In order to have safe and economy construction, different sources of uncertainty should be properly characterized and considered in structural design and verification. Commonly, reliability analysis is being used to evaluate the consistency of design process, including the uncertainty. A full probabilistic approach is an appropriate tool to consider the aleatory portion of uncertainty. However, in dealing with epistemic uncertainty in reliability analysis, modern mathematical tools such as fuzzy logic is required. Unreinforced masonry (URM) is known as sustainable building material and is on the top of worldwide building materials consumed in residential buildings. In this thesis, based on the available information on material, probabilistic models including the involved uncertainty for masonry properties has been provided for reliability study. Furthermore, a new experimental procedure for initial shear and friction coefficient, which theoretically reduces epistemic uncertainty, has been introduced. The unreinforced masonry walls that are important structural members in typical masonry buildings have been chosen as the cases of reliability study. Different verification methods for combination of in-plane shear and compression according to various codes has been collected and direct deterministic equations to predict the capacity has been extracted. In order to identify uncertainty (inaccuracy in design models), the observed (experimental results) load-carrying capacity are compared with predicted values and then the relevant uncertainty in models has been derived. Several reliability analysis using only stochastic method and using fuzzy-randomness technique has been conducted. The effect of uncertainty on assessed reliability has been highlighted. Additionally, the distinction between linear and non-linear application of partial safety factors has been investigated. Finally, by means of 3D graphs the actual reliability level of various masonry walls designed according to the latest German National Annex code DIN EN 1996-1-1 /NA :2012-05 on different load situation has been illustrated. / Analyse von unbewehrten Mauerwerkskonstruktionen mit unscharfen Daten Ingenieurmethoden zur Überprüfung von unbewehrten Mauerwerkswänden unter Scheibenschub (Probabilistische und Fuzzy-Methoden) Um sichere und ökonomische Konstruktionen zu planen, sollten die Datenunschärfe mit ihren verschiedenen Quellen bzw. Ursachen richtig charakterisiert und bei der Tragwerksplanung bzw. dem Nachweis berücksichtigt werden. Üblicherweise wird eine Zuverlässigkeitsanalyse angewendet, um einen konsistenten Tragwerksentwurf einschließlich der Unschärfe zu beurteilen. Eine vollständige probabilistische Näherungslösung ist ein brauchbares Werkzeug, um aleatorische Unschärfe zu berücksichtigen. Für die Erfassung der epistemischen Unschärfe bei der Zuverlässigkeitsanalyse sind moderne mathematische Werkzeuge wie z.B. die Fuzzy-Set-Theorie erforderlich. Unbewehrtes Mauerwerk (URM) ist als nachhaltiges Baumaterial bekannt und weltweit an der Spitze der verbauten Baumaterialien in Wohngebäuden. In dieser wissenschaftlichen Arbeit, die sich auf die verfügbaren Informationen über das Material stützt, werden probabilistische Modelle, einschließlich der zugehörigen wahrscheinlichkeitsbasierten Unschärfen der Mauerwerkseigenschaften, für die Zuverlässigkeitsstudie zur Verfügung gestellt. Außerdem wurde ein neues experimentelles Verfahren für die Ermittlung der Haftscherfestigkeit und den Reibungsbeiwert eingeführt, um die (epistemische) Unschärfe zu reduzieren. Unbewehrte Mauerwerkswände, die wichtige Tragglieder in typischen Mauerwerksgebäuden sind, wurden für die Zuverlässigkeitsstudie ausgewählt. Verschiedene Nachweismethoden für die Kombination von Scheibenschub und Druckbeanspruchung wurden nach verschiedenen Normen zusammengestellt und deterministische Gleichungen, zur Ermittlung der Tragfähigkeit herausgearbeitet. Um Unschärfe zu identifizieren (Ungenauigkeit der Modelle), werden die beobachtete Tragfähigkeit (experimentelle Ergebnisse) mit rechnerischen Werten verglichen, und daraus relevante Aussagen zur Modellunschärfe abgeleitet. Verschiedene Zuverlässigkeitsanalysen wurden zunächst mit stochastischen Methoden und danach mit einem fuzzy-randomness basierten Vorgehen geführt. Die Auswirkung der Unschärfe auf die bewertete Zuverlässigkeit wird herausgestellt. Zusätzlich wurde der Unterschied zwischen der linearen und nichtlinearen Anwendung zur Bestimmung von Teilsicherheitsfaktoren untersucht. Schließlich wird mithilfe von 3D-Graphen das Zuverlässigkeitsniveau von verschiedenen bemessenen Mauerwerkswänden nach dem letzten deutschen nationalen Anhang, DIN EN 1996-1-1/NA:2012-05, für verschiedene Lastsituationen dargestellt.

In order to have safe and economy construction, different sources of uncertainty should be properly characterized and considered in structural design and verification. Commonly, reliability analysis is being used to evaluate the consistency of design process, including the uncertainty. A full probabilistic approach is an appropriate tool to consider the aleatory portion of uncertainty. However, in dealing with epistemic uncertainty in reliability analysis, modern mathematical tools such as fuzzy logic is required. Unreinforced masonry (URM) is known as sustainable building material and is on the top of worldwide building materials consumed in residential buildings. In this thesis, based on the available information on material, probabilistic models including the involved uncertainty for masonry properties has been provided for reliability study. Furthermore, a new experimental procedure for initial shear and friction coefficient, which theoretically reduces epistemic uncertainty, has been introduced. The unreinforced masonry walls that are important structural members in typical masonry buildings have been chosen as the cases of reliability study. Different verification methods for combination of in-plane shear and compression according to various codes has been collected and direct deterministic equations to predict the capacity has been extracted. In order to identify uncertainty (inaccuracy in design models), the observed (experimental results) load-carrying capacity are compared with predicted values and then the relevant uncertainty in models has been derived. Several reliability analysis using only stochastic method and using fuzzy-randomness technique has been conducted. The effect of uncertainty on assessed reliability has been highlighted. Additionally, the distinction between linear and non-linear application of partial safety factors has been investigated. Finally, by means of 3D graphs the actual reliability level of various masonry walls designed according to the latest German National Annex code DIN EN 1996-1-1 /NA :2012-05 on different load situation has been illustrated. / Analyse von unbewehrten Mauerwerkskonstruktionen mit unscharfen Daten Ingenieurmethoden zur Überprüfung von unbewehrten Mauerwerkswänden unter Scheibenschub (Probabilistische und Fuzzy-Methoden) Um sichere und ökonomische Konstruktionen zu planen, sollten die Datenunschärfe mit ihren verschiedenen Quellen bzw. Ursachen richtig charakterisiert und bei der Tragwerksplanung bzw. dem Nachweis berücksichtigt werden. Üblicherweise wird eine Zuverlässigkeitsanalyse angewendet, um einen konsistenten Tragwerksentwurf einschließlich der Unschärfe zu beurteilen. Eine vollständige probabilistische Näherungslösung ist ein brauchbares Werkzeug, um aleatorische Unschärfe zu berücksichtigen. Für die Erfassung der epistemischen Unschärfe bei der Zuverlässigkeitsanalyse sind moderne mathematische Werkzeuge wie z.B. die Fuzzy-Set-Theorie erforderlich. Unbewehrtes Mauerwerk (URM) ist als nachhaltiges Baumaterial bekannt und weltweit an der Spitze der verbauten Baumaterialien in Wohngebäuden. In dieser wissenschaftlichen Arbeit, die sich auf die verfügbaren Informationen über das Material stützt, werden probabilistische Modelle, einschließlich der zugehörigen wahrscheinlichkeitsbasierten Unschärfen der Mauerwerkseigenschaften, für die Zuverlässigkeitsstudie zur Verfügung gestellt. Außerdem wurde ein neues experimentelles Verfahren für die Ermittlung der Haftscherfestigkeit und den Reibungsbeiwert eingeführt, um die (epistemische) Unschärfe zu reduzieren. Unbewehrte Mauerwerkswände, die wichtige Tragglieder in typischen Mauerwerksgebäuden sind, wurden für die Zuverlässigkeitsstudie ausgewählt. Verschiedene Nachweismethoden für die Kombination von Scheibenschub und Druckbeanspruchung wurden nach verschiedenen Normen zusammengestellt und deterministische Gleichungen, zur Ermittlung der Tragfähigkeit herausgearbeitet. Um Unschärfe zu identifizieren (Ungenauigkeit der Modelle), werden die beobachtete Tragfähigkeit (experimentelle Ergebnisse) mit rechnerischen Werten verglichen, und daraus relevante Aussagen zur Modellunschärfe abgeleitet. Verschiedene Zuverlässigkeitsanalysen wurden zunächst mit stochastischen Methoden und danach mit einem fuzzy-randomness basierten Vorgehen geführt. Die Auswirkung der Unschärfe auf die bewertete Zuverlässigkeit wird herausgestellt. Zusätzlich wurde der Unterschied zwischen der linearen und nichtlinearen Anwendung zur Bestimmung von Teilsicherheitsfaktoren untersucht. Schließlich wird mithilfe von 3D-Graphen das Zuverlässigkeitsniveau von verschiedenen bemessenen Mauerwerkswänden nach dem letzten deutschen nationalen Anhang, DIN EN 1996-1-1/NA:2012-05, für verschiedene Lastsituationen dargestellt.

'Independent component analysis' (ICA) ist ein Werkzeug der statistischen Datenanalyse und Signalverarbeitung, welches multivariate Signale in ihre Quellkomponenten zerlegen kann. Obwohl das klassische ICA Modell sehr nützlich ist, gibt es viele Anwendungen, die Erweiterungen von ICA erfordern. In dieser Dissertation präsentieren wir neue Verfahren, die die Funktionalität von ICA erweitern: (1) Zuverlässigkeitsanalyse und Gruppierung von unabhängigen Komponenten durch Hinzufügen von Rauschen, (2) robuste und überbestimmte ('over-complete') ICA durch Ausreissererkennung, und (3) nichtlineare ICA mit Kernmethoden. / Independent component analysis (ICA) is a tool for statistical data analysis and signal processing that is able to decompose multivariate signals into their underlying source components. Although the classical ICA model is highly useful, there are many real-world applications that require powerful extensions of ICA. This thesis presents new methods that extend the functionality of ICA: (1) reliability and grouping of independent components with noise injection, (2) robust and overcomplete ICA with inlier detection, and (3) nonlinear ICA with kernel methods.

In geotechnical engineering, the uncertainties such as the variability and uncertainty inherent in the geotechnical properties have caught more and more attentions from researchers and engineers. They have found that a single “Factor of Safety” calculated by traditional deterministic analyses methods can not represent the slope stability exactly. Recently in order to provide a more rational mathematical framework to incorporate different types of uncertainties in the slope stability estimation, reliability analyses and non-deterministic methods, which include probabilistic and non probabilistic (imprecise methods) methods, have been applied widely. In short, the slope non-deterministic analysis is to combine the probabilistic analysis or non probabilistic analysis with the deterministic slope stability analysis. It cannot be regarded as a completely new slope stability analysis method, but just an extension of the slope deterministic analysis. The slope failure probability calculated by slope non-deterministic analysis is a kind of complement of safety factor. Therefore, the accuracy of non deterministic analysis is not only depended on a suitable probabilistic or non probabilistic analysis method selected, but also on a more rigorous deterministic analysis method or geological model adopted. In this thesis, reliability concepts have been reviewed first, and some typical non-deterministic methods, including Monte Carlo Simulation (MCS), First Order Reliability Method (FORM), Point Estimate Method (PEM) and Random Set Theory (RSM), have been described and successfully applied to the slope stability analysis based on a numerical simulation method-Strength Reduction Method (SRM). All of the processes have been performed in a commercial finite difference code FLAC and a distinct element code UDEC. First of all, as the fundamental of slope reliability analysis, the deterministic numerical simulation method has been improved. This method has a higher accuracy than the conventional limit equilibrium methods, because of the reason that the constitutive relationship of soil is considered, and fewer assumptions on boundary conditions of slope model are necessary. However, the construction of slope numerical models, particularly for the large and complicated models has always been very difficult and it has become an obstacle for application of numerical simulation method. In this study, the excellent spatial analysis function of Geographic Information System (GIS) technique has been introduced to help numerical modeling of the slope. In the process of modeling, the topographic map of slope has been gridded using GIS software, and then the GIS data was transformed into FLAC smoothly through the program built-in language FISH. At last, the feasibility and high efficiency of this technique has been illustrated through a case study-Xuecheng slope, and both 2D and 3D models have been investigated. Subsequently, three most widely used probabilistic analyses methods, Monte Carlo Simulation, First Order Reliability Method and Point Estimate Method applied with Strength Reduction Method have been studied. Monte Carlo Simulation which needs to repeat thousands of deterministic analysis is the most accurate probabilistic method. However it is too time consuming for practical applications, especially when it is combined with numerical simulation method. For reducing the computation effort, a simplified Monte Carlo Simulation-Strength Reduction Method (MCS-SRM) has been developed in this study. This method has estimated the probable failure of slope and calculated the mean value of safety factor by means of soil parameters first, and then calculated the variance of safety factor and reliability of slope according to the assumed probability density function of safety factor. Case studies have confirmed that this method can reduce about 4/5 of time compared with traditional MCS-SRM, and maintain almost the same accuracy. First Order Reliability Method is an approximate method which is based on the Taylor\'s series expansion of performance function. The closed form solution of the partial derivatives of the performance function is necessary to calculate the mean and standard deviation of safety factor. However, there is no explicit performance function in numerical simulation method, so the derivative expressions have been replaced with equivalent difference quotients to solve the differential quotients approximately in this study. Point Estimate Method is also an approximate method involved even fewer calculations than FORM. In the present study, it has been integrated with Strength Reduction Method directly. Another important observation referred to the correlation between the soil parameters cohesion and friction angle. Some authors have found a negative correlation between cohesion and friction angle of soil on the basis of experimental data. However, few slope probabilistic studies are found to consider this negative correlation between soil parameters in literatures. In this thesis, the influence of this correlation on slope probability of failure has been investigated based on numerical simulation method. It was found that a negative correlation considered in the cohesion and friction angle of soil can reduce the variability of safety factor and failure probability of slope, thus increasing the reliability of results. Besides inter-correlation of soil parameters, these are always auto-correlated in space, which is described as spatial variability. For the reason that knowledge on this character is rather limited in literature, it is ignored in geotechnical engineering by most researchers and engineers. In this thesis, the random field method has been introduced in slope numerical simulation to simulate the spatial variability structure, and a numerical procedure for a probabilistic slope stability analysis based on Monte Carlo simulation was presented. The soil properties such as cohesion and friction angle were discretized to continuous random fields based on local averaging method. In the case study, both stationary and non-stationary random fields have been investigated, and the influence of spatial variability and averaging domain on the convergence of numerical simulation and probability of failure was studied. In rock medium, the structure faces have very important influence on the slope stability, and the rock material can be modeled as the combination of rigid or deformable blocks with joints in distinct element method. Therefore, much more input parameters like strength of joints are required to input the rock slope model, which increase the uncertainty of the results of numerical model. Furthermore, because of the limitations of the current laboratory and in-site testes, there is always lack of exact values of geotechnical parameters from rock material, even the probability distribution of these variables. Most of time, engineers can only estimate the interval of these variables from the limit testes or the expertise’s experience. In this study, to assess the reliability of the rock slope, a Random Set Distinct Element Method (RS-DEM) has been developed through coupling of Random Set Theory and Distinct Element Method, and applied in a rock slope in Sichuan province China.

In geotechnical engineering, the uncertainties such as the variability and uncertainty inherent in the geotechnical properties have caught more and more attentions from researchers and engineers. They have found that a single “Factor of Safety” calculated by traditional deterministic analyses methods can not represent the slope stability exactly. Recently in order to provide a more rational mathematical framework to incorporate different types of uncertainties in the slope stability estimation, reliability analyses and non-deterministic methods, which include probabilistic and non probabilistic (imprecise methods) methods, have been applied widely. In short, the slope non-deterministic analysis is to combine the probabilistic analysis or non probabilistic analysis with the deterministic slope stability analysis. It cannot be regarded as a completely new slope stability analysis method, but just an extension of the slope deterministic analysis. The slope failure probability calculated by slope non-deterministic analysis is a kind of complement of safety factor. Therefore, the accuracy of non deterministic analysis is not only depended on a suitable probabilistic or non probabilistic analysis method selected, but also on a more rigorous deterministic analysis method or geological model adopted. In this thesis, reliability concepts have been reviewed first, and some typical non-deterministic methods, including Monte Carlo Simulation (MCS), First Order Reliability Method (FORM), Point Estimate Method (PEM) and Random Set Theory (RSM), have been described and successfully applied to the slope stability analysis based on a numerical simulation method-Strength Reduction Method (SRM). All of the processes have been performed in a commercial finite difference code FLAC and a distinct element code UDEC. First of all, as the fundamental of slope reliability analysis, the deterministic numerical simulation method has been improved. This method has a higher accuracy than the conventional limit equilibrium methods, because of the reason that the constitutive relationship of soil is considered, and fewer assumptions on boundary conditions of slope model are necessary. However, the construction of slope numerical models, particularly for the large and complicated models has always been very difficult and it has become an obstacle for application of numerical simulation method. In this study, the excellent spatial analysis function of Geographic Information System (GIS) technique has been introduced to help numerical modeling of the slope. In the process of modeling, the topographic map of slope has been gridded using GIS software, and then the GIS data was transformed into FLAC smoothly through the program built-in language FISH. At last, the feasibility and high efficiency of this technique has been illustrated through a case study-Xuecheng slope, and both 2D and 3D models have been investigated. Subsequently, three most widely used probabilistic analyses methods, Monte Carlo Simulation, First Order Reliability Method and Point Estimate Method applied with Strength Reduction Method have been studied. Monte Carlo Simulation which needs to repeat thousands of deterministic analysis is the most accurate probabilistic method. However it is too time consuming for practical applications, especially when it is combined with numerical simulation method. For reducing the computation effort, a simplified Monte Carlo Simulation-Strength Reduction Method (MCS-SRM) has been developed in this study. This method has estimated the probable failure of slope and calculated the mean value of safety factor by means of soil parameters first, and then calculated the variance of safety factor and reliability of slope according to the assumed probability density function of safety factor. Case studies have confirmed that this method can reduce about 4/5 of time compared with traditional MCS-SRM, and maintain almost the same accuracy. First Order Reliability Method is an approximate method which is based on the Taylor\'s series expansion of performance function. The closed form solution of the partial derivatives of the performance function is necessary to calculate the mean and standard deviation of safety factor. However, there is no explicit performance function in numerical simulation method, so the derivative expressions have been replaced with equivalent difference quotients to solve the differential quotients approximately in this study. Point Estimate Method is also an approximate method involved even fewer calculations than FORM. In the present study, it has been integrated with Strength Reduction Method directly. Another important observation referred to the correlation between the soil parameters cohesion and friction angle. Some authors have found a negative correlation between cohesion and friction angle of soil on the basis of experimental data. However, few slope probabilistic studies are found to consider this negative correlation between soil parameters in literatures. In this thesis, the influence of this correlation on slope probability of failure has been investigated based on numerical simulation method. It was found that a negative correlation considered in the cohesion and friction angle of soil can reduce the variability of safety factor and failure probability of slope, thus increasing the reliability of results. Besides inter-correlation of soil parameters, these are always auto-correlated in space, which is described as spatial variability. For the reason that knowledge on this character is rather limited in literature, it is ignored in geotechnical engineering by most researchers and engineers. In this thesis, the random field method has been introduced in slope numerical simulation to simulate the spatial variability structure, and a numerical procedure for a probabilistic slope stability analysis based on Monte Carlo simulation was presented. The soil properties such as cohesion and friction angle were discretized to continuous random fields based on local averaging method. In the case study, both stationary and non-stationary random fields have been investigated, and the influence of spatial variability and averaging domain on the convergence of numerical simulation and probability of failure was studied. In rock medium, the structure faces have very important influence on the slope stability, and the rock material can be modeled as the combination of rigid or deformable blocks with joints in distinct element method. Therefore, much more input parameters like strength of joints are required to input the rock slope model, which increase the uncertainty of the results of numerical model. Furthermore, because of the limitations of the current laboratory and in-site testes, there is always lack of exact values of geotechnical parameters from rock material, even the probability distribution of these variables. Most of time, engineers can only estimate the interval of these variables from the limit testes or the expertise’s experience. In this study, to assess the reliability of the rock slope, a Random Set Distinct Element Method (RS-DEM) has been developed through coupling of Random Set Theory and Distinct Element Method, and applied in a rock slope in Sichuan province China.

In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<p<oo oder p=1 bzw. p=oo, durchgeführt, wobei der Fall p=oo der üblichen sup-Norm entspricht. Darüber hinaus wird im Fall unabhängiger Zufallsgrößen der lokale Grenzwertsatz für Dichten betrachtet. Mittels der elektronischen Datenverarbeitung neue numerische Resultate zu erhalten. Die Arbeit wird abgerundet durch verschiedene Hinweise auf praktische Anwendungen. / In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1 <p <oo or p = 1 or p = oo, where in the case p = oo it is the usual sup-norm. In addition, in the case of independent random variables the local limit theorem for densities is considered. By means of electronic data processing new numerical results are obtained. The work is finished by various references to practical applications.

In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1<p<oo oder p=1 bzw. p=oo, durchgeführt, wobei der Fall p=oo der üblichen sup-Norm entspricht. Darüber hinaus wird im Fall unabhängiger Zufallsgrößen der lokale Grenzwertsatz für Dichten betrachtet. Mittels der elektronischen Datenverarbeitung neue numerische Resultate zu erhalten. Die Arbeit wird abgerundet durch verschiedene Hinweise auf praktische Anwendungen. / In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1 <p <oo or p = 1 or p = oo, where in the case p = oo it is the usual sup-norm. In addition, in the case of independent random variables the local limit theorem for densities is considered. By means of electronic data processing new numerical results are obtained. The work is finished by various references to practical applications.