<p> The objective of this thesis was to undertake an uncertainty analysis on the outputs from a two-dimensional hydrodynamic model. The analysis utilized an application of the Resource Management Associates' RMA2 model for the Upper St. Lawrence River in Ontario, Canada. Two uncertainty analysis methods, First-Order Second Moment (FOSM) and Monte Carlo analysis, are applied to calculate the uncertainty in water levels and velocities computed by the model.</p> <p> Both uncertainty analysis methods can be applied together with two-dimensional hydrodynamic modelling, but based on the findings of this work, the FOSM method is preferred. First, FOSM estimates of uncertainty are slightly larger than those obtained using Monte Carlo analysis. Thus, FOSM provides a conservative estimate of the uncertainty, a positive characteristic. Second, the FOSM method is simpler to apply than Monte Carlo analysis, requiring less information to describe the model inputs, fewer model executions and computations to calculate the uncertainty. Third, FOSM provides an immediate indication of the primary contributors to the uncertainty in the output, where Monte Carlo analysis requires additional effort to do the same.</p> <p> The model input that contributed the most to the uncertainty in the model outputs is the bottom resistance represented in RMA2 using Manning's n. The uncertainty in Manning's n is large and the model is sensitive to the parameter. As a result, a significant amount of uncertainty in the model outputs is contributed by this parameter.</p> <p> Uncertainty analysis is a practical addition to the two-dimensional hydrodynamic modelling process. The effort required to complete an uncertainty analysis using the FOSM method is minimal and the resulting insight is meaningful. It provides information to the model developer, quantifying how good the model actually is. It also provides a measure of the accuracy of the model for future model users or clients using hydrodynamic modelling outputs.</p> / Thesis / Master of Applied Science (MASc)

[pt] Este trabalho apresenta a utilização de métodos de equilíbrio limite em conjunto com métodos numéricos pela abordagem probabilística, para avaliação da estabilidade de uma barragem de rejeitos de mineração de ferro situada no quadrilátero ferrífero, no Estado de Minas Gerais. Foram conduzidas análises fazendo uso dos métodos de Bishop Simplificado (1955) e Morgenstern e Price (1965), além de análises numéricas de tensão vs deformação pelo método dos elementos finitos. Os campos de poropressão gerados no modelo numérico em dois momentos distintos da obra foram exportados para os modelos de equilíbrio limite, de forma que ambas análises fossem realizadas em condições piezométricas similares. Para as análises probabilísticas foram adotados os métodos FOSM (First Order Second Moment) e Monte Carlo. São apresentandos os conceitos básicos de análises de estabilidade determinísticas e probabilísticas, abrangendo os métodos de cálculo de FS, fundamentos de probabilidade e estatística e conceitos de modelagem numérica. Este estudo aborda também os princípios construtivos de barragens de rejeito de mineração, além da aplicação dos métodos probabilísticos para avaliação da estabilidade de uma barragem de rejeitos, construída pelo método de jusante. Os resultados indicaram que a consideração de uma superfície de ruptura livre foi fator preponderante na influência dos parâmetros na PR (Probabilidade de Ruptura). Concluiu-se também que os método de cálculo de FS teve maior influência no valor da PR do que as condições piezométricas da fundação. / [en] This work presents the use of limit equilibrium (LEM) and numerical methods by the probabilistic approach, in order to evaluate the stability of an iron ore tailings dam located in Minas Gerais s iron quadrangle. LEM analyzes were carried out using the Simplified Bishop (1955) and Morgenstern and Price (1965) methods, in addition to stress vs strain analyzes with the finite element method. The pore pressure fields generated in the numerical model at two distinct construction stages were exported to the limit equilibrium models in order to guarantee that both analyzes were performed on the same piezometric conditions. For the probabilistic analyzes the FOSM (First Order Second Moment) and Monte Carlo methods were applied. The basic concepts of deterministic and probabilistic stability analysis are presented together with the fundamentals of statistics and probability, FoS (Factor of Safety) calculation methods, as well as numerical modeling concepts. This work also addresses the definition and construction principles of tailings dams. This study presents an application of probabilistic methods to assess the stability of a tailings dam built and heightened by the downstream method. The results indicate that the consideration of a free failure surface was the decisive factor to the magnitude of the influence of each parameter on the PF (Probability of Failure). It was also concluded that the FoS calculation methods had a greater influence on the PF than the foundation s piezometric conditions.

[pt] Na prática geotécnica, a estabilidade de taludes é atualmente estudada apenas com análises determinísticas, obtendo-se o valor do Fator de Segurança (FS) da estrutura geotécnica. Estas análises são simplificadas, pois fornecem valores de FS sem considerar a variabilidade intrínseca dos solos e rochas. Desprezar as incertezas dos parâmetros geotécnicos pode levar a resultados pouco confiáveis sobre a segurança de taludes. Análises fundamentadas em conceitos estatísticos, chamadas probabilísticas, passam ser mais frequentes na geotecnia por permitirem considerar efeitos da variabilidade inerente aos materiais. Estas análises incorporam elementos estatísticos que possibilitam tratar FS como uma função e estudar suas propriedades. Como resultados finais, os métodos probabilísticos fornecem o índice de confiabilidade (beta) e a probabilidade de ruptura (Pr) da estrutura averiguada. Este trabalho aplica análises determinísticas e probabilísticas de um talude de 200m de altura da Mina do Cauê, Itabira, MG. A estabilidade do talude foi analisada por cinco métodos determinísticos usuais e três probabilísticos (FOSM, EP e MC). Os resultados indicam que a fixação da superfície crítica de ruptura fornece, em geral, valores de beta e Pr semelhantes aos obtidos quando a superfície pode variar livremente. Com a superfície crítica fixa observou-se, também, que os resultados de beta e Pr do talude variam significativamente com o método de equilíbrio limite adotado. Após comparação dos resultados, pode-se recomendar o uso de análises probabilísticas FOSM com base no método de Morgenstern-Price em análises semelhantes ao caso estudado. / [en] In current geotechnical practice, slope stability assessments are usually carried out only based on deterministic methods, obtaining a value of Safety Factor (FS). These analyses are simplified because the FS values do not consider the natural variability of soils and rocks. Disregarding the uncertainties inherent to geotechnical parameters may lead to unreliable results of slope safety. Probabilistic analyses, based on statistical concepts, have become more frequent in geotechnical practice, as they allow incorporating the materials intrinsic variability. These analyses are based on statistical elements that allow treating the FS as a function and studying its properties. The probabilistic methods indicate the reliability index (beta) and the probability of failure (Pr) of the verified geotechnical structure. This work presents deterministic and probabilistic analyses of a 200m high slope at the Cauê Mine, located in Itabira, Minas Gerais, Brazil, with basis on five usual deterministic methods and three probabilistic techniques (FOSM, Point Estimates and Monte Carlo). The results indicate that fixing the critical deterministic surface generally gives values of beta and Pr similar to those achieved when the surface is free to vary. It was also concluded that, with a fixed critical surface, beta and Pr results change significantly, when different Limit Equilibrium methods are adopted. It is recommended to use probabilistic FOSM analysis with Morgenstern and Price stability method in analyses similar to the one presented herein.

[pt] O presente trabalho propõe um método estatístico computacionalmente eficiente (chamado Green-FOSM) para consideração de incertezas em problemas geomecânicos, com o objetivo de melhorar o processo de tomada de decisão ao analisar problemas associados com o processo de injeção ou depleção de fluídos. A novidade do método proposto está associada com a utilização do método das funções de Green (GFA), que, com o auxílio do método estatístico first-order second-moment (FOSM), é utilizado para propagar as inerentes incertezas associadas às propriedades mecânicas do material para o campo de deslocamento da formação geológica. Além disso, através dos conceitos de grid estocástico e função de autocorrelação, o método proposto permite a consideração da variabilidade espacial de variáveis aleatórias de entrada que representam essas propriedades mecânicas. O GFA utiliza as soluções fundamentais da mecânica clássica (solução fundamental de Kelvin, solução fundamental de Melan, entre outras) e o teorema da reciprocidade para determinar o campo de deslocamento de uma formação geológica com geometria irregular e diferentes tipos de materiais. A grande vantagem deste método em relação ao clássico método dos elementos finitos (MEF) é que ele não requer a imposição de condições de contorno e a análise do problema pode ser realizada considerando apenas o domínio do reservatório ou outras regiões de interesse. Esta estratégia de modelagem diminui os graus de liberdade do modelo e o tempo de processamento da análise. Desta forma, como o GFA requer menos esforço computacional, este método torna-se ideal para ser utilizado na propagação de incertezas em problemas geomecânicos. Inicialmente, baseado no método das funções de Green original proposto por Peres et al. (2021), foi proposto uma versão iterativa do método Green-FOSM, que apresenta resultados estatísticos semelhantes aos encontrados através da clássica simulação de Monte Carlo (SMC). Nesta versão original, o campo de deslocamento é PUC-Rio - Certificação Digital Nº 1912634/CA calculado usando um esquema numérico iterativo que diminui o desempenho computacional do método e pode gerar problemas de convergência. Tais limitações tem dificultado a aplicação do GFA original e do método Green-FOSM iterativo em problemas reais. Assim, o presente trabalho desenvolveu uma nova versão do GFA que utiliza um esquema numérico não-iterativo. Para os problemas de validação analisados, o método não-iterativo demonstra ser até 17.5 vezes mais rápido do que a versão original. Além disso, esta versão demonstra ser capaz de expandir a aplicabilidade do GFA, pois os problemas de convergência foram eliminados e os resultados obtidos por este método, ao analisar um perfil geológico representativo do pré-sal brasileiro, são semelhantes aos encontrados via MEF. Por fim, a partir do GFA não-iterativo foi proposta uma versão não-iterativa do método Green-FOSM. Esta versão não-iterativa é capaz de analisar probabilisticamente formações geológicas complexas, como é o caso das formações geológicas do présal brasileiro. Utilizando os mesmos recursos computacionais, o método GreenFOSM não-iterativo é no mínimo 200 vezes mais rápido que o método iterativo. De forma geral, os resultados encontrados nas análises realizadas (determinísticas e probabilísticas) são próximos dos resultados obtidos pelo método de referência (MEF e SMC, respectivamente). / [en] The present work proposes a computationally efficient stochastic statistical method (called Green-FOSM) that considers uncertainties in geomechanical problems, with the objective of improving the decision-making process related to problems associated with the process of fluid injection or depletion. The novelty of the method lies in the use of the Green s function approach (GFA), which, together, with the first-order second-moment statistical method (FOSM), is used to propagate uncertainties associated with the mechanical properties of material to the displacement field of the geological formation. Furthermore, using the concepts of stochastic grid and autocorrelation function, the proposed method allows the consideration of the spatial variability of random variables that represent these mechanical properties. The GFA uses the fundamental solutions of classical mechanics (Kelvin fundamental solution, Melan fundamental solution, among others) and the reciprocity theorem to calculate the displacement field of a geological formation with irregular geometry, and different types of materials. The great advantage of this method compared to the classical finite element method (FEM) is that it does not require the imposition of boundary conditions and the analysis of the problem can be performed considering only the reservoir or other regions of interest. This modeling strategy decreases the degrees of freedom of the model and the CPU time of the deterministic analysis. In this way, as the GFA requires less computational effort, this approach becomes ideal for propagating the uncertainties in geomechanical problems. Initially, an iterative version of the Green-FOSM method was proposed, which presents statistical results similar to those found through the classic Monte Carlo simulation (MCS). In this initial version, the displacement field is calculated using an iterative numerical scheme, which decreases the computational performance of the method and can generate convergence problems. Such limitations would restrict the application of the PUC-Rio - Certificação Digital number 1912634/CA original GFA and the iterative Green-FOSM method in real problems. Thus, the present work also developed a new version of the GFA, which uses a non-iterative numerical scheme. For the proposed validation problems, the non-iterative method proved to be up to 17.5 times faster than the original version. This version is able to expand the applicability of the GFA, since the convergence problems were eliminated and the results obtained by this method, when analyzing a representative geological profile of the Brazilian pre-salt, are similar to those found via FEM. Finally, based on the non-iterative GFA, a non-iterative version of the Green-FOSM method was proposed. This non-iterative version is capable of probabilistically analyzing complex geological formations, such as the Brazilian pre-salt geological formations. Using the same computational resources, the non-iterative GreenFOSM method is at least 200 times faster than the iterative Green-FOSM method. In general, the results found in the investigated analyzes (deterministic and probabilistic) are close to the results obtained by the reference method (FEM and MCS, respectively).

Comúnmente los estudios geotécnicos de análisis de estabilidad de taludes se realizan mediante métodos determinísticos, en los cuales se obtiene como resultado un factor de seguridad (FS) de la estructura analizada. Sin embargo, estos métodos no cuantifican la incertidumbre en las variables de entrada (parámetros de resistencia del suelo), en el análisis y tampoco muestran los detalles de las variables más influyentes en la estabilidad del talud. Es por ello los métodos probabilísticos permiten superar estos problemas. Esta investigación tiene como objetivo principal realizar el análisis de estabilidad de taludes mediante el método probabilístico Primer Orden Segundo Momento (FOSM) dos recomendaciones de aplicabilidad, que ofrecen los autores Sandroni & Sayao (1992) y Mostyn y Li (1993). Se analizó un caso común y muy importantes porque fueron claves en la realización de métodos probabilísticos en la ingeniería geotécnica. El caso, es una de las carreteras más importantes de Brasil, el cual sufrió deslizamiento 2 años después de su construcción. Se realizó el análisis probabilístico dando como resultado que los parámetros de confiabilidad de la estructura eran muy bajos y la probabilidad de rotura muy alta. Se plantea la propuesta de solución de estabilidad con una probabilidad de rotura mínima. La investigación se realizó con datos realizados en Brasil, por la cantidad de datos que requieren los estudios probabilísticos. Los tipos de suelos que se estudiaron fueron cohesivos, los cuales se pueden encontrar también en la selva peruana. Sin embargo, no se está tomando en cuenta proyectos nacionales por lo que no cuentan con la información necesaria para la aplicación de esta metodología. / Geotechnical studies on slope stability analyzes are performed using deterministic methods, from which are obtained as a result of a safety factor (FS) of the structure analyzed. However, these methods do not quantify the uncertainty in the input variables (parameters of soil resistance), analysis and do not show the details of the most influential variables in the stability of the slope. Therefore, probabilistic methods can overcome these problems. This research has as main objective the analysis of slope stability by probabilistic first order second moment method (FOSM) by two applicability recommendation that we provide the authors Sandroni & Sayao (1992) and Mostyn and Li (1993); and a comparison with the results obtained. Resulting in the second method is more accurate by being number of analysis performed. The case is one of the most important roads in Brazil which suffered a landslide two years after its construction. probabilistic analysis result that reliability parameters of the structure were very low and very high probability of rupture was performed. It was decided to make the solution stability with a minimum breaking probability. The research was carried out with studies of Brazil, for the amount of data that is required when probabilistic studies are conducted. Soil types found were generally cohesive, which can be found in the Peruvian jungle. However, there is no necessary data projects with this amount of data is required. / Tesis

<div>In this thesis, the role of location-specific microstructural features in the fatigue performance of the safety-critical aerospace components made of Nickel (Ni)-base superalloys and linear friction welded (LFW) Titanium (Ti) alloys has been studied using crystal plasticity finite element (CPFE) simulations, energy dispersive X-ray diffraction (EDD), backscatter electron (BSE) images and digital image correlation (DIC).</div><div><br></div><div>In order to develop a microstructure-sensitive fatigue life prediction framework, first, it is essential to build trust in the quantitative prediction from CPFE analysis by quantifying uncertainties in the mechanical response from CPFE simulations. Second, it is necessary to construct a unified fatigue life prediction metric, applicable to multiple material systems; and a calibration strategy of the unified fatigue life model parameter accounting for uncertainties originating from CPFE simulations and inherent in the experimental calibration dataset. To achieve the first task, a genetic algorithm framework is used to obtain the statistical distributions of the crystal plasticity (CP) parameters. Subsequently, these distributions are used in a first-order, second-moment method to compute the mean and the standard deviation for the stress along the loading direction (σ_load), plastic strain accumulation (PSA), and stored plastic strain energy density (SPSED). The results suggest that an ~10% variability in σ_load and 20%-25% variability in the PSA and SPSED values may exist due to the uncertainty in the CP parameter estimation. Further, the contribution of a specific CP parameter to the overall uncertainty is path-dependent and varies based on the load step under consideration. To accomplish the second goal, in this thesis, it is postulated that a critical value of the SPSED is associated with fatigue failure in metals and independent of the applied load. Unlike the classical approach of estimating the (homogenized) SPSED as the cumulative area enclosed within the macroscopic stress-strain hysteresis loops, CPFE simulations are used to compute the (local) SPSED at each material point within polycrystalline aggregates of 718Plus, an additively manufactured Ni-base superalloy. A Bayesian inference method is utilized to calibrate the critical SPSED, which is subsequently used to predict fatigue lives at nine different strain ranges, including strain ratios of 0.05 and -1, using nine statistically equivalent microstructures. For each strain range, the predicted lives from all simulated microstructures follow a log-normal distribution; for a given strain ratio, the predicted scatter is seen to be increasing with decreasing strain amplitude and are indicative of the scatter observed in the fatigue experiments. Further, the log-normal mean lives at each strain range are in good agreement with the experimental evidence. Since the critical SPSED captures the experimental data with reasonable accuracy across various loading regimes, it is hypothesized to be a material property and sufficient to predict the fatigue life.</div><div><br></div><div>Inclusions are unavoidable in Ni-base superalloys, which lead to two competing failure modes, namely inclusion- and matrix-driven failures. Each factor related to the inclusion, which may contribute to crack initiation, is isolated and systematically investigated within RR1000, a powder metallurgy produced Ni-base superalloy, using CPFE simulations. Specifically, the role of the inclusion stiffness, loading regime, loading direction, a debonded region in the inclusion-matrix interface, microstructural variability around the inclusion, inclusion size, dissimilar coefficient of thermal expansion (CTE), temperature, residual stress, and distance of the inclusion from the free surface are studied in the emergence of two failure modes. The CPFE analysis indicates that the emergence of a failure mode is an outcome of the complex interaction between the aforementioned factors. However, the possibility of a higher probability of failure due to inclusions is observed with increasing temperature, if the CTE of the inclusion is higher than the matrix, and vice versa. Any overall correlation between the inclusion size and its propensity for damage is not found, based on inclusion that is of the order of the mean grain size. Further, the CPFE simulations indicate that the surface inclusions are more damaging than the interior inclusions for similar surrounding microstructures. These observations are utilized to instantiate twenty realistic statistically equivalent microstructures of RR1000 – ten containing inclusions and remaining ten without inclusions. Using CPFE simulations with these microstructures at four different temperatures and three strain ranges for each temperature, the critical SPSED is calibrated as a function of temperature for RR1000. The results suggest that critical SPSED decreases almost linearly with increasing temperature and is appropriate to predict the realistic emergence of the competing failure modes as a function of applied strain range and temperature.</div><div><br></div><div>LFW process leads to the development of significant residual stress in the components, and the role of residual stress in the fatigue performance of materials cannot be overstated. Hence, to ensure fatigue performance of the LFW Ti alloys, residual strains in LFW of similar (Ti-6Al-4V welded to Ti-6Al-4V or Ti64-Ti64) and dissimilar (Ti-6Al-4V welded to Ti-5Al-5V-5Mo-3Cr or Ti64-Ti5553) Ti alloys have been characterized using EDD. For each type of LFW, one sample is chosen in the as-welded (AW) condition and another sample is selected after a post-weld heat treatment (HT). Residual strains have been separately studied in the alpha and beta phases of the material, and five components (three axial and two shear) have been reported in each case. In-plane axial components of the residual strains show a smooth and symmetric behavior about the weld center for the Ti64-Ti64 LFW samples in the AW condition, whereas these components in the Ti64-Ti5553 LFW sample show a symmetric trend with jump discontinuities. Such jump discontinuities, observed in both the AW and HT conditions of the Ti64-Ti5553 samples, suggest different strain-free lattice parameters in the weld region and the parent material. In contrast, the results from the Ti64-Ti64 LFW samples in both AW and HT conditions suggest nearly uniform strain-free lattice parameters throughout the weld region. The observed trends in the in-plane axial residual strain components have been rationalized by the corresponding microstructural changes and variations across the weld region via BSE images. </div><div><br></div><div>In the literature, fatigue crack initiation in the LFW Ti-6Al-4V specimens does not usually take place in the seemingly weakest location, i.e., the weld region. From the BSE images, Ti-6Al-4V microstructure, at a distance from the weld-center, which is typically associated with crack initiation in the literature, are identified in both AW and HT samples and found to be identical, specifically, equiaxed alpha grains with beta phases present at the alpha grain boundaries and triple points. Hence, subsequent fatigue performance in LFW Ti-6Al-4V is analyzed considering the equiaxed alpha microstructure.</div><div><br></div><div>The LFW components made of Ti-6Al-4V are often designed for high cycle fatigue performance under high mean stress or high R ratios. In engineering practice, mean stress corrections are employed to assess the fatigue performance of a material or structure; albeit this is problematic for Ti-6Al-4V, which experiences anomalous behavior at high R ratios. To address this problem, high cycle fatigue analyses are performed on two Ti-6Al-4V specimens with equiaxed alpha microstructures at a high R ratio. In one specimen, two micro-textured regions (MTRs) having their c-axes near-parallel and perpendicular to the loading direction are identified. High-resolution DIC is performed in the MTRs to study grain-level strain localization. In the other specimen, DIC is performed on a larger area, and crack initiation is observed in a random-textured region. To accompany the experiments, CPFE simulations are performed to investigate the mechanistic aspects of crack initiation, and the relative activity of different families of slip systems as a function of R ratio. A critical soft-hard-soft grain combination is associated with crack initiation indicating possible dwell effect at high R ratios, which could be attributed to the high-applied mean stress and high creep sensitivity of Ti-6Al-4V at room temperature. Further, simulations indicated more heterogeneous deformation, specifically the activation of multiple families of slip systems with fewer grains being plasticized, at higher R ratios. Such behavior is exacerbated within MTRs, especially the MTR composed of grains with their c-axes near parallel to the loading direction. These features of micro-plasticity make the high R ratio regime more vulnerable to fatigue damage accumulation and justify the anomalous mean stress behavior experienced by Ti-6Al-4V at high R ratios.</div><div><br></div>