**Reliability analysis of existing structures using dynamic state estimation methods**
**Technical Report**
**By**
**B Radhika and C S Manohar**
**April 2010**
**Department of Civil Engineering**
**Indian Institute of science**
**Bangalore 560 012 India**
**Abstract**
The problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations is considered. The study assumes that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes’ theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. In all these studies, the governing stochastic differential equations are discretized using the strong forms of Ito-Taylor’s discretization schemes. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplified by considering the reliability analysis of a few low dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on limited amount of pertinent Monte Carlo simulations.
**Keywords **
Dynamic state estimation; reliability of existing structures; extreme value analysis; Ito-Taylor expansion.
**1. Introduction**
The present study considers the problem of reliability assessment of existing structural systems which typically carry dynamic loads during their normal operation. We take that a realistic representation of these loads demands the application of probabilistic models. One could consider an existing railway bridge as an example of this class of structures. Here the formation loads are dynamic in nature and, given the uncertainties that are invariably involved in train speeds, formation lengths, payloads, and track unevenness, the loads could realistically be modeled as being random in nature. The present study further considers that the reliability assessment must take into account a set of measurements made on the existing structure in its operating conditions and the availability of an acceptable mathematical model for the structure and the applied loads. The measurements are typically taken to include time histories of components of structural strains, displacements and (or) accelerations and to be, in general, spatially incomplete and noisy in nature. In certain applications, such as earthquake response of structures, it could be possible to measure external actions, while, in problems of vehicle structure interactions or wind loaded structures, the applied forces could remain unmeasured. The present study allows for the two contingencies of being able or not being able to measure the external forces. Furthermore, the mathematical model for the structural system itself could be nonlinear in nature and allowance is also made for possible imperfections in formulating the mathematical model. The parameters of the mathematical model are assumed to have been already identified through diagnostic measurements in a previous system identification step. The reliability metric could be defined in terms of performance functions which are solely dependent on measured response quantities, or, alternatively, and, more realistically, could involve a mix of measured and unmeasured response states. In either case, the problem on hand consists of updating the structural reliability model by assimilating the measured response. The present study develops a framework, based on the Kalman and particle filtering strategies, to tackle this problem.
The basic motivation for undertaking this study has arisen due to the present authors’ involvement in a project aimed at condition assessment of existing railway bridges in India with a view to evaluate their ability to carry increased axle loads and faster and longer moving formations. The field work conducted as a part of this investigation has resulted in large amount of static and dynamic response data on structural strains, translations, rotations and accelerations under various loading conditions. One of the data set that has been collected in these studies has been the response of the bridge structure to ambient traffic load over a period of about 72-96 hours. Questions on the best utilization of such data towards establishing updated model for reliability of the structure that takes into account the measurements made, do not have clear answers and it is hoped that the issues addressed in the present study offer useful directions in this regard. As a prelude to the development of the ideas, we briefly review the relevant published literature.
**2. A review of literature**
The study of reliability of existing structures has been the subject of an early research monograph (Yao 1985) and the topic is also covered in individual chapters in the books by Ditlevsen and Madsen (1996) and Melchers (1999). Typically, the reliability analysis here involves Bayesian updating of postulated models for the mechanical behaviour of the structure, probability distribution functions (PDF-s) of the loads and structural resistance characteristics based on a set of measurements on the structure under controlled (and hence measured) loads or under (mostly unmeasured) ambient loads. Additional information based on non-destructive testing could also be available. The main sources of uncertainties here include: (a) noise in measurement of structural responses and applied actions, (b) imperfections in the mathematical model governing the system states, (c) imperfections in mathematical model that relates measured quantities to the system states, (d) assessment of present condition of the system that includes models for structural resistance allowing for the degradation that may have taken place during the completed life of the structure, and (e) mathematical model for external actions. Moreover, a step involving structural system identification typically precedes the reliability analysis which would have lead to optimal characterization of properties such as elastic constants, boundary conditions, mass, and damping properties.
We briefly mention some of the studies in the existing literature in this area of research. The work of Shah and Dong (1984) addresses questions on strengthening existing structures to meet higher seismic demands and discusses a probabilistic framework to achieve this goal. Diamantidis (1987) explores the application of first order reliability methods to assess reliability of existing structures. The problem of combining system identification tools and reliability methods with knowledge-based diagnostics has been investigated by Yao and Natke (1994). Issues related to the development of reliability based condition assessment criteria have been examined by Ellingwood (1996), Mori and Nonaka (2000) and Val and Stewart (2001). The papers by Melchers (2001) and Catbas *et al.,* (2008) provide overviews on current state of art in this area of research. Recently Ching *et al.*, (2007) have used Kalman filters for estimating hidden states in randomly excited systems and have developed a simulation based strategy to update the reliability of existing structures based on sparse measurements. Their study employs importance sampling schemes and also includes a step involving identification of external forces. The reliability here is essentially defined with respect to an unobserved response of the structure exceeding a prescribed threshold during a loading episode that has already occurred. The major differences between the study by Ching *et al.*, (2007) and the present study are that (a) we are focusing on updating the model for structural reliability against future episodes of loading, (b) the study takes into account nonlinear models for structural behavior and measurement models, and (c) the updating here is based on use of nonlinear filtering tools.
We consider linear/nonlinear multi degree of freedom (mdof) dynamical systems which are randomly excited. It is assumed the measurements on response of this structure have been made for a limited number of loading episodes and at a (possibly) sparse set of points. The study allows for the likely inability to measure time histories of applied actions. A random process model for the excitation that permits digital simulation of samples of excitation is assumed to be available whether or not samples of applied excitations are measured. In case external excitations are not measured, the study proceeds based on the basis of simulated samples from the assumed stochastic model for excitations. On the other hand, if samples of excitation are measured, the ensemble of future excitations is simulated using conditional simulation strategies. A validated mathematical model for the structure is also taken to be available. The study proposes that methods of dynamic state estimation, which employ Monte Carlo simulations, be used to assimilate the measured responses into the mathematical model. This in turn facilitates the simulation of sample realizations of the system states conditioned on the episodes of measurements. These estimated trajectories are subsequently used to obtain updated reliability models.
For the case of linear state space models with additive Gaussian noises, it is shown that an approximate analytical solution to the problem of reliability model updating is obtainable. This is based on the application Poisson counting process model to the threshold crossings of the expected response conditioned on the measurements. For more general class of models, involving nonlinear process and measurement equations and (or) non-Gaussian noises, we propose a procedure that combines particle filtering strategy (to obtain updated system states) with a data based extreme value analysis (to develop models for extreme responses). Here it is postulated that the PDF of the highest response over a given duration agrees with one of the classical asymptotic extreme value PDF-s, namely, Gumbel, Weibull or Frechet distributions. This analysis involves two steps: (a) identification of basin of attraction to which the PDF of extremes of the estimated response belongs to; this, in turn, is based on hypothesis tests such as those discussed by Castillo (1988, attributed to Pickands and Galambos) or Hasofer and Wang (1992), and (b) estimation of parameters of the extreme value distribution ascertained in the previous step using limited number of samples of the estimated response. The proposed approach is exemplified by considering a few low dimensional dynamical systems subjected to stochastic excitations. The governing equations are taken to have cubic, tangent and (or) hysteretic nonlinear terms. The excitations considered include white and band limited random excitations. The efficacy of the results obtained is assessed with the help of limited large-scale Monte Carlo simulation studies.
Thus, the present study draws on knowledge base available in the existing literature in the areas of Kalman and particle filtering (Kalman 1960, Gordon *et al.,* 1993, Tanizaki 1996, Doucet *et al.,* 2000 and Ristic *et al.*, 2004), extreme value analysis (Castillo 1988, Kotz and Nadarajah 2000), numerical solution of stochastic differential equations (SDE-s) (Kloeden and Platen 1992) and conditional simulation of random processes using techniques of dynamic state estimation (Vanmarcke and Fenton 1991, Kameda and Morikawa 1994 and Hoshiya 1995). It may be noted that particle filtering methods provide statistical solutions to problems of dynamic state estimation and are capable of treating nonlinear and (or) non-Gaussian state space models. Their applications in problems of structural engineering are recently being explored (Ching and Beck 2006, Manohar and Roy 2006, Sajeeb *et al.*, 2007 Namdeo and Manohar 2007, and Ghosh* et al.,* 2008). The application of statistical methods to estimate the PDF-s of extremes of random processes over specified duration is widely studied (Castillo 1988, Alves and Neves 2006). Hasofer and Wang (1992) have proposed a test statistic to test the hypothesis that a sample comes from a distribution in the domain of attraction of one of the classical extreme value distributions. This test has been used by Dunne and Ghanbari (2001) to investigate the extremes of response of nonlinear beams based on measured random response. Recently, Radhika *et al., *(2008) have investigated the problem of determining the model for extremes of nonlinear structural response using data based extreme value analysis and Monte Carlo simulation method.
**3. Problem statement**
In this study we consider structural dynamical systems governed by the equation of the form
…(1)
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