laquelle s'avère même supérieure à celle de codes commerciaux récents de programmation conique. Enfin, nous présentons une approche de décomposition, elle aussi originale, des problèmes de bornes supérieures en analyse limite. Cette approche est basée à la fois sur la méthode cinématique mixte et l'algorithme de point intérieur précédents, et elle est conçue pour utiliser jusqu'à des champs de vitesse quadratiques discontinus. Détaillée dans le cas de la déformation plane, cette approche apparaît très rapidement convergente, ainsi que nous le vérifions sur le problème du barreau comprimé de von Mises/Tresca dans le cas de champs de vitesse linéaires continus. Puis elle est appliquée, dans le cas de champs quadratiques discontinus, au problème classique de la stabilité du talus vertical de Tresca, avec des résultats particulièrement remarquables puisqu'ils améliorent nettement les solutions cinématiques connues jusqu'à présent dans la littérature sur le sujet. Cette caractéristique de forte convergence qualifie particulièrement cette méthode de décomposition comme algorithme de base pour une parallélisation directe— ou récursive — de l'approche par éléments finis de l'analyse limite. Abstract Firstly, the main notions of the theory of Limit analysis (LA) in Mechanics —or collapse load theory – is presented. Then is proposed an Interior Point method to solve convex programming problems raised by the static method of LA, in order to obtain lower bounds to the collapse (or limit) load of a mechanical system. We explain the main features of this Interior Point method, describing in particular its typical iteration. Secondly, we show and analyze the results of its application to a practical Limit Analysis problem, for a wide range of sizes, and we compare them for validation with existing results and with those of linearized versions of the static problem. Classical problems are also analyzed for Gurson materials to which linearization or conic programming does not apply. The second part of this work focuses on the kinematical method of Limit Analysis, aiming this time to provide upper bounds on collapse loads. In a first step, we detail the equivalence between the classical an general mixed approaches, starting from an earlier variational approach of Radenkovic and Nguyen. In a second step, keeping in mind numerical formulation requirements, an original purely kinematical mixed method—using linear or quadratic, continuous or discontinuous velocity fields as virtual variables—is proposed. Its practical modus operandi is deduced from the Karush-Kuhn-Tucker optimality conditions, providing an example of crossfertilization between mechanics and mathematical programming. The method is tested on classical problems for von Mises/tresca and Gurson plasticity criteria. Using only the yield criterion as material data, it appears very efficient and robust, even more reliable than recent conic commercial codes. Furthermore, both static and kinematic present approaches give rise to the first solutions of problem for homogeneous Gurson materials. Finally, an original decomposition approach of the upper bound method of limit analysis is proposed. It is based on both previous kinematical approach and interior point solver, using up to discontinuous quadratic velocity. Detailed in plane strain, this method appears very rapidly convergent, as verified in the von Mises/Tresca compressed bar problem in the linear continuous velocity case. Then the method is applied, using discontinuous quadratic velocity fields, to the classical problem of the stability of a Tresca vertical cut, with very significant results as they notably improved the kinematical solutions of the literature. Moreover its strong convergence qualifies this decomposition scheme as a suitable algorithm for a direct—or recursive—parallelization of the LA finite element approach.

The reliability analysis of plates and shells with respect to plastic collapse or to inadaptation is formulated on the basis of limit and shakedown theorems. The loading, the material strength and the shell thickness are considered as random variables. Based on a direct definition of the limit state function, the nonlinear problems may be efficiently solved by using the First and Second Order Reliability Methods (FORM/SORM). The sensitivity analyses in FORM/SORM can be based on the sensitivities of the deterministic shakedown problem. The problem of reliability of structural systems is also handled by the application of a special barrier technique which permits to find all the design points corresponding to all the failure modes. The direct plasticity approach reduces considerably the necessary knowledge of uncertain input data, computing costs and the numerical error. / Die Zuverlässigkeitsanalyse von Platten und Schalen in Bezug auf plastischen Kollaps oder Nicht-Anpassung wird mit den Traglast- und Einspielsätzen formuliert. Die Lasten, die Werkstofffestigkeit und die Schalendicke werden als Zufallsvariablen betrachtet. Auf der Grundlage einer direkten Definition der Grenzzustandsfunktion kann die Berechnung der Versagenswahrscheinlichkeit effektiv mit den Zuverlässigkeitsmethoden erster und zweiter Ordnung (FROM/SORM) gelöst werden. Die Sensitivitätsanalysen in FORM/SORM lassen sich auf der Basis der Sensitivitäten des deterministischen Einspielproblems berechnen. Die Schwierigkeiten bei der Ermittlung der Zuverlässigkeit von strukturellen Systemen werden durch Anwendung einer speziellen Barrieremethode behoben, die es erlaubt, alle Auslegungspunkte zu allen Versagensmoden zu finden. Die Anwendung direkter Plastizitätsmethoden führt zu einer beträchtlichen Verringerung der notwendigen Kenntnis der unsicheren Eingangsdaten, des Berechnungsaufwandes und der numerischen Fehler.

[pt] O presente trabalho tem por objetivo desenvolver um otimizador eficiente de grande escala, que permita a aplicabilidade prática da Análise Limite Numérica pelo MEF, para resolver problemas reais da Engenharia Geotécnica. Para isto, foi desenvolvido um otimizador para o programa GEOLIMA (GEOtechnical LIMit Analysis) (Carrión, 2004) baseado no algoritmo de Pontos Interiores, computacionalmente mais eficiente que os otimizadores comerciais existentes. Pelo fato das propriedades do solo serem de natureza aleatória, a possibilidade de aplicar Análise de Confiabilidade com a Análise Limite pelo método FORM em problemas geotécnicos é pesquisada também. Sendo a grande vantagem do método FORM a possibilidade de se aplicar para funções de falha quaisquer e variáveis com distribuição quaisquer. Inicialmente, são apresentados os fundamentos da teoria de Análise Limite e sua formulação numérica pelo MEF (Método dos Elementos Finitos). A seguir, é investigada a possibilidade de se usar otimizadores comerciais para resolver o problema matemático resultante da aplicação de Análise Limite com o MEF e são descritos os fundamentos teóricos do otimizador implementado baseado no algoritmo de Pontos Interiores. Um resumo dos fundamentos teóricos da Análise de Confiabilidade é apresentado. É descrito o processo de cálculo pelo método FORM e dois exemplos de aplicação são realizados. Finalmente, análises de diferentes problemas resolvidos com o otimizador implementado são apresentados indicando o grande potencial da Análise Limite Numérica, na solução de problemas reais da Engenharia Geotécnica. / [en] This work has, as its main objective, the development of an efficient and large scale optimizer, that allows the practical application of Numerical Limit Analysis (NLA) with Finite Element Method (FEM) to solve real problems in Geotechnical Engineering. For that purpose, an optimizer was developed for GEOLIMA (GEOtechnical LIMit Analysis) program (Carrión, 2004), based on Interior Points algorithm, computationally more efficient than the existing commercial optimizers. Due to the fact that soils have random properties, the possibility to apply Reliability Analysis with Limit Analysis using the FORM method was also investigated. Initially, Limit Analysis theory was presented together with its numerical formulation using the FEM. In sequence, the use of commercial optimizers was investigated in order to solve the resulting mathematical problem. Subsequently, the theorical foundations of the developed optimizer, based on the Interior Points algorithm were described. A summary of Reliability Analysis was also presented together with a description of computational procedures using FORM and two examples were developed. Finally, analyses of different problems solved with developed optimizer were presented. The obtained results demonstrated the great potential of Numerical Limit Analysis (NLA), in the solution of real problems in Geotechnical Engineering.

[pt] O presente trabalho tem como principal objetivo mostrar a aplicabilidade prática da análise limite pelo método de elementos finitos na avaliação de problemas de estabilidade de talude, sendo este colocado como um problema de programação matemática, no qual se precisa realizar um processo de otimização para a solução do problema. Apresenta-se um método para obter a solução do problema de estabilidade de taludes utilizando para isso a programação matemática, e fazendo ênfase na utilidade da programação cônica da segunda ordem (SOCP). Inicialmente faz uma revisão das formulações da análise limite, via o método de elementos finitos, encontradas na literatura existente. A seguir é descrita a formulação da análise limite numérica partindo do principio do trabalho virtual para sua formulação, e utilizando a ferramenta dos elementos finitos para realizar a implementação numérica. São propostas diferentes formas de trabalhar com o critério de resistência do material, sendo a de melhor desempenho, em termos de tempo de processamento a forma cônica quadrática que permite acoplar a programação cônica da segunda ordem (SOCP) na ferramenta numérica. É acoplada a técnica da redução dos parâmetros de resistência do material com a finalidade de encontrar o fator de segurança da estrutura do talude (FS). Finalmente são apresentados exemplos de validação e aplicação, os quais permitem visualizar a eficiência da ferramenta desenvolvida em termos de tempo de processamento ao utilizar a programação cônica da segunda ordem (SOCP). Os resultados sugerem viabilidade da utilização da técnica estudada na solução de problemas relacionada à estabilidade de taludes. / [en] The main objective of this work is to show the practical applicability of limit analysis by finite element method in the evaluation of slope stability problems, and this placed as a mathematical programming problem, which you need to perform an optimization process to solve the problem. We present a method to obtain the solution of the problem of slope stability using for this mathematical programming, and making emphasis on the usefulness of the second order conic programming (SOCP). Initially, a review of formulations Limit Analysis via Finite Element Method, found in the existing literature. Then is described the Numerical Limit Analysis formulation starting from virtual work principle their formulation, and using Finite Element Method as a tool to carry out the numerical implementation. We propose different ways of working with the yield criterion of the material, being the best performing in terms of processing time the conic quadratic form that allows to coupling to the second order conic programming (SOCP) in numerical implementation. It is coupled to the technique of reducing the strength parameters of the material in order to find the safety factor of the slope of the structure (FS). Finally, examples are presented for validation and application, which allow you to view the efficiency of the developed implementation in terms of processing time with the use of second order conic programming (SOCP). The results suggest the feasibility of using the technique studied in the solution of problems related to Slope Stability.

[pt] Este trabalho de dissertação de mestrado apresenta a implementação da análise limite numérica com formulação mista-fraca, baseada no teorema do límite inferior, e sua aplicação em problemas de estabilidade axissimétricos. Aformulação com elementos finitos foi implementada no software Matlab, onde se estabelece o problema de otimização que compreende a definição da equação de equilibrio e a adaptação dos criterios de ruptura de Drucker-Prager e Mohr-Coulomb às programações cônica de segunda ordem e semidefinida, respectivamente, e que posteriormente é resolvido com o algoritmo Mosek Aps 9.2. Como resultado do problema de otimização o fator de colapso e o campo de velocidades podem ser obtidos, permitindo identificar o mecanismo de ruptura. O presente trabalho foca-se na análise de estabilidade de um poço que é executada em 3 fases, em função das condições consideradas no modelo. Os resultados obtidos da análise axissimétrica foram validados mediante analises em modelos tridimensionais e comparados com resultados dos softwares Plaxis 2D e Optum G2, também foram incluídos os resultados da modelagem MPM, com o sotware MPM-PUCRio. Por fim foi estudado o caso da capacidade de carga de uma fundação circular rasa, cujos resultados foram comparados com os apresentados por outros autores. / [en] This work dissertation presents the implementation of numerical limit analysis with mixed-weak formulation, based on the the lower bound limit theorem and its application in axisymmetric stability problems. The finite element formulation was implemented in Matlab, where the optimization problem is established, which comprises the definition of the equilibrium equation and the adaptation of the Drucker-Prager and Mohr-Coulomb rupture criteria to the second-order cone programming and semidefined programming, respectively, and which is later solved with the Mosek Aps 9.2 algorithm. As a result of the optimization problem, the collapse factor and the speed field can be obtained, allowing to identify the rupture mechanism.The present work focuses on the stability analysis of a well that is carried out in 3 phases, depending on the conditions considered in the model. The results obtained in the axissymmetric analysis were validated through analysis in three-dimensional models and compared with results of plaxis 2D and Optum G2 software, also included the results of MPM modeling, with the software MPM-PUCRio. Finally, the case of the load capacity of a shallow circular foundation is studied, the results of which are compared with those presented by other authors.

Masonry spires are a typical part of church architecture. Since it is rare that masonry is used as a load-bearing material in the western world today, it is important to maintain and increase the knowledge of modelling masonry structures both from a maintenance point of view and to build new masonry structures. The purpose of this master thesis is to look at and evaluate some different methods to model masonry spires exposed to common loads such as gravity, settlement and wind. The spire of the Salisbury Cathedral is used as a template regarding geometry and mechanical properties for the modelling methods. Two modelling methods are used in the master’s thesis. The first one is the limit analysis method applied to masonry. It is used to calculate a critical thickness for the masonry of the spire for a severe wind load. The second method is the Finite Element Method (FEM). The commercial finite element software Abaqus is used to create the model and the discretization used with the FE modelling is the macro-modelling approach. Concrete Damage Plasticity (CDP) in Abaqus is used as the material model and adapted to masonry. The finite element model consists of the spire itself along with the supporting structure beneath it down to the piers. Four different simulations (jobs) are run with varying wind direction and two of them have settling piers. The results from the finite element simulations indicate that the membrane stresses in the spire faces for the various jobs were not significantly different from one another. One of the jobs with settling piers could not be completed because the tensile stresses in the arches reached the tensile strength capacity of the material. The other simulation with a settlement that did complete did not have any significant difference in stress compared with the simulations without settlements. While the arches and the piers underwent plastic straining the spire itself did not. The stress levels there remained in the linear range for all the completed simulations. The finite element results also agree with the limit analysis. These findings call into question some of the modelling choices. The inclusion of the structure beneath the spire in the finite element model, as a way to study the effect of settlements, did not give more insight into the spire’s behaviour. Furthermore, the method to implement settlements was too inaccurate and another approach should be used to study the effect of settlements on the state of spires. Further work needs to be done on that topic. Improvements can also be made regarding how CDP was adapted for masonry. / Murade tornspiror är en vanlig takkonstruktion inom kyrkoarkitekturen. Eftersom det numera är sällsynt att murverk fungerar som lastbärande material i västvärlden, är det viktigt att upprätthålla och utöka kunskapen om murverkskonstruktioner för både underhåll och nybyggnation. Syftet med denna masteruppsats är att betrakta och utvärdera några olika modelleringsmetoder för murade tornspiror som är utsatta för några typiska laster såsom egentyngd, sättningar och vind. Katedralen i Salisbury används som en modelleringsmall i uppsatsen med avseende på katedralens geometri och materialegenskaper. Två modelleringsmetoder används i uppsatsen. Den första är gränsanalys tillämpad på murverkskonstruktioner. Den används för att beräkna en kritisk tjocklek för tornspiran under en stor vindlast. Den andra metoden är Finita Elementmetoden (FEM). Den kommersiella finita elementprogramvaran Abaqus används för finita elementanalysen och diskretiseringen som används för murverket i finita elementmodellen är makromodellering. Concrete Damage Plasticity (CDP) i Abaqus används som materialmodell och anpassas för murverk. Finita elementmodellen består utav själva tornspiran inklusive de bärande delarna under spiran och ned till pelarna. Fyra olika simuleringar ("jobb") körs med vindlast som angriper från olika riktningar och två av simuleringarna har pelare som sätter sig. Resultaten från simuleringarna visar att membranspänningarna i tornspirans väggar, för de olika jobben, inte skilde sig i någon betydelig grad från varandra. Ett av jobben med pelare som satte sig kunde inte köras klart eftersom dragspänningarna i valvbågarna överskred draghållfastheten på murverket i modellen. Den andra simuleringen med sättningar som kördes klart uppvisade inte några avsevärda skillnader i spänningar i tornspiran jämfört med simuleringarna utan sättningar. Medan plastiska töjningar uppkom i både valvbågarna och pelarna i modellen, uppkom de inte i tornspiran. Spänningsnivåerna i tornspiran var inom det linjära intervallet för alla simuleringar. Resultaten från finita elementanalysen stämde överens med resultaten från gränsanalysen. Analysresultaten ifrågasätter vissa av modelleringsvalen. Att inkludera de bärande delarna under tornspiran i finita elementmodellen, för att undersöka effekten av sättningar, gav inte en större insikt i hur sättningar påverkar tornspiran. Dessutom, var metoden för att tillämpa sättningar för oprecis och en annan metod borde användas. Mer arbete måste utföras vad gäller det ämnet. Sättet att tillämpa CDP för murverk kan också förbättras.