The reliability of structures subjected to time-invariant or time-variant random loads is considered herein. This is an important field of engineering, as it provides the framework for assessing whether newly designed or existing structural systems meet their design requirements in a given lifetime, or whether they experience what is termed “structural failure”. An important aspect of reliability analysis is the study of structures subjected to multiple time-varying loads. For this class of systems, it is well-known that by modelling the loads as (time-variant) random processes, the reliability may be evaluated by considering the outcrossing of a vector process out of a safe domain. However, due to the possibility that the loads may not be fully-dependent, all loads may not necessarily contribute to structural failure. To account for this the treatment of vector-outcrossings may need to allow for the possibility of outcrossings being caused by individual loads, as distinct from combinations of all loads. The procedure used to analyse combinations of loads depends on the stochastic process model used to represent the loads. Two well-known load models have been presented in the literature—they are referred to herein as the ‘on-off’ model and the ‘standard’ model. The ‘on-off’ model typically assumes loads are non-negative, and are either ‘on’ (eg their value is non-zero) or ‘off’ (eg their value is strictly zero). They can contribute to failure only when they are ‘on’. This model is represented by a somewhat artificial ‘composite’ probability distribution, obtained by modifying the original load probability density function (pdf) so that a ‘finite’ non-zero probability represents explicitly the possibility that the load is ‘off’. To implement this model in time-variant analysis, it is necessary to consider all possible combinations of loads being ‘on’ and ‘off’. In contrast, the ‘standard’ model (which is the more commonly used) typically allows loads to be negative; it is also typically represented solely by the original load pdf, and therefore effectively assumes each load is always ‘on’. To allow for the possibility of one or more loads not to cause failure, herein the value of such loads is held ‘constant’ at the time of failure, when the value of all loads actually causing failure is allowed to change. Use of the ‘standard’ model is examined herein. The “Directional Simulation in the Load Space (DS-LS)” approach is a tool used to perform reliability analysis. It is particularly suitable for time-variant analysis, as it allows loads to be represented as random processes, and to be modelled properly. DS-LS has so far been shown to work well for relatively simple structures subjected to one or more time-invariant random loads, and has been used to examine vector outcrossings in systems comprising either discrete or continuous loads. To enable the proper consideration of load combinations, and to provide some improvements in the formulation of the technique, a generalisation of the DS-LS approach is proposed herein. The generalisation is achieved in two stages. The first involves modifying the time-invariant and time-variant DS-LS formulation to allow for the possibility of positioning the origin of DS-LS not only in the ‘safe’ region of the load space (which the formulation currently requires) but in the ‘failure’ region, or even ‘exactly’ on the boundary separating the safe and failure regions. The modifications are necessary because for even simple structures, the ‘exact’ location of the safe and failure region is not always known explicitly ‘a priori’. The second involves developing the time-variant DS-LS formulation to consider explicitly outcrossings caused by combinations of one or more loads, during analysis of systems comprising stationary continuous gaussian loads. To do this, the direction of the load process vector is ‘fixed’ at each point of outcrossing, to physically represent the particular combination of loads causing the outcrossing. By considering each possible load combination, all loads not causing an outcrossing are then held constant during radial integration, thereby modelling those that do not contribute to each outcrossing. The proposed formulation differs from most load combination analysis techniques (which, evidently, simplify the analysis) as it is analytically ‘exact’, and it considers explicitly all possible combinations of loads. The concepts and formulations proposed herein may provide further understanding of reliability analysis performed by DS-LS (or other techniques) and may aid their future development. / PhD Doctorate

Uncertainty analysis of a system response is an important part of engineering probabilistic analysis. Uncertainty analysis includes: (a) to evaluate moments of the response; (b) to evaluate reliability analysis of the system; (c) to assess the complete probability distribution of the response; (d) to conduct the parametric sensitivity analysis of the output. The actual model of system response is usually a high-dimensional function of input variables. Although Monte Carlo simulation is a quite general approach for this purpose, it may require an inordinate amount of resources to achieve an acceptable level of accuracy. Development of a computationally efficient method, hence, is of great importance. First of all, the study proposed a moment method for uncertainty quantification of structural systems. However, a key departure is the use of fractional moment of response function, as opposed to integer moment used so far in literature. The advantage of using fractional moment over integer moment was illustrated from the relation of one fractional moment with a couple of integer moments. With a small number of samples to compute the fractional moments, a system output distribution was estimated with the principle of maximum entropy (MaxEnt) in conjunction with the constraints specified in terms of fractional moments. Compared to the classical MaxEnt, a novel feature of the proposed method is that fractional exponent of the MaxEnt distribution is determined through the entropy maximization process, instead of assigned by an analyst in prior. To further minimize the computational cost of the simulation-based entropy method, a multiplicative dimensional reduction method (M-DRM) was proposed to compute the fractional (integer) moments of a generic function with multiple input variables. The M-DRM can accurately approximate a high-dimensional function as the product of a series low-dimensional functions. Together with the principle of maximum entropy, a novel computational approach was proposed to assess the complete probability distribution of a system output. Accuracy and efficiency of the proposed method for structural reliability analysis were verified by crude Monte Carlo simulation of several examples. Application of M-DRM was further extended to the variance-based global sensitivity analysis of a system. Compared to the local sensitivity analysis, the variance-based sensitivity index can provide significance information about an input random variable. Since each component variance is defined as a conditional expectation with respect to the system model function, the separable nature of the M-DRM approximation can simplify the high-dimension integrations in sensitivity analysis. Several examples were presented to illustrate the numerical accuracy and efficiency of the proposed method in comparison to the Monte Carlo simulation method. The last contribution of the proposed study is the development of a computationally efficient method for polynomial chaos expansion (PCE) of a system's response. This PCE model can be later used uncertainty analysis. However, evaluation of coefficients of a PCE meta-model is computational demanding task due to the involved high-dimensional integrations. With the proposed M-DRM, the involved computational cost can be remarkably reduced compared to the classical methods in literature (simulation method or tensor Gauss quadrature method). Accuracy and efficiency of the proposed method for polynomial chaos expansion were verified by considering several practical examples.

Neste trabalho, são desenvolvidas novas técnicas aproximadas de análise de confiabilidade para grelhas de concreto armado levando-se em consideração as probabilidades de falha de vários modos importantes. Realiza-se um acoplamento entre os métodos de Monte Carlo, elementos finitos e procedimentos de otimização para considerar esses modos de falha importantes e classificá-los. Esse acoplamento também permite a redução do número de chamadas ao modelo de elementos finitos. Os cenários de falha são caracterizados como o encurtamento excessivo do concreto e o alongamento do aço. Estes cenários determinam a capacidade última da estrutura, e podem ser representados por um coeficiente escalar que multiplica todas as ações presentes na estrutura. Para a determinação desses estados estruturais últimos, um procedimento incremental-iterativo é utilizado. A análise de confiabilidade é realizada em diferentes conjuntos de realizações aleatórias das variáveis de projeto. O conjunto de respostas estruturais e de realizações permite a determinação dos coeficientes da superfície de respostas da estrutura. O acoplamento realizado permite também o tratamento com estruturas de concreto com elevado número de modos de falha. Aplicam-se as técnicas em exemplos de grelhas de concreto armado / In this work, new local approaches of reliability analysis applied to reinforced concrete grid structures are developed, taking into account several critical cross-section failure probabilities. Monte Carlo simulations are coupled with finite element analyses and optimization techniques with techniques to take into account the failure in the most important cross-sections, in order to classify the severity of failure modes. The failure scenario is depicted when either a concrete fiber or a steel bar reaches the predefined conventional limit. This scenario gives the structural ultimate capacity, which can be represented by a scalar coefficient multiplying all the loads acting on the structure. To achieve the failure scenario, an incremental and iterative procedure is used. To carry out the reliability analysis, the mechanical analysis has to be performed for different sets of random variable realizations of the mechanical, material and geometrical properties. The set of ultimate coefficients obtained from several mechanical analyses defines the response surface. The coupling between Monte Carlo simulations and response surface techniques applied in this work aims to reduce significantly the number of the finite element model calls, and hence to deal with real, or high-scale, reinforced concrete grids where large number of failure components can be found. The proposed procedure is then applied to reinforced concrete grids in order to show some more complex reinforced concrete examples

Neste trabalho, são desenvolvidas novas técnicas aproximadas de análise de confiabilidade para grelhas de concreto armado levando-se em consideração as probabilidades de falha de vários modos importantes. Realiza-se um acoplamento entre os métodos de Monte Carlo, elementos finitos e procedimentos de otimização para considerar esses modos de falha importantes e classificá-los. Esse acoplamento também permite a redução do número de chamadas ao modelo de elementos finitos. Os cenários de falha são caracterizados como o encurtamento excessivo do concreto e o alongamento do aço. Estes cenários determinam a capacidade última da estrutura, e podem ser representados por um coeficiente escalar que multiplica todas as ações presentes na estrutura. Para a determinação desses estados estruturais últimos, um procedimento incremental-iterativo é utilizado. A análise de confiabilidade é realizada em diferentes conjuntos de realizações aleatórias das variáveis de projeto. O conjunto de respostas estruturais e de realizações permite a determinação dos coeficientes da superfície de respostas da estrutura. O acoplamento realizado permite também o tratamento com estruturas de concreto com elevado número de modos de falha. Aplicam-se as técnicas em exemplos de grelhas de concreto armado / In this work, new local approaches of reliability analysis applied to reinforced concrete grid structures are developed, taking into account several critical cross-section failure probabilities. Monte Carlo simulations are coupled with finite element analyses and optimization techniques with techniques to take into account the failure in the most important cross-sections, in order to classify the severity of failure modes. The failure scenario is depicted when either a concrete fiber or a steel bar reaches the predefined conventional limit. This scenario gives the structural ultimate capacity, which can be represented by a scalar coefficient multiplying all the loads acting on the structure. To achieve the failure scenario, an incremental and iterative procedure is used. To carry out the reliability analysis, the mechanical analysis has to be performed for different sets of random variable realizations of the mechanical, material and geometrical properties. The set of ultimate coefficients obtained from several mechanical analyses defines the response surface. The coupling between Monte Carlo simulations and response surface techniques applied in this work aims to reduce significantly the number of the finite element model calls, and hence to deal with real, or high-scale, reinforced concrete grids where large number of failure components can be found. The proposed procedure is then applied to reinforced concrete grids in order to show some more complex reinforced concrete examples

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