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The finite element method (FEM) is commonly used to design the reinforcement in concrete slabs. In order to simplify the analysis and to be able to utilize the superposition principle for evaluating the effect of load combinations, a linear analysis is generally adopted although concrete slabs normally have a pronounced non-linear response. This type of simplification in the modeling procedure will generally lead to unrealistic concentrations of cross-sectional moments and shear forces. Concrete cracks already at service loads, which leads to redistribution of moments and forces. The moment- and force-peaks, obtained through linear finite element analysis, can be redistributed to achieve a distribution more similar to what is seen in reality. The topic of redistribution is however poorly documented and design codes, such as the Eurocode for concrete structures, do not give descriptions of how to perform this in practice. In 2012, guidelines for finite element analysis for the design of reinforced concrete slabs were published in a joint effort between KTH Royal Institute of Technology, Chalmers University of Technology and ELU consulting engineers, which was financially supported by the Swedish Transport Administration. These guidelines aim to include the non-linear response of reinforced concrete into a linear analysis. In this thesis, the guidelines mentioned above are followed to obtain reinforcement plans based on crack control, for a fictitious case study bridge by means of a 3D finite element model. New models were then constructed for non-linear analyses, where the reinforcement plans were implemented into the models by means of both shell elements as well as a mixture of shell and solid elements. The results from the non-linear analyses have been compared to the assumptions given in the guidelines. The results from the non-linear analyses indicate that the recommendations given in the aforementioned guidelines are indeed reasonable when considering crack width control. The shell models yield crack widths equal to approximately half the design value. The solid models, however, yielded cracks widths that were 15 - 20$\%$ lower than the design value. The results show that many factors attribute to the structural behavior during cracking, most noticeably the fracture energy, a parameter not featured in the Eurocode for concrete structures. Some limitations of the models used in this thesis are mentioned as well as areas for further improvement.

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Béla Bartók's Piano Sonata Sz.80 is known for its integration of modernist language with traditional elements. However, due to Bartók's radical style of writing, it remains challenging to precisely define the piece's motives, voice-leading, and structure, even though pianists who perform it may intuitively comprehend them. Therefore, this study aims to elucidate the Piano Sonata's motivic and tonal structure, genesis and development. First, this study demonstrates Bartók's use of linear motives and progressions to elucidate the Piano Sonata's large-scale structure and demonstrate its internal coherence. Second, by comparing the published score with the facsimile of the Budapest Manuscript, it is possible to shed light on the significance of the changes that Bartók made, facilitating a better understanding of his intentions. Lastly, this study suggests interpretive decisions based on the analysis and manuscripts, thus providing performers with a more thorough understanding of the piece.

Orientador: Jefferson Sidney Camacho / Resumo: Este trabalho tem como objetivo principal analisar o comportamento linear estático e dinâmico de placas, com carregamento perpendicular ao seu plano médio, realizando a discretização estrutural com o elemento finito prismático regular linear. Na dedução das matrizes de rigidez e de massas do elemento finito em questão, utiliza-se a formulação com parâmetros generalizados e com coordenadas homogêneas, cujas funções aproximadoras com vinte e quatro monômios, respectivamente, foram extraídos do polinômio algébrico cúbico em “x”, “y” e “z”. Para a consideração do amortecimento utiliza-se o Método de Rayleigh e para a integração numérica ao longo do tempo utiliza-se o Método de Newmark, via algoritmo previsor / corretor. Ao final deste trabalho foram elaborados exemplos elucidativos visando uma análise quantitativa e qualitativa dos resultados obtidos, que foram comparados com os valores determinados do elemento finito de placa retangular. Como conclusões finais, para placas muito delgadas deve-se utilizar o elemento finito de placa retangular; em placas delgadas é possível a utilização dos elementos finitos de placa retangular e o prismático regular linear; e para placas espessas deve-se utilizar o elemento finito prismático regular linear. / Abstract: The main objective of this work is to analyze the static and dynamic linear behavior of plates with perpendicular loading to its mean plane, performing the structural discretization with the regular linear prismatic finite element. In the deduction of the stiffness and mass matrices of the finite element, it is used the formulation with generalized parameters and with homogeneous coordinates, whose approximate functions with twenty-four monomials, respectively, were extracted from the cubic algebraic polynomial in x, "y" and "z". For the damping consideration, it is used the Rayleigh Method and for the numerical integration by along the time it is used the Newmark Method, via forecaster / corrector algorithm. At the end of this work, elucidative examples were elaborated aiming a quantitative and qualitative analysis of the obtained results which were compared with the finite element determined values of rectangular plate. As final conclusions, for very thin plates must use the rectangular plate finite element; in thin plates it is possible to use the rectangular plate finite elements and the linear regular prismatic; and for thick plates the linear regular prismatic finite element must be used. / Mestre

Neste trabalho analisam-se grelhas de concreto armado, adaptando-se modelos não-lineares baseados na teoria mecânica do dano contínuo, com a incorporação da distorção da seção transversal ao estado de deformações. Discutem-se os modelos do CEB-158 (1985), da NBR-6118, para o concreto armado e modelos elastoplásticos para as barras da armadura longitudinal. Estuda-se a teoria de vigas de Timoshenko, onde são incorporadas as distorções da seção transversal à matriz de rigidez do elemento finito de viga. Discute-se o modelo de dano de MAZARS(1984), mostrando a sua formulação e parâmetros. Dedica-se especial atenção também à análise não-linear de grelhas de concreto armado, enfatizando-se as técnicas de solução dos sistemas de equações não-lineares. Por fim, faz-se uma revisão sobre o modelo da treliça clássica de Mörsch, tomando-a como base para a formulação de um modelo de resistência de esforços cortantes em uma seção transversal de concreto armado. / In this work, Reinforced concrete grids are analysed, assuming non-linear models based on the damage theory, and also incorporating into shear deformations. The proposed CEB-158 and the NBR-6118 models for reinforced concrete members are discussed, as well as the elastoplastic model assumed for the reinforcement. The Timoshenko's beam theory is studied and adopted to modify the beam stiffness matrix to take into account the shear deformation. The MAZARS' damage model is discussed, presenting the main aspects and parameters. Particular attention has been given to the reinforced concrete grid, pointing out the usual non-linear system solution techniques. Finally, the Mörsch truss model is revised and adopted to formulated an algorithm to deal with reinforced concrete beam elements, dividing the shear forces into two parts: the concrete and the reinforcement resultants.

abstract: A comprehensive study was performed on non-proprietary ultra-high-performance concrete (UHPC) material and several design methods were suggested based on numerous experimental results. Several sets of compression tests, direct tensile tests, and flexural tests were performed on UHPC to provide a better understanding of the mechanisms involved in the mechanical behavior of the fiber reinforced material. In addition to compressive tests, flexural tests, based on ASTM C1609 and EN 14651, were performed. The effect of the strain rate on the UHPC material was also investigated through the high-speed tensile tests at different strain rates. Alongside the usual measurement tools such as linear variable differential transformers (LVDT) and clip gages, digital image correlation (DIC) method was also used to capture the full-range deformations in the samples and localized crack propagations. Analytical approaches were suggested, based on the experimental results of the current research and other research groups, to provide design solutions for different applications and design approaches for UHPC and hybrid reinforced concrete (HRC) sections. The suggested methods can be used both in the ultimate limit state (ULS) and the serviceability limit state (SLS) design methods. Closed form relationships, based on the non-linear design of reinforced concrete, were used in the calculation of the load-deflection response of UHPC. The procedures were used in obtaining material properties from the flexural data using procedures that are based on back-calculation of material properties from the experimental results. Model simulations were compared with other results available in the literature. Performance of flexural reinforced UHPC concrete beam sections tested under different types of loading was addressed using a combination of fibers and rebars. The same analytical approach was suggested for the fiber reinforced concrete (FRC) sections strengthened (rehabilitated) by fiber reinforced polymers (FRP) and textile reinforced concrete (TRC). The objective is to validate the proper design procedures for flexural members as well as connection elements. The proposed solutions can be used to reduce total reinforcement by means of increasing the ductility of the FRC, HRC, and UHPC members in order to meet the required flexural reinforcement, which in some cases leads to total elimination of rebars. / Dissertation/Thesis / Masters Thesis Civil, Environmental and Sustainable Engineering 2018

Two methods for overcoming locking phenomena in isogeometric beams are presented. The first method blends the rotation of a Timoshenko beam with the rotation of a Bernoulli beam to produce realistic displacements in straight beams. The second method uses mixed variational principles, specifically the Hu-Washizu Principle, to produce realistic displacements as well as realistic strains without post-processing.

Denna avhandling är fokuserad på modellering av elkraftsystemkomponenter och deras representation vid simuleringar av elkraftsystem. Avhandlingen jämför olika linjäriseringstekniker. Dessa tekniker är såväl numeriska som analytiska och används vid linjärisering av ett dynamiskt system. Efter en linjärisering är det möjligt att beräkna egenvärdena av det linjäriserade systemet samt använda andra verktyg ämnade för studier av linjära system. I avhandlingen visas hur olika linjäriseringtekniker influerar egenvärdesberäkningen av det linjära systemet. I avhandlingen tas fram bandviddsreducerade linjära modeller av en kraftsystemkomponent med hjälp av två tekniker. Senare görs simuleringar med de linjära modellerna tillsammans med ett introducerat gränssnitt. Den studerade kraftsystemkomponenten är en tyristorstyrd seriekondensator (TCSC). En fördel med att använda en linjär representation av en kraftsystemkomponent är att det förenklar simuleringarna. Storleken på komplexiteten av en simulering vid lösandet av ekvationerna minskar och den konsumerade fysiska tiden att simulera minskar. En nackdel med en linjär modell är att dess giltighet kan vara begränsad. Behovet av att bygga linjära modeller av kraftsystemkomponenter torde även finnas i framtiden. Med dagens horisont (år 2006) finns behov av att bygga linjära modeller utgående från detaljerade modeller av bl a högspända likströmslänkar (HVDC-länkar), reaktiva effektkompensatorer (SVC) samt tyristorstyrda seriekondensatorer (TCSC). Hur skall dessa representeras när vi vill studera dynamiken av ett helt kraftsystem och det då är nödvändigt att reducera deras komplexitet? Denna frågeställning uppkommer när vi vill genomföra tidsdomänsimuleringar på en inte alltför detaljerad nivå av de individuella kraftsystemkomponenterna eller när vi vill linjärisera kraftsystemet för att studera dess stabilitet med hjälp av småsignalanalys. / This thesis is focused in modelling of power system components and their representation in simulations of power systems. The thesis compares different linearization techniques. These techniques are both numerical as well as analytical and are utilized when linearization of a dynamic system is desired. After a linearization it is possible to calculate the eigenvalues of the linearized system as well as to perform other applicable activities on a linear system. In the thesis it is shown how the linearization techniques influence the calculation of eigenvalues of the linear system. In the thesis bandwidth-reduced linear models of a power system component are developed using two techniques. The simulations with the linear models are done with an introduced interface system. The studied power system component is a Thyristor-Controlled Series Capacitor (TCSC). One advantage with using a linear representation of a power system component is that it simplifies the simulations. The size of the complexity of a simulation when solving the equations decreases and the consumed physical time to simulate becomes shorter. A disadvantage of a linear model is that its validity might be limited. The need of building linear models of power systems will continue to attract interest in the future. With the horizon of today (year 2006) there is a need of among other models to build linear models of detailed models of High Voltage Direct Current-links (HVDC-links), Static Var Compensators (SVCs), as well as Thyristor-Controlled Series Capacitors (TCSCs). How should these be represented when we want to study the dynamics of a whole power system and it is necessary to reduce their complexity? This question rises when we want to perform time-domain simulations with a not too detailed level of complexity of each individual power system component or if we want to linearize the power system and study it within small-signal stability analysis. / QC 20100915

La mayoría de las estructuras de hormigón armado se someten a solicitaciones combinadas de esfuerzos axiles, flexión, cortante y torsión. La fisuración del hormigón, plastificación de las armaduras y otros efectos no-lineales hacen que las secciones transversales de estos elementos presenten un comportamiento anisótropo que deriva en el acoplamiento de los esfuerzos normales y tangenciales. Es decir, esfuerzos normales o momentos flectores pueden producir deformaciones de corte y vice versa. Aunque en algunas ocaciones, esta interacción es considerada de forma simplificada en el dimensionamiento de estructuras, hasta el momento no se ha realizado un análisis profundo de los efectos acoplados en secciones de forma arbitraria bajo cargas 3D generales utilizando modelos de fibras.El objetivo principal de esta tesis es generalizar el análisis de secciones de hormigón armado mediante fibras, de forma que se pueda reproducir la res-puesta no-lineal acoplada frente a esfuerzos normales y tangenciales bajo solicitaciones tridimensionales generales. De igual forma, se pretende obtener, para los esfuerzos cortantes y torsión, la misma capacidad de representación de geometrías y combinación de materiales que ofrecen los modelos de fibras para esfuerzos de flexo-compresión.La primera problemática estriba en representar adecuadamente la cinemática de la sección transversal. Con la excepción de las deformaciones normales contenidas en el plano de la sección, no existe una teoría cinemática que a priori pueda dar la distribución del resto de deformaciones o tensiones en la sección, sin dejar de satisfacer las condiciones de equilibrio interno o continuidad entre las fibras que componen la misma.Por otra parte, para materiales anisótropos, como el hormigón fisurado, en general todos los esfuerzos internos pueden estar acoplados. Además, es preciso considerar la distorsión de la sección transversal para satisfacer el equilibrio entre fibras.El problema se aborda de forma general, considerando una sección de forma y materiales cualesquiera. Se parte del problema diferencial de equilibrio de un sólido con el que se ha podido deducir un sistema de equilibrio entre fibras (equilibrio a nivel sección). Se puede demostrar que éste es complementario al problema estándar de vigas. El sistema complementario permite recuperar información tridimensional que normalmente se pierde al resolver un problema de vigas.Posteriormente, se propone una solución interna del problema complementario, en la que el alabeo y la distorsión de la sección quedan expresados como una función de las deformaciones generalizadas de una viga: deformaciones axil y cortantes, curvaturas de flexión y torsión. No son necesarios grados de libertad adicionales a nivel estructura ni hipótesis a-priori sobre la forma de los campos de deformación o tensión interna.A partir de la formulación teórica, se desarrolla un modelo de elementos finitos plano de la sección transversal. El modelo está preparado para servir como respuesta constitutiva de cualquier tipo de elemento viga en sus puntos de integración. %Se evita así la necesidad de realizar un modelo de elementos sólidos de toda la barra para estudiar la respuesta frente a una combinación general de esfuerzos normales y tangenciales.Se implementan una serie de modelos constitutivos para distintos materiales. En particular, se implementa un modelo constitutivo triaxial para hormigón fisurado, considerando la anisotropía inducida por la fisuración e incluyendo la superficie de rotura según un criterio multiaxial.La formulación seccional es validada mediante varios casos de estudio teóricos y experimentales. La respuesta no-lineal acoplada bajo diversas combinaciones de esfuerzos normales y tangenciales es reproducida con precisión, lo cual queda patente tanto en las curvas esfuerzo-deformación obtenidas como en las matrices de rigidez seccionales.Finalmente, se recopilan las conclusiones derivadas de la presente investigación y seofren recomendaciones para futuros trabajos. / Most RC structures are subjected to combined normal and tangential forces, such as bending, axial load, shear and torsion. Concrete cracking, steel yielding and other material nonlinearities produce an anisotropic sectional response that results in a coupling between the effects of normal and shear forces, i.e. normal force or bending moments may produce shear strains and vice versa. Although this interaction is sometimes taken into account, in a simplified manner, in the design of RC structures, a deep analysis of the coupling effects of RC sections using fiber models has not yet been made for arbitrary shape sections under general 3D loading.The main objective of this thesis is to generalize the fiber-like sectional analysis of reinforced concrete elements, to make it capable of considering the coupled non-linear response under tangential and normal internal forces, from a general 3D loading.Similarly, it is desired to obtain, for torque and shear forces, the same capacity and versatility in reproducing complex geometry and materials combination that fiber-like sectional representations offers for bending and stretching.The first problematic lies in finding a proper representation of the section's kinematics under such general loading. Except for in-plane normal strains, there is no single kinematical theory capable of a-priori representing the correct distribution of the others strains or stresses satisfying, at the same time, inter-fiber equilibrium and continuity. On the other hand, for rather anisotropic materials, such as cracked concrete, all internal forces are, in general, coupled. It is also required that distortion is allowed for the section's kinematics in order to guarantee satisfaction of internal equilibrium.The problem is dealt in a general form considering arbitrary shaped sections and any material behaviour. Starting from the differential equilibrium of a solid, an inter-fiber equilibrium system (equilibrium at the sectional level) was deduced. This system, which is complementary to the standard equilibrium problem of a beam-column, allows to recuperate information of the three-dimensional problem that is generally lost when solving a beam problem.Further, a solution of the equilibrium at the sectional level is proposed in which the section's warping and distortion are posed as a function of the generalized beam-column strains (axial and shear strains, bending and torsion curvatures). No additional degrees of freedom are required at the structural level nor a-priori hypotheses on the distribution of the internal strains or stresses.After the theoretical formulation, a planar finite element model for cross-sectional analysis is developed. The model can be used as a constitutive law for general beam column elements at their integration points.A series of constitutive models have been implemented for several materials. In particular, a triaxial constitutive model for cracked concrete is implemented considering crackinduced anisotropy and a multiaxial failure criterion.The sectional formulation is validated by means of various theoretical and experimental case studies. Non-linear coupled response under normal and tangential internal forces is reproduced with accuracy, as can be seen both in the predicted internal force-strain curves and in the sectional stiffness matrixes.Finally, the conclusions drawn from the current research are summarized andrecomendations for future works are given.