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This study highlights the continued need for numerical simulation methods to predict the earthquakeresponse and damage of masonry mosques, despite recent advances in research. TheKamanl

Circular/spherical rigid particle models that were initially applied to rock fracture studieswere not able to match the ratio of the compressive strength to tensile strength that occurs in rock.In addition, the predicted macroscopic friction angle was much lower than the known hard rockexperimental values. Several enhancements have been proposed to address these issues, namelythe use of a clumped particle logic or the adoption of polygonal/polyhedral grain structures, eitherrigid or flexible. In this work, a flexible 2D DEM based particle model (PM) that allows deformableparticles to interact in a simplified way is presented. The proposed flexible PM model keeps thecontact interaction simplicity and the reduced computational costs characteristic of circular rigidparticle models. The PM model is tested using biaxial tests and Brazilian tests. A discussion re-garding the influence of the grain deformability on the macroscopic elastic and strength response ispresented. It is shown that, when compared with a rigid model, the proposed flexible PM modelpredicts more reasonable indirect tensile strength to direct tensile strength ratio and requires asmaller value of contact fracture energy to give a good agreement with known experimental data.It is also shown that the proposed flexible PM model can predict a behaviour similar to that ob-tained using a flexible PM model through inner particle discretization that is more computationallydemanding.

Circular and spherical particle models are a class of discrete elements (DEM) that have been in-creasingly applied to fracture studies of quasi-brittle materials, such as rock and concrete, due to their proven ability to simulate fracture processes through random particle assemblies repre-senting quasi-brittle materials at the grain scale. More recently, DEM models have been applied to old stone masonry fracture studies. In order to extend its applicability to structures of larger di-mensions, an enhanced hybrid particle model is here proposed that allows finite elements with a given surface roughness, provided by the discretization of the element boundary with particles, to interact with the particulate media in which they are embedded. The performance of the hybrid model is compared with that of a traditional all-particle model under uniaxial testing. It is shown that similar results are obtained, namely in the elastic phase, figures of rupture, pre-peak and post-peak behaviour, while the hybrid model allows a significant computational run time re-duction of 20% to 25% in the coarse particle assemblies. Finally, the proposed hybrid model is applied in the simulation of shear tests of stone masonry walls, dry and mortared joints, providing a reasonably good agreement with both experimental results and predictions. For the rubble masonry tests, the hybrid model allows a computation run time reduction of around 40%, when compared with an all-particle model.

Discrete element models are widely used in the analysis of masonrystructures, as they provide an effective approximation of the observedbehaviour, namely for modelling collapse modes governed by thediscontinuities. Assessment of the response under earthquakes is an importantapplication of these models, which perform time domain dynamic analysisusing explicit algorithms. Rayleigh damping is generally used, but itsmass-proportional component has potential adverse effects, and the small stepsrequired by explicit solutions lead to large computational costs. An alternativedamping formulation employing Maxwell elements in parallel with the jointstiffness is proposed and tested. Problems involving linear elastic analysis ofcolumn bending, rocking of rigid blocks, and out-of-plane failure of masonrywalls under seismic loading are examined. The proposed damping approachdisplayed a sound mechanical performance and is computationally efficient.These early results justify further investigation of the technique for a broaderapplication in masonry structural dynamics.

This research explores the mechanical behavior of dry-joint masonry subjected to combined shear

Concrete gravity dams are mass concrete structures, often built on rock mass foundations, conceived to rely upon their weight for stability. To prevent sliding, these structures are usually keyed/embedded into the foundation, a good construction practice particularly relevant in medium to high intensity seismic zones. In stability analysis, the extra strength obtained by keying the dam into the foundation is usually either neglected or taken as a passive resistance, which, such as explored in this paper, do not reflect the real structural response in pre-collapse situations. Limit state philosophy requires the ultimate equilibrium conditions to be expressed as accurately as possible. In this paper, the rigid-body equilibrium of a wedgy model representing the dam and a downstream rock wedge is analyzed according to the large displacement regime. Failure mechanisms were identified, analytically described and numerically validated. Application to two Portuguese large concrete gravity dams led to safety factors considerably larger than those computed assuming the usual practice. The proposed approach is intended to support probabilistic and/or semi-probabilistic methodologies for safety assessment of concrete gravity dams, in the design and feasibility phases, in which the limit state approach is inherently followed.

Concrete gravity dams are mass concrete structures, often built on rock mass foundations, conceived to rely upon their weight for stability. To prevent sliding, these structures are usually keyed/embedded into the foundation, a good construction practice particularly relevant in medium to high intensity seismic zones. In stability analysis, the extra strength obtained by keying the dam into the foundation is usually either neglected or taken as a passive resistance, which, such as explored in this paper, do not reflect the real structural response in pre-collapse situations. Limit state philosophy requires the ultimate equilibrium conditions to be expressed as accurately as possible. In this paper, the rigid-body equilibrium of a wedgy model representing the dam and a downstream rock wedge is analyzed according to the large displacement regime. Failure mechanisms were identified, analytically described and numerically validated. Application to two Portuguese large concrete gravity dams led to safety factors considerably larger than those computed assuming the usual practice. The proposed approach is intended to support probabilistic and/or semi-probabilistic methodologies for safety assessment of concrete gravity dams, in the design and feasibility phases, in which the limit state approach is inherently followed.

A detailed modelling approach to represent masonry at the meso-scale is proposed, basedon the discrete element method, considering the nonlinear behavior of the joints and the units. Thefracture of units is represented by the bonded-block concept, in which a random network of potentialcracks is created, allowing the progressive development of failure mechanisms. For simplicity, onlythe 2D case is presented, but the extension to 3D is straightforward. A key component of the proposedmodel is a framework for a joint or interface constitutive model, including the post-peak softeningrange, taking into account the experimental fracture energies. In this model, the softening curves intension or shear are defined by piecewise linear segments, calibrated to reproduce the most commonmasonry constitutive models. The essential issues involved in the application of bonded-blockmodels to masonry are examined, namely the block shape, either Voronoi polygons or triangles; size;deformability; and the influence of the main constitutive parameters. Uniaxial compression tests areanalyzed in detail. The simulation of a well-known experiment of a brick panel under shear showsthe good performance of the proposed approach. The investigation results demonstrate the model

To predict the structural behaviour of ancient stone masonry walls is still a challenging taskdue to their strong heterogeneity. A rubble-stone masonry modeling methodology using a 2D particlemodel (2D-PM), based on the discrete element method is proposed given its ability to predict crackpropagation by taking directly into account the material structure at the grain scale. Rubble-stone(ancient) masonry walls tested experimentally under uniaxial compression loading conditions arenumerically evaluated. The stone masonry numerical models are generated from a close mappingprocess of the stone units and of the mortar surfaces. A calibration procedure for the stone-stoneand mortar-mortar contacts based on experimental data is presented. The numerical studies showthat the 2D-PM wall models can predict the formation and propagation of cracks, the initial stiffnessand the maximum load obtained experimentally in traditional stone masonry walls. To reducethe simulation times, it is shown that the wall lateral numerical model adopting a coarser mortardiscretization is a viable option for these walls. The mortar behaviour under compression with lateralconfinement is identified as an important micro-parameter, that influences the peak strength and theductility of rubble-masonry walls under uniaxial loading.

This paper is focused on the study of the dynamic behaviour of two large arch dams, and it presents some innovations for the improvement of Seismic and Structural Health Monitoring (SSHM) systems for dams. The work describes a methodology based on the integrated use of software for automatic monitoring data analysis and of computational 3D finite element (3DFE) models for dam dynamic behaviour simulation. The monitoring data analysis software was developed for automatic modal identification, in order to obtain natural frequencies and mode shapes, for automatic detection of vibrations induced by seismic events, to be distinguished from those caused by other operational sources, and for comparison between results retrieved from measured vibrations and numerical results from 3DFE modelling. The numerical simulations are carried out using a 3DFE program developed for dynamic analysis of dam-reservoir-foundation systems, based on a solid–fluid coupled formulation and considering the dam-water dynamic interaction, including calculation modules for complex modal analysis and for linear and non-linear seismic analysis. The case studies are two large arch dams that have been under continuous dynamic monitoring over the last ten years: Cabril dam (132 m high), the highest dam in Portugal, and Cahora Bassa dam (170 m high), in Mozambique, one of the highest dams in Africa. The SSHM systems installed in both dams have similar schemes and were designed to continuously record accelerations in several locations at the upper part of the dam body and near the dam-foundation interface, using uniaxial and triaxial accelerometers. The most significant experimental results from continuous dynamic monitoring are presented and compared with numerical results for both dams, with emphasis on the evolution of natural frequencies over time, including the vibration mode shapes for various water levels, and on the measured accelerations during low-intensity seismic events. Furthermore, the main results of non-linear seismic response simulations are provided, considering the effects due to joint movements and tensile and compressive concrete damage, aiming to assess the seismic performance of both dams based on the Endurance Time Analysis method.

Neste trabalho apresentam-se os primeiros resultados obtidos com um sistema de observação do comportamento dinâmico instalado na barragem do Cabril, em que se mostra a evolução no tempo das primeiras quatro frequências naturais da obra durante um período de observação de cerca de um ano. Descreve-se um conjunto de rotinas implementadas para a identificação automática das frequências naturais da estrutura; a filosofia de gestão dos dados que são adquiridos em contínuo; o controlo remoto do sistema e a transmissão de dados via internet, entre o centro de controlo do sistema instalado em obra e o centro de análise situado no Laboratório Nacional de Engenharia Civil (LNEC). Este sistema, baseado na medição de vibrações em contínuo, permite a caracterização do comportamento dinâmico desta obra ao longo do tempo, tendo em conta os diversos tipos de excitação, tais como o vento, a vibração provocada pelo funcionamento dos grupos de produção de energia e dos órgãos de descarga, micro tremores e sismos de média e elevada intensidade.

Rock blasting in urban settings is always extremely demanding in what regards the control of ground vibrations and other environmental impacts. Extensive underwater blasting was performed in the Leixões Harbour, an important harbour surrounded by a dense urban mesh. Macro-seismometers were installed in the perimeter and recorded more than one thousand blasts. After completion of the works classical curve fitting statistical analysis of peak particle velocity (PPV) as a function of charge and distance to the blast was performed. PPV spatial distributions were determined for the duration of the construction and maps of ground peak vibration velocity drawn. The results showed that the accuracy of the attenuation law is within the usual range for this kind of phenomenon. However, factors like terrain morphology or geology variations cannot easily be included in a regression law but can be treated in a simple numerical model. A dynamic finite-difference model was developed and the influence of these features was studied thus increasing the understanding of the problem and providing valuable insights into some aspects of the usually large dispersion in the PPV.

The presented work explores a combination of strain localization methods and strong discontinuity ap-proaches to remove the flaws (stress-locking and mesh bias dependence) of the classical strain localization methods, and reduce the sophistication of the strong-discontinuity based methods. The concept of strong discontinuity is initially substituted by that of weak discontinuity capturing the displacement jump through its distribution (smearing) in a finite length: the oriented with of a finite element through which the fracture passes in a rather diffuse manner. In other words, the classical strain localization concept is brought to the approach, no strong discontinuity enrichment is made and standard constitutive models are used in a classi-cal strain localization setting. The standard mesh refinement recovers the strong discontinuity concept by taking the element width to zero. However, in order to remove the spurious mesh orientation dependence, in a second stage specific localization modes are injected, via mixed finite elements formulations, to the path of elements that are going to capture the cracks. These modes, as well as their injection time, are character-ized from the information provided by the SDA. Then, the obtained approach enjoys the benefits of the strong discontinuities one, at a complexity similar to the classical, and simpler, localization methods.

A 3D spherical rigid particle discrete element formulation based on the Discrete Element Method, DEM, which has been used to model concrete, is described. The model has been shown to give a good agreement in terms of peak strength, fracture localization and crack patterns for various load conditions. An explicit formulation for the reinforcement model which allows the model to be applied to reinforced concrete is presented. The reinforced rigid particle model is validated against known shear transfer tests due to dowel action. It is shown that the model is able to predict the influence of the dowel stiffness and the localized fracture in the vicinity of the initial crack.

The paper presents the 2 main methods for the determination of the in-situ state of stress in rock masses, developed by the LNEC, and refers to a practical case in which they were used, at an underground powerhouse in northern Portugal.

Abstract. The application of discrete element models based on rigid block formulations to theanalysis of masonry walls under horizontal out-of-plane loading is discussed. The problemsraised by the representation of an irregular fabric by a simplified block pattern are addressed.A test problem provides a comparison of various regular and random block patterns, showingtheir influence on the failure loads. An example of application of a rigid block model to a wallcapacity problem is presented.

This paper presents a study, which describes the influence of the dynamic behaviour of the intake tower on the experimental results obtained from operational modal analysis of Cabril dam. A 3D finite element model, for the intake tower of Cabril dam, was developed in order to study the dynamic behaviour of this neighbour structure of the dam. The main dynamic parameters computed and evaluated experimentally for this structure are summarized and discussed, to understand how they can interfere in the modal identification of Cabril dam.

The paper aims at presenting some achievements of joint statistics. The different statistical distributions used for the description of the main parameters of the joint sets (attitude, intensity, area, and aperture) are introduced. For the attitude, a comparison of the existing distribution functions is made. For the intensity, the formulas for its determination are given, the applicable statistical models are presented, and the Poisson model for the occurrence of the discontinuities in rocks is discussed in depth. The statistical models for the distribution of the volume of the blocks of a rock mass are introduced. For the area, again, the formulas for its determination are given, and the applicable statistical models presented. Finally, the distribution function for the aperture is referred to, as well as a note on the roughness and waviness.

A libertação do estado de tensão inicial nos maciços rochosos é a principal acção a considerar em escavações subterrâneas. Como o estado de tensão é muito influenciado pela topografia e por condições geológicas particulares de cada local, a sua determinação a partir de ensaios de campo necessita de uma abordagem particular que, tendo em conta a natureza pontual dos ensaios e a sua variabilidade, permita estimar o estado de tensão global no maciço. Apresentam-se os ensaios realizados para o projecto de uma central subterrânea, com destaque para a modelação numérica que permitiu integrar os resultados dos ensaios e determinar o estado de tensão global mais provável. Os modelos usados permitiram considerar no estado de tensão inicial o efeito erosivo do rio ao escavar o vale e da abertura das galerias onde os ensaios foram executados.

This paper deals with issues related to the evaluation of design parameters from some types of in situ and laboratory tests performed, during the geotechnical investigations, to characterize the most relevant geotechnical properties of the rock mass and also the actions. Some considerations regarding safety evaluation and limit state design are presented. Several types of Rock Mechanics tests are introduced in some detail, and a few application examples covering different Rock Engineering projects are presented to show how test results can be analysed to determine their values to be used in the design. The following examples are shown: triaxial tests to determine the strength of the intact rock, a joint network study for a dam foundation, laboratory shear tests to assess the shear strength of rock joints, flat jack and overcoring tests to estimate the in situ state of stress for an underground cavern, and borehole dilatometer and large flat jack tests to evaluate the deformability of a concrete dam foundation.

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.