Jorge E. Hurtado

Neural-network-based reliability analysis: a comparative study

Abstract A study on the applicability of different kinds of neural networks for the probabilistic analysis of structures, when the sources of randomness can be modeled as random variables, is summarized. The networks are employed as numerical devices for substituting the finite element code needed by Monte Carlo simulation. The comparison comprehends two network types (multi-layer perceptrons and radial basis functions classifiers), cost functions (sum of square errors and cross-entropy), optimization algorithms (back-propagation, Gauss–Newton, Newton–Raphson), sampling methods for generating the training population (using uniform and actual distributions of the variables) and purposes of neural network use (as functional approximators and data classifiers). The comparative study is performed over four examples, corresponding to different types of the limit state function and structural behaviors. The analysis indicates some recommended ways of employing neural networks in this field.

Equivalent linearization of the Bouc–Wen hysteretic model

Abstract The smooth endochronic hysteretic Bouc–Wen model is studied from the point of view of random vibration. The sources of the errors of the method of equivalent linearization applied to this model using the hypothesis of joint Gaussian behaviour are examined. The method of linearization for softening hysteretic models proposed by the authors, which is based on a combination of Dirac and Gauss densities, is developed and applied to the Bouc–Wen model under a variety of conditions. It is shown that the method gives excellent estimations of the response statistics without increasing the computational effort required by the conventional technique.

Analysis of one-dimensional stochastic finite elements using neural networks

Abstract This paper explores the applicability of neural networks for analyzing the uncertainty spread of structural responses under the presence of one-dimensional random fields. Specifically, the neural network is intended to be a partial surrogate of the structural model needed in a Monte Carlo simulation, due to its associative memory properties. The network is trained with some pairs of input and output data obtained by some Monte Carlo simulations and then used in substitution of the finite element solver. In order to minimize the size of the networks, and hence the number of training pairs, the Karhunen–Loeve decomposition is applied as an optimal feature extraction tool. The Monte Carlo samples for training and validation are also generated using this decomposition. The Nystrom technique is employed for the numerical solution of the Fredholm integral equation. The radial basis function (RBF) network was selected as the neural device for learning the input/output relationship due to its high accuracy and fast training speed. The analysis shows that this approach constitutes a promising method for stochastic finite element analysis inasmuch as the error with respect to the Monte Carlo simulation is negligible.

Improved stochastic linearization method using mixed distributions

A new procedure for the random vibration analysis of hysteretic structures using stochastic equivalent linearization is reported. Its aim is to improve the prediction of the response obtained by conventional Gaussian linearization technique. To this purpose, mixed discrete-continuous Gaussian distributions are used taking into account the bounded nature of the non-linear restoring force. The simple but important property of the mixed distribution is its linearity, which allows the use of the previous results obtained by the Gaussian hypothesis, avoiding the need of employing non-Gaussian continuous distributions or other time-consuming techniques such as local Monte Carlo simulations. Closed-form expressions of the new linearization coefficients for the Bouc-Wen-Baber model are then provided. The relative weights of the discrete and Gaussian distributions are calculated in dependence of the degree of non-linearity in each time step. The comparison of the results with previously published ones obtained by simulation shows a good agreement, providing a substantial improvement of the method with respect to the conventional Gaussian technique with the same calculation effort.

Neural networks in stochastic mechanics

SummaryA state of art on the application of neural networks in Stochastic Mechanics is presented. The use of these Artificial Intelligence numerical devices is almost exclusively carried out in combination with Monte Carlo simulation for calculating the probability distributions of response variables, specific failure probabilities or statistical quantities. To that purpose the neural networks are trained with a few samples obtained by conventional Monte Carlo techniques and used henceforth to obtain the responses for the rest of samples. The advantage of this approach over standard Monte Carlo techniques lies in the fast computation of the output samples which is characteristic of neural networks in comparison to the lengthy calculation required by finite element solvers. The paper considers this combined method as applied to three categories of stochastic mechanics problems, namely those modelled with random variables, random fields and random processes. While the first class is suitable to the analysis of static problems under the effect of values of loads and resistances independent from time and space, the second is useful for describing the spatial variability of material properties and the third for dynamic loads producing random vibration. The applicability of some classical and special neural network types are discussed from the points of view of the type of input/output mapping, the accuracy and the numerical efficiency.

Capacity, fragility and damage in reinforced concrete buildings: a probabilistic approach

The main goals of this article are to analyze the use of simplified deterministic nonlinear static procedures for assessing the seismic response of buildings and to evaluate the influence that the uncertainties in the mechanical properties of the materials and in the features of the seismic actions have in the uncertainties of the structural response. A reinforced concrete building is used as a guiding case study. In the calculation of the expected spectral displacement, deterministic nonlinear static methods are simple and straightforward. For not severe earthquakes these approaches lead to somewhat conservative but adequate results when compared to more sophisticated procedures involving nonlinear dynamic analyses. Concerning the probabilistic assessment, the strength properties of the materials, concrete and steel, and the seismic action are considered as random variables. The Monte Carlo method is then used to analyze the structural response of the building. The obtained results show that significant uncertainties are expected; uncertainties in the structural response increase with the severity of the seismic actions. The major influence in the randomness of the structural response comes from the randomness of the seismic action. A useful example for selected earthquake scenarios is used to illustrate the applicability of the probabilistic approach for assessing expected damage and risk. An important conclusion of this work is the need of broaching the fragility of the buildings and expected damage assessment issues from a probabilistic perspective.

Reanalysis of linear and nonlinear structures using iterated Shanks transformation

Reanalysis of structures is a common task in two important fields of structural engineering, namely optimization and reliability. It implies the solution of the kinematic equations for each new set of data. In order to simplify this task several approximate methods have been developed. In this paper a simple method for reanalysis, applicable to either linear or nonlinear structures, is proposed. It is based on the improvement of the Neumann resolvent series, which diverges beyond a certain radius, by means of the iterated Shanks transformation. The method is compared with two relevant proposals intended to improve the convergence of the Neumann series, namely the Pade series and the scaled approximation methods. The accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures.

Analytical Characterization of Hysteresis Loops Described by the Bouc-Wen Model

The Bouc-Wen model is able to capture, in an analytical form, a range of shapes of hysteretic cycles which match the behavior of a wide class of nonlinear systems. The obtained models have been used either to predict the behavior of the physical hysteretic elements or for control purposes. The Bouc-Wen model belongs to a class of smooth analytical models described by differential equations. This gives an appropriate mathematical framework to develop analytical knowledge on relevant properties, which can be exploited for a better practical use, in particular for identification and control purposes. In this context, recent papers have studied input-output and dissipative properties and obtained analytical expressions for hysteretic limit cycles and parameter identification formulae. In this direction, this paper presents new results in the form of analytical expressions which allow to understand the direct influence of the model parameters in the shape of the hysteretic loops, thus giving a theoretically based insight for a precise use of the model in practical applications.

Experimental and analytical research on seismic vulnerability of low-cost ferrocement dwelling houses

Analytical and experimental research for determining the seismic vulnerability of ferrocement houses, which is a low-cost system using precast thin panels, is presented. The research comprises the following tasks: (a) a field survey on the ferrocement dwelling houses actually built in seismic regions in Colombia; (b) a cyclic load test on 1:1 scale models; (c) identification of the parameters of the Bouc–Wen hysteretic model for the observed panel behaviour; and (d) Monte Carlo simulations of models intended to estimate the fragility curves of the houses. Despite the models exhibiting stiffness degradation, it was concluded that the system is adequate for regions of moderate to high seismicity.

Fourier-based maximum entropy method in stochastic dynamics

Abstract As the recent research by Trȩbicki and Sobczyk has demonstrated 1 , 2 , 3 the principle of maximum entropy is a powerful tool for solving stochastic differential equations. In particular, its use in connection with the moment equations generated by the Ito formula provides accurate estimations of the probability density evolution of some oscillators for which conventional methods such as the diverse closure schemes are not applicable. A major computational requirement of the method, however, lies in the need of calculating a large number of multidimensional integrals at each time step—a numerical task for which both accurate and economic algorithms are required. In this paper it is shown that conventional economic integration techniques often lead to numerical collapse of the solution, especially when dealing with higly nonlinear oscillators. A strategy that overcomes this difficulty is proposed. In essence, the integrals are reformulated in terms of multidimensional Fourier transforms, which are solved by an ad hoc FFT algorithm aimed at obtaining only one single “frequency” point. It is demonstrated that the numerical stability and the accuracy of the proposed algorithm are superior to those afforded by other integration schemes.