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The mobilized shear strength ( τ f m ) of a soil is the most important design parameter governing ultimate limit states. The authors have proposed some models and equations to characterize several useful aspects of the mean, variance, and probability density function of τ f m for a two dimensional (2D) spatially variable clay specimen. This paper demonstrates that the models and equations developed for 2D can be extended to three-dimensional (3D) cases. An important class of 3D cases are examined in detail: two horizontal scales of fluctuation (SOF) are equal, and both are larger than the vertical SOF. A key observation for 3D cases is that the averaging effect over the potential slip planes is significantly stronger than that for 2D cases. The upshot is that the phenomenon of the critical SOF (e.g., SOF producing the lowest mean value of τ f m ) is less clear for 3D cases. [ABSTRACT FROM AUTHOR]

In this paper we present a framework to model the effect of multiple sources of degradation on deteriorating systems, and to find easy-to-evaluate expressions for important reliability quantities. By considering the system deterioration as a general increasing Lévy process (i.e., a subordinator), which is a process with independent, stationary and non-negative increments, the proposed methodology allows the modeling of shock-based degradation (in the form of compound Poisson processes) with different distributions of shock sizes, progressive degradation as deterministic linear drift plus a Gamma process, and multiple sources of degradation by the superposition of any of these models into the same mathematical formalism. In addition, we obtain expressions for the reliability quantities (reliability function, and probability density and moments of the lifetime L ), and the mean and central moments of the deterioration process X t . Moreover, we present efficient and accurate numerical methods to compute the reliability quantities and to simulate sample paths. Several deterioration models are compared in terms of their reliability quantities, the simulated sample paths and the feasible moments of the deterioration X t . Furthermore, we propose a method to mix different Lévy models that extend the available moments with possible applications to data fitting. The results demonstrate the generality, versatility, efficiency and accuracy of the proposed Lévy degradation framework, which can open a new productive research field in the area of probabilistic degradation models. [ABSTRACT FROM AUTHOR]

The analysis of reliability of complex engineering systems remains a challenge in the field of reliability. It will be even more difficult if correlated random vectors are involved, which is generally the case as practical engineering systems invariably contain parameters that are mutually correlated. A new method for transforming correlated distributions, involving the Nataf transformation, is proposed that avoids the solution of integral equations; the method is based on the Taylor series expansion of the probability density function (PDF) of a bivariate normal distribution resulting in an explicit polynomial equation of the equivalent correlation coefficient. The required numerical results can be obtained efficiently and accurately. The proposed method for transformation of correlated random vectors is useful for developing a method for system reliability including complex systems with correlated random vectors. Based on the complete system failure process (originally defined as the development process of nonlinearity) and the fourth-moment method, the analysis of system reliability for elastic-plastic material avoids the identification of the potential failure modes of the system and their mutual correlations which are required in the traditional methods. Finally, four examples are presented – two examples to illustrate the potential of the new method for transformation of correlated random vectors, and two examples to illustrate the application of the proposed more effective method for system reliability. [ABSTRACT FROM AUTHOR]