Publicações

A monumental tetrastyle canopy (four columns topped by a hollow cap overlaying a statue), built in1948 at the centre of Kathmandu, Nepal, has suffered full collapse during the April 25, 2015 Mw7.8earthquake. Two video camera recordings let us recover the most important movements of thestructure. At a distance of 2.9 km from the monument, an accelerometric station (KATNP) recordedthe motion which we used as input of a model structure made with DEM. Geometry and dimensionsof main elements were mostly obtained from satellite images. Concrete and steel in columns wereconsidered as the common practice at that time. We performed some sensitivity analyses varyingthe mechanical properties due to the possibility of aging alterations in concrete and steel and toincorporate uncertainties on the geometric characteristics. Albeit the response has a significantvariability, results show that it is possible to reproduce well the video images, attesting theimportance of video cameras to capture the dynamic performance of structures, providing insightinto their behaviour and data to support numerical investigations.

In this study, the in-plane structural behavior, capacity, and performance of dry-joint stone masonry walls (DJSMWs)and the effects of the vertical stress level on these factors are investigated via a stochastic discontinuumanalysis that considers the material uncertainty. A discontinuum type of analysis is performed based on thediscrete element method (DEM), where each stone masonry unit is explicitly represented in the computationalmodel. To better simulate the cracking and shear failure modes within the stone units, a coupled fracture energybasedcontact constitutive model is implemented into a commercial discrete element code, 3DEC. First, theproposed modeling approach is validated by comparing to experimental findings in literature. Then, theapproach is used to explore the failure mechanism and the force

The installation of vibration-based structural health-monitoring systems relying in the study of natural frequencies to assess the condition of civil engineering structures is becoming more and more common. The extraction of these monitoring features can be achieved through automated operational modal analysis, combining output-only identification methods with cluster analyses, and comparing the identified modal properties with references, in a process known as modal tracking. However, changing environmental and operational conditions affect the dynamic behaviour of structures, disturbing the process of modal tracking, which may lead to the loss of important information and to misidentifications. In this context, this paperproposes a new methodology for modal tracking within the scope of automated operational modal analysis, especially prepared for scenarios with strong external influence on modal properties. A concrete arch dam with large variations of some natural frequencies is used as case study, and the proposed methodology is compared to a standard procedure using a quite unique monitoring data set continuously collected for 3 years.

Neste artigo apresenta-se a formulação e resultados da aplicação de dois modelos descontínuos tridimensionais que simulam a interação hidromecânica. Nestes modelos o comportamento mecânico é simulado de forma idêntica mas o comportamento hidráulico é simulado com duas abordagens diferentes. O primeiro modelo baseia-se numa formulação que admite o escoamento através de elementos planos de interface. O segundo modelo, que se propõe neste artigo, baseia-se numa discretização unidimensional, ocorrendo o escoamento através de elementos de canais. Os modelos hidromecânicos implementados no módulo computacional Parmac3D Fflow são verificados e validados através de exemplos simples, e o modelo proposto é calibrado de modo a serem obtidos, com os dois modelos, os mesmos valores de pressão e de caudal. Foi desenvolvido um modelo hidromecânico de um conjunto barragem/fundação, tendo em consideração a existência de cortina de impermeabilização e de sistema de drenagem na fundação. Salientam-se as vantagens do modelo proposto no estudo do comportamento hidromecânico de fundações de barragens.

Earthquake damage in historic masonry buildings is generally caused by the collapse of individual portions thatbecome detached form the structure and fail by overturning. These mechanisms are mainly governed by thediscrete nature and geometry of the block units. Analyses based on explicit micro-modelling in which each blockis considered separately are therefore of increasing importance. This paper offers a robust tool for the seismicassessment of masonry structures under either, quasi-static or dynamic loading. Algorithms for performing dynamicpulses and pushover analyses through the Discrete Element Method are developed and described, takinginto account the actual discrete nature and geometry of masonry. A numerical procedure is proposed thatautomatically detects the collapse mode and follows the evolution of the analysis until collapse. The implementedmethod is able to give a reliable estimate of the expected failure mechanism, providing the seismicacceleration required to trigger the motion and the ultimate displacement beyond which the collapse occurs.Finally, the implemented algorithms are applied to two case studies and the results are compared with thetraditional analysis based on rigid-block kinematics to outline the features and potentialities of the proposedapproach.

Conservation of the European-built heritage is nowadays closely related to interventions in masonry construction. Since a significant part of these structures are located in earthquake prone zones, designing protective measures that ensure their preservation requires the assessment of the associated seismic vulnerability and the understanding of their degradation mechanisms. However, due to the presence of discontinuities between its constituent elements, the numerical simulation of the non-linear response of masonry structures subjected to strong earthquakes is a cumbersome task, especially when performed using the traditional finite element method. As an alternative, the discrete element method is particularly suitable for the numerical analysis of such structures under large, damaging loads. By allowing discrete displacements/rotations of individual bodies, and by automatically recognising new contacts between bodies, while eliminating obsolete ones, this numerical approach can simulate the complete detachment of discrete bodies initially in contact.The present paper illustrates the use of the discrete element method to assess the seismic vulnerability of stone arches. A probabilistic approach allows to consider the variability of theseismic action and the uncertainties in the definition of the material properties. First, an unreinforced masonry arch is considered. Based on non-linear multi-directional incremental dynamic analyses, five damage states are proposed and the respective fragility curves are elaborated. Toimprove the arch

A seismic assessment study of two historical masonry arch bridges, with different geometries and spans, ispresented. The seismicity of the two locations was analyzed in order to define the seismic action to be applied.Artificial ground motion records were generated accordingly, and real ground motions were also used. The twonumerical models were created with a 3D discrete element code. The accuracy of the deformability of the blockmodels was verified by comparison with a finite element model. In situ measurements of the natural frequencieswere available for one of the bridges, providing further means for model calibration. Time domain dynamicanalyses were performed with each set of ground motions, allowing an evaluation of damage and failure modesunder seismic action. The shorter span bridge was found to withstand the regulatory records with minimumdamage. In contrast, the longer span bridge displayed a significant vulnerability to earthquake action. The studyshows that rigid block models have the ability to represent effectively the dynamic elastic response of the bridgesand can be calibrated by ambient vibration measurements. Furthermore, the discrete element models provide apowerful tool to investigate the nonlinear dynamic response and failure of these structures, which has to be basedon a comprehensive set of dynamic records.

Stability analyses of geotechnical structures in rock are traditionally performed using deterministic methods. In Europe, Eurocode7, introduced in the beginning of the 21st century, adopts limit state design and semi-probabilistic methods, using partial factors for the design of geotechnical structures. Meanwhile, reliability-based design, using probabilistic methods, is becoming more common in practical cases. The paper considers an intentionally simple case study?the analysis of a slope in a rock mass with one discontinuity, considered in a discrete way, forming a rock block to be stabilised by anchors?to compare the results obtained with the different methods. The objective is to calculate the force applied by the anchors so that the ultimate limit states of sliding of the rock block is verified. Deterministic-based design optimization considering both the traditional global safety factor approach and the partial factor approach following the Eurocode 7 are first applied. A reliability-based design optimization procedure?which takes geometrical and mechanical properties of the discontinuities as random variables?is then used, and the results are compared to the former ones. A discussion is presented concerning the consistency of the obtained results.

Civil engineering structures are generally supposed to have long operation life spans during which they aresubmitted to many factors, both environmentally and human-induced, that degrade their condition and performance,gradually leading to the occurrence of damages and malfunctions. Therefore, systems capable ofmonitoring and assessing structural performance automatically are highly desirable.In this context, this paper describes the vibration-based structural health monitoring of Baixo Sabor arch dam,in Portugal, detailing results from the several processing stages needed to achieve the ability to detect damage.Automated operational modal analysis is applied to the data obtained during the first three years of monitoring,after which environmental and operational conditions affecting modal properties are studied and their effectsminimised through the combined application of weighted regression models and principal components analysis.Finally, damages are simulated in a numerical model of the dam, and the results are used to test the ability ofdamage detection tools to find these anomalies.The present paper uses a quite unique set of experimental data to test for the first time the ability of avibration-based SHM system to identify realistic damages in an arch dam.

The paper investigates the effectiveness of two-dimensional Discrete Element Method as a tool forstructural analysis of arches provided with buttresses and backfill and strengthened with externallybonded reinforcement. Masonry vault and buttresses are modelled as an assembly of 2Drigid blocks interacting through non-linear contact joints, the backfill is discretized into deformableelastic-plastic triangular elements, while the reinforcement is modelled by means of trusselements bonded to the substrate through non-linear springs. A parametric analysis on theinfluence of material parameters and the effect of the discretization of the backfill is carriedout, and the outcome of different reinforcement systems, consisting of Steel Reinforced Groutapplied either at the intrados or at the extrados, are analysed. The comparison with experimentalfull-scale tests proved the ability of the numerical approach to capture hinges position, loadbearing capability, as well as the increase in deflection and load capacity provided by thereinforcement.

Os modelos detalhados de partículas, MP, ao terem em consideração a estrutura granular da rocha incluem de forma explícita a aleatoriedade do material e introduzem um tamanho limite à localização do dano. Neste trabalho é apresentado um modelo de partículas rígidas esféricas que tem sido desenvolvido tendo em vista o estudo da fratura em betão e em rocha. A interação entre partículas é definida com base na estrutura de Voronois dos centros de gravidade das partículas esféricas. Adota-se um modelo de contacto múltiplo, no qual a área da superfície de contacto é dada pela área da faceta comum dos Voronois associados a cada partícula. Vários estudos paramétricos são apresentados de modo a demonstrar a influência dos parâmetros micromecânicos, elásticos e resistentes, na resposta macroscópica dos sistemas de partículas.

This paper presents the results of in situ stress measurements and the application of a global interpretation model for evaluation of the most likely in situ stress field in the rock mass of the underground powerhouse cavern of the Salamonde II re-powering scheme, located in a mountainous zone of the North of Portugal. The objective of the study was to obtain the stress field prior to the powerhouse cavern excavation, for design purposes. Overcoring tests were performed in boreholes, before construction started. During early construction stages, when direct access to the vicinity of the cavern was created, flat jack tests were performed. For estimating the most likely state of stress in the rock mass, a global inverse analysis was carried out, which combined the overcoring and the flat jack tests results, using an appropriate methodology and a three dimensional numerical model. In order to assess the influence of time dependent effects on the in situstresses, a viscoelastic gravitational analysis was also performed using the same numerical model. The resultsobtained are compared and discussed.

Neste trabalho apresenta-se um modelo interpretativo tridimensional, que permite estimar o estado de tensão mais provável num maciço rochoso, integrando os resultados provenientes de ensaios de medição do estado de tensão in situ. Apresenta-se o caso de estudo da caverna da central subterrânea do reforço de potência do empreendimento hidroelétrico de Salamonde, localizado no rio Cávado, em Portugal. Neste caso de estudo, foiefetuada uma análise global integrando os resultados de dois tipos de ensaios, realizados em diferentes locais do maciço rochoso: i) ensaios de sobrecarotagem (STT), executados em furos de sondagens, antes de se iniciarem as obras do reforço de potência e; ii) ensaios de reposição de tensões com macacos planos (SFJ), realizados já em fase de execução da obra. Foi utilizado um modelo numérico tridimensional e uma metodologia apropriada, incluindo a geometria das escavações na altura da execução dos ensaios e a topografia do terreno, de forma a permitir uma estimativa do estado de tensão mais provável na zona da caverna da central.

Identification of potential failure mechanisms defined by the rock mass discontinuities or the concrete-rockinterface plays an important role in the safety assessment of concrete dam foundations. The paper presentsinstances of application of discrete elements deformable block models for analysis of these failure scenarios,representing the rock mass foundation and the concrete dam. The rock mass characterization needs and themethodologies for obtaining the input parameters for these models are discussed.

Os ensaios de macacos planos de grande área são utilizados para medir a deformabilidade de maciços rochosos e fornecem dados importantes para o projeto de grandes obras. Este ensaio consiste na abertura de um rasgo no maciço rochoso no qual é inserido um macaco plano de grande área. O macaco aplica pressão às paredes do rasgo através de óleo bombeado para o seu interior e são medidos os deslocamentos no maciço. A maior dificuldade na interpretação do ensaio reside no efeito provocado pela abertura de uma fenda de tração durante a aplicação de pressão pelo macaco plano. Para melhorar a compreensão sobre os efeitos da fenda de tração e assim melhorar a interpretação dos resultados do ensaio, desenvolveu-se um modelo numérico tridimensional do ensaio. O modelo permite estabelecer relações entre a tensão in situ perpendicular aos rasgos, a resistência à tração do maciço, as pressões aplicadas e a profundidade da fenda.

A methodology is proposed for dealing with the unbalanced stresses that arise from numerical models due to topography effects when horizontal stresses are applied at the vertical boundaries. The proposed methodology is used for understanding the role of topography on the distribution of in situ stresses due to both gravity and tectonic loadings. In particular, the influence of Poisson

Recent regulations introduced a limit state approach to geotechnical design by using representative values of the actions and of the strength parameters, partial safety factors that affect them, and by including safety margins in the calculation models. Moreover, Eurocode 7 stresses the importance of making use of test results for establishing ground parameters, and so rock joint shear tests are set to play a relevant role in the assessment of the shear strength required in the design of important projects, such as concrete dams, large bridge foundations, slopes, or underground excavations. In this keynote, joint shear tests are described, along with a presentation of their equipments and different procedures. Test results and calculations for the assessment of relevant shear strength parameters are illustrated, and several topics regarding sampling and variability are discussed. Opportunity is taken to present a practical apparatus allowing to perform simple shear tests (push tests) under very low normal stresses with advantages over tilt or pull tests.

The New Alto Ceira Dam is a 41 m high and 133 m long at the crest concrete arch dam, located in the middle zone of Portugal. The objective of the dam

Este documento descreve a rotura ocorrida na barragem indiana de Bhandardara,constituída por alvenaria, cujas causas estão relacionadas com problemas defissuração e acção da pressão da água no seu interior. O documento apresentatambém uma avaliação da segurança estrutural, com base em modelos numéricoselaborados através do Método dos Elementos Discretos, das obras de reabilitaçãopreconizadas que incluíram a utilização de ancoragens e pregagens.

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Masonry is a combination of units such as stones, bricks or blocks usually laid in a cementitious or limemortar. It is probably the oldest material used in construction and has proven to be both simple to buildand durable. Over the years, existing masonry constructions have inevitably suffered damage with time.Earthquakes, soil settlements, material degradation and lack of maintenance are the main reasons forthat. Careful and periodic assessment of such structures is necessary in order to evaluate their structuralcapacity and safety levels. However, performing the structural analysis of masonry construction is notan easy task. A review is presented of the main models based on the discrete element method and the available related numerical techniques that have been proposed for the analysis of masonry. The essential assumptions adopted by these models and numerical implementation issues are discussed. Differences between available models are illustrated by applications to various masonry problems including static and dynamic analysis of masonry arch bridges, walls, vaults, domes and ancient colonnades.This book is composed of 17 chapters authored/co-authored by 25 outstanding researchers from 11countries (Canada, France, Greece, Hungary, Iran,Italy, Mexico, Norway, Portugal, UK, USA), whichwere reviewed by 60 referees.

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This chapter consists of two parts. First a general overview is given: the exact definition of the Discrete Element Method is provided, and the main types of existing models are introduced; then the theoretical fundaments of the most important approaches available today for masonry analysis are given. The second part of the chapter focuses on 3DEC, the most widespread commercial DEM code for masonry analysis. This second part begins with a theoretical overview. Application issues for masonry structures are then considered (block material behaviour, contact representation, structural elements like cables or bars etc.). Among the practical examples arches, barrel vaults, walls, coloumns and complete structures (like houses) are presented, including the issues of quasi-static versus dynamic analysis and of block fracture as well. The readers can find helpful advice on how to simulate their own problems most effectively.

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Discrete element models have become a major tool for masonry analysis, allowing a suitable representation of its discontinuous nature and marked nonlinear behaviour. The chapter reviews the fundamental assumptions of the main DEM formulations available. It focuses more specifically on the widely used UDEC and 3DEC codes, explaining their relation with the other approaches, and the most relevant issues arising in their application to masonry. Applications to the main types of structures are reviewed, considering both static and dynamic analyses. Recent research providing comparisons of the numerical models with experimental and field data is particularly covered. The discussion of the different examples is aimed at clarifying the key capabilities of DEM and demonstrating the most effective ways of using it in a variety of masonry analysis situations.

This chapter is focused on the Portuguese experience on the developmentand exploration of systems for continuously monitoring dam vibrations, usingaccelerometers. The pioneer system for seismic and structural health monitoring(SSHM) installed in Cabril dam (the highest Portuguese arch dam: 132 m high) isdescribed in detail. The design of this system was the result of a long-term LNECresearch program, still ongoing. These monitoring systems should include softwaredeveloped to automatically perform the analysis of collected data, including theautomatic comparison with numerical results from 3DFE models. In view of thegood results obtained with the system in operation in Cabril dam since 2008, similarsystems have been installed in other large dams in Portugal, particularly in recentlybuilt dams. Finally, Baixo Sabor dam is presented as an example of a new Portuguesedam with a complete SSHM system, in operation since 2015. Themain experimentalresults obtained for both dams are shown, namely the evolution of natural frequenciesover time, mode shapes and the measured seismic response to earthquake events.

Para verificação do cumprimento das exigências de segurança das barragens,tanto relativamente à funcionalidade (cenários correntes), como em relação àsegurança ao colapso (cenários de rotura), é essencial utilizar modelos de simulação oude interpretação (no caso de obras existentes) do seu comportamento. Estes modelossão idealizações (simplificações) da realidade, tanto no que diz respeito às acções(modelos das acções), como no que respeita à componente estrutural (modelos estruturais).A solução destes modelos, através de adequados métodos de análise, quantificaparâmetros da resposta das estruturas que permitem avaliar a sua segurança, utilizandocritérios de segurança previamente definidos.A validação destes modelos pode ser feita, no caso de cenários correntes,através da sua comparação com os resultados da observação do comportamento debarragens. No caso de cenários de rotura, não existe muita experiência de comparaçãodestes resultados com situações reais: é pequeno o número de acidentes ocorrido e,mesmo nestes, o volume de informação é reduzido. Desta forma, o recurso a métodosexperimentais, para o estudo de cenários de rotura, tem uma importância acrescida.Assim o principal objectivo deste trabalho é contribuir para melhorar a capacidade deavaliação da segurança de barragens de betão envolvendo cenários de rotura pela fundação.Neste trabalho desenvolveram-se dois ensaios experimentais em modelosfísicos: um primeiro envolvendo uma barragem abóbada, para o estudo de cenários derotura da fundação para acções estáticas; e um segundo de uma barragem gravidadeensaiado em mesa sísmica, com vista a estudar a rotura pela fundação para acçõesdinâmicas.A previsão e interpretação dos ensaios foram efectuadas com modelosmatemáticos adequados, que permitiram a sua validação e posterior utilização no estudode outros cenários.

Neste trabalho apresenta-se o sistema para monitorização em contínuo do comportamento dinâmico da barragem do Cabril que foi instalado em 2008, pelo LNEC e pela EDP, referindo alguns dos principais resultados que têm sido obtidos a partir da análise dos dados recolhidos, quer sob excitação ambiente/operacional, quer sob ações sísmicas. Salienta-se a importância do desenvolvimento do software para tratamento, análise e gestão automática dos dados e apresentam-se os programas que têm vindo a ser desenvolvidos no LNEC, em MATLAB, para apoio a este tipo de sistemas, nomeadamente programas de identificação modal (Modal_ID2.0 e Modal_ID_Auto2.0) e programas de elementos finitos tridimensionais para análise do comportamento dinâmico de sistemas barragem-fundação-albufeira (DamDySSA2.0).

This document presents the research plan for the doctoral thesis of LNEC's Junior Research Fellow Margarida Espada. This work, to be submitted to the Faculty of Engineering of the University of Porto (FEUP), will be developed in the Modelling and Rock Mechanics Unit of the Concrete Dams Department, and it is part of the on-going P2I research project DEMRock6m. This work aims at developing more efficient methodologies to perform safety studies and seismic analysis of rock engineering structures, based on discrete element models and in the integrated use of probabilistic models for the geometric and shear strength parameters and for the water pressures in rock discontinuities. This will allow a more realistic definition of the rock mass in discrete element models and an adequate identification of the potential failure mechanisms.

Neste relatório é apresentado o método de Euler para calcular as órbitas do sistema n-corpos,por exemplo, o dos planetas do sistema Solar, tendo em conta a propagação das forçasgravíticas entre o Sol e os planetas e entre os planetas entre si. Apenas requer como dadosiniciais uma observação anterior das órbitas em termos de posição e velocidade. A correcçãoda interacção entre planetas não é suavizada, o que se justifica pelos seus valores. A correcçãorelativista é importante.

As equações de Navier-Stokes são equações de derivadas parciais. As incógnitas nocaso de incompressibilidade consistem no campo de velocidades. As pressões numa 1ªaproximação resultam de um equilíbrio estático.A principal causa de problemas relacionados com a solução destas equações é devida àparcela das acelerações convectivas que está na origem da turbulência.Pesando estas equações e integrando duas vezes por partes obtém-se uma forma fracadestas equações.Em problemas 3D as variáveis independentes são 3 espaciais e 1 temporal. Admite-seque o campo de velocidades pode ser desenvolvido em série de distribuições de Delta deDirac espaciais e suas derivadas com coeficientes funções do tempo. Com funções deteste polinomiais obtém-se a formulação distribucional proposta das equações deNavier-Stokes.

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O objectivo deste relatório é o estudo da matemática aplicada à solução de problemasfísicos de propagação de ondas esféricas com operador espacial biharmónico,consideradas quânticas.As técnicas matemáticas usadas incluem a transformada de Laplace da função de ondaf e o cálculo de sua inversa tal como sugerida por Pipes e Harvill.O método matemático usado para descrever as ondas esféricas quânticas é compequenas adaptações o método causal não-linear descrito por Croca.

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A análise do comportamento de barragens de betão com base em modelos de elementos finitos tridimensionais exige a definição da discretização a adoptar para o conjunto barragem-fundação, ou barragem-fundação-albufeira. A realização desta tarefa sem o recurso a programas de geração automática é muito dispendiosa pelo que tem vindo a ser desenvolvido no NMMF um módulo computacional para geração automática de malhas de elementos finitos tridimensionais especialmente adaptado para a geração de malhas tridimensionais de barragens abóbada. Inicialmente desenvolvido em FORTRAN, o referido módulo computacional para geração de malhas de elementos finitos tridimensionais tem vindo a ser objecto de diversos melhoramentos, os mais recentes centrados ao nível da interface com o utilizador. Assim, apresenta-se neste relatório o módulo computacional

This document presents a back analysis procedure for identification of the elastic parameters of transversely isotropic rock cores, containing an overcoring triaxial strain probe, from the strains measured during a biaxial test. A three-dimensional finite element model was developed to simulate the biaxial test on the overcored rock specimen and to compute the strains at the location of the strain gauges. Different optimisation algorithms were tested and the most suitable one was selected. The back analysis procedure was tested for identification of the five elastic parameters and the two orientation angles that characterise a transversely isotropic rock core. In spite that, with the developed methodology, convergence was reached and all those parameters could be identified, sensitivity analyses demonstrated that the results obtained were not stable and therefore they were not reliable. By introducing constrains based on common practice and previous experience, a stable and robust methodology was achieved: the three elastic parameters, E1, E2 and ?2, are reliably identified using the value of G2 calculated with Saint-Venant’s expression and a fixed value of ?1, while the orientation parameters are obtained from observation of overcored rock. Analysis of the results shows that application of this methodology represents an enormous step forward when compared with the traditional use of isotropy. Besides, the methodology is general and can also be used with other types of overcoring equipment. The five elastic parameters and the two orientation angles obtained can then be used, together with the overcoring strains, to compute the complete in situ state of stress.

This document presents the work developed by the PhD student Margarida Espada in the scope of the course unit Risk Analysis in Geotechnical Works of the Doctoral Program in Civil Engineering of the Faculty of Engineering of the University of Porto (FEUP). This work presents a reliability analysis, using the response surface method (RSM), for the case study of the left bank excavation slopes of the Baihetan arch dam foundation, which is one of the case studies of Margarida Espada's PhD thesis. The aim was to approximate an implicit limit state function by computing safety factors, using a discrete element model of the Baihetan left bank developed in 3DEC software, where the shear strength parameters of two important rock discontinuities were considered as random variables. The probability of failure was then obtained through an iterative process, using approximation methods. This work presents the 3DEC model developed to analyse the stability conditions in the Baihetan left bank and the results of the reliability analysis.

O problema da solução das equações de Navier Stokes é abordado neste trabalhopropondo-se uma técnica de representação das incógnitas deslocamentos e pressõesvariáveis no espaço e no tempo em série de Delta de Dirac e suas derivadas no tempocujos coeficientes se determinam com base na solução de sistemas de equaçõesdiferenciais definidos no espaço (problema com três variáveis espaciais independentesque pode ser estudado pelo método de Monge).

Neste relatório apresentam-se as equações básicas da elastodinâmica linear, as soluçõesfundamentais e o teorema da representação no domínio de tempo e de frequência.Para movimentos finitos de casca formula-se o teorema dos trabalhos virtuais e a suaimplementação pelo método dos elementos finitos.

Neste relatório apresentam-se os principais aspectos da modelação sísmica de barragensabóbada. Assim propõe-se: modelos numéricos de geração de sismos por uma falha; aforma como o sismo actua na barragem; modelos para estudar a influência da albufeirana vibração do conjunto; métodos para solução das equações do movimento; e novasformulações de elementos finitos para o problema estrutural.

In pressure tunnels and shafts, in addition to the mechanical effect that excavation has on the rock mass and the disturbance that it introduces in the water flow, also the contained water can interact from the mechanical and hydraulic points of view with the rock mass. Seepage in fractured rock masses is dominated by the flow that takes place through the joints, and this is highly d(;pendent on their properties, namely aperture and roughness. Joint aperture, in tum, i::; very sensitive to stress changes. Special attention must therefore be paid to the joints in hydromechanical analyses. In this report are presented some studies that were carried out for the hydromechanical characterization of joints of the Alto Lindoso rock mass and numerical simulations of the hydromechanical behaviour of two selected sections of the high pressure circuit.