Reliability Analysis of Dynamical Systems
The use of reduced-order models in the context of reliability analysis of dynamical systems under stochastic excitation is explored in this chapter. A stochastic excitation model based on a point-source model is introduced, and it is used for the generati
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Reliability Analysis of Dynamical Systems
Abstract The use of reduced-order models in the context of reliability analysis of dynamical systems under stochastic excitation is explored in this chapter. A stochastic excitation model based on a point-source model is introduced, and it is used for the generation of ground motions. The corresponding reliability analysis represents a high-dimensional reliability problem whose solution is carried out by an advanced simulation technique. Two application problems are considered in order to evaluate the effectiveness of the proposed model reduction technique. The first example consists of a two-dimensional frame structure, while the second example considers an involved nonlinear finite element building model. The results show that an important reduction in computational effort can be achieved without compromising the accuracy of the reliability estimates.
4.1 Motivation Reliability analysis allows the possibility of accounting for the unavoidable effects of uncertainty over the performance of a structure. In this context, the level of safety of a structure can be measured in terms of the reliability, which is a metric of plausibility that the structure fulfills certain performance requirements during its lifetime. The complement of the reliability is the probability of failure, that is, the probability that a structure violates prescribed performance criteria. Thus, reliability can be incorporated as one of the performance criteria in the analysis and design of structures to explicitly address the effects of uncertainty [24, 25, 29, 33, 42, 49, 58]. In this framework, it is assumed that the external force vector f(t) (see Eq. (1.1)) is modeled as a non-stationary stochastic process and characterized by a random variable vector z ∈ Ωz ⊂ R n z . This vector is defined in terms of a probability density function p(z). Furthermore, consider a vector θ ∈ Ωθ ⊂ R n θ of uncertain model parameters. These parameters are characterized in a probabilistic manner by means of a joint probability density function q(θ ). It is noted that alternative approaches for modeling uncertainties do exist, as well. For example, methodologies based on non-traditional uncertainty models can be very useful in a number of cases [8, 12, 27, 47, 48]. How© Springer Nature Switzerland AG 2019 H. Jensen and C. Papadimitriou, Sub-structure Coupling for Dynamic Analysis, Lecture Notes in Applied and Computational Mechanics 89, https://doi.org/10.1007/978-3-030-12819-7_4
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4 Reliability Analysis of Dynamical Systems
ever, the focus here is on probabilistic approaches. The performance of the structural system due to the excitation is characterized by means of n r responses of interest ri (t, z, θ ) , i = 1, . . . , n r , t ∈ [0, T ]
(4.1)
where T is the duration of the excitation. Clearly, the aforementioned responses ri are functions of time t (due to the dynamic nature of the loading), and functions of the system parameter vector θ and random variable vector z. The response functions ri (t, z, θ ), i
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