Methods for the analysis of structural systems subjected to seismic acceleration modelled as stochastic processes

In the framework of the design and of the reliability assessment of fixed structures, among the static and dynamic loads that have to be considered, certainly the most important one is the seismic load, due to its terrible and disastrous consequences, not only in terms of the breakdown of the structure but also for the preservation of life. In fact during the past decades Italy has been the scene of terrible earthquakes, that destroyed whole cities and with a lot of human victims. First of all, in terms of magnitude and, unfortunately, a large number of deaths, Messina earthquake, in 1908, caused about 120000 victims, between Messina and Reggio Calabria, with an estimated magnitude of 7.1 (Richter scale). Then Irpinia earthquake, in 1980 (2914 victims, 6.5 Ricther), L’Aquila earthquake, in 2009 (309 victims, 5.9 Ricther) and the last events in 2016 in the centre of Italy, see Amatrice (299 victims, 6 Richter), Ussita (5.9 Richter), and Norcia (2 victims, 6.1 Ricther). Due to the difficulty in the prevision of the seismic event, one of the most important and hard problem in seismic engineering is the correct characterization of the ground motion acceleration; in fact it has been demonstrated that it is possible to increase the reliability level of the structures defining in a suitable way the seismic input and shaping realistically the structure. Nowadays, from the analysis of the large amount of data of recorded events, it is possible to study the main characteristics of real earthquakes and reproduce them with analytical models. In particular, because of the randomness of the seismic event, in terms of energy distribution and intensity, propagation path of the seismic waves through any specified location from the earthquake focus to the epicenter, etc…, it has been shown that it should be modelled as a stochastic process. On the other hand, once the input has been defined, the second problem in the seismic engineering is the reliability assessment of the structures subjected to the ground motion acceleration. It is obvious that, if the excitations are modelled as random processes, the dynamic responses are random processes too, and the structural safety needs to be evaluated in a probabilistic sense. Among the models of failure, the simplest one, which is also the most widely used in practical analyses, is based on the assumption that a structure fails as soon as the response at a critical location exits a prescribed safe domain for the first time. In random vibration theory, the problem of probabilistically predicting this event is termed first passage problem. Unfortunately, this is one of the most complicated problem in computational stochastic mechanics. Therefore, several approximate procedures have been proposed. These procedures lead to the probabilistic assessment of structural failure as a function of barrier crossing rates, distribution of peaks and extreme values. The latter quantities can be evaluated, for non-stationary input process, as a function of the well-known Non-Geometric Spectral Moments (NGSMs). Aim of this thesis is to propose a novel procedure to obtain closed form solutions of the spectral characteristics of the response of linear structural systems subjected to seismic acceleration modelled as stochastic processes. The proposed method is a powerful tool in the analysis of both classically and non-classically damped systems, in reliability assessment problems and takes into account also the case of multi-correlated forcing input. In Chapter 1 the preliminary definitions of probability theory are outlined, starting from the concept of random variable and stochastic process, analysing the stationary Gaussian random process with its statistics, with a short discussion on the probability distribution for maxima. Chapter 2 focuses on the characterization of the ground motion acceleration, thanks’ to a statistical analysis of a set of real earthquakes; the different strategies to model the ground motion acceleration stochastic process will be investigated. Furthermore, in order to follow the prescriptions of the building codes, a procedure to generate artificial fully non-stationary spectrum-compatible accelerograms will be proposed. The spectral characteristics of the response of linear structural systems, subjected to non-stationary excitation, will be obtained in Chapter 3 and, in Chapter 4, closed form solutions of the Time-Frequency varying Response (TFR) vector function will be proposed. In particular the main steps of the proposed approach are: i) the use of modal analysis, or the complex modal analysis, to decouple the equation of motion; ii) the introduction of the modal state variable in order to evaluate the NGSMs, in the time domain, as element of the Pre-Envelope Covariance (PEC) matrix; iii) the determination, in state variable, by very handy explicit closed-form solutions, of the TFR vector functions and of the Evolutionary Power Spectral Density (EPSD) matrix function of the structural response for the most common adopted models of the seismic input in the framework of stochastic analysis; iv) the evaluation of the spectral characteristics of the stochastic response by adopting the closed-form expression of the EPSD matrix function. Finally, in Chapter 5 the reliability assessment of structural systems will be performed; in particular two different approaches for the first passage probability problem will be used: the method requiring the evaluation of the mean up-crossing rate of given thresholds, considered independent or occurring in clumps, and the method requiring censored closures of the non-stationary extreme value random response process. Several numerical applications will be done in order to test the effectiveness of the proposed procedure; in particular the presented results will be compared with the Monte Carlo simulation method, that will confirm the validity and the generality of the proposed method.