Keywords: radial basis function network, implicit performance function, first order reliability method.

Structural reliability methods deal with the statistical nature of many basic variables in structural safety analysis and design. In many cases of practical importance, particularly for complicated structures, the performance function is generally not available as an explicit, closed-form function in terms of the random variables [1]. Implicit performance functions are often encountered when costly physical experiments or computationally intensive numerical methods such as finite element methods are adopted for the mechanical analysis of a structural system. The first-order reliability method (FORM) has been widely used to estimate the failure probability of structural systems, and key steps of FORM are to compute the derivatives. When the performance function is implicit, however, its derivatives are not readily available, and each evaluation of the performance function could be time consuming. A few approaches have been developed to compute the derivatives of implicit performance functions, such as finite difference method, classical perturbation methods, iterative perturbation analysis techniques [1], and multilayer perceptron network [2].

Although the radial basis function (RBF) network found some applications for deterministic engineering problems, reports on its application to a structural reliability problem are scarce in the literatures to our knowledge. This paper presents a RBF based reliability analysis method, i.e. a RBF based FORM, which employes a radial basis function network technique to simultaneously approximate the implicit performance functions and derivatives. The RBF network is trained on a small set of data with different input values. Using the optimal parameters and weights of RBF network, it is possible to develop a mathematical expressions relating the input and output variables that simultaneously approximates the implicit performance function and its derivatives. Next FORM can then be performed. The RBF based FORM differentiates itself from other FORMs in that it employs an RBF to simultaneously compute the values and gradients of implicit performance function.

Examples are given in the paper to illustrate how the proposed RBF based structural reliability analysis can be carried out. The present results are compared well with those obtained by the other reliability methods such as multilayer perceptron network [2] and the FORM method 2 [1]. Results can be made that both FORM methods gave similar results. However, the number of computation cycles are different: The multilayer perceptrons method has six computation cycles, while the multiquadrics RBF method has four cycles, the inverse multiquadrics and Gaussians RBF methods have both five cycles, and the analytical derivatives methods have only four cycles. The computation cycles of multiquadrics were relatively small. This may probably be attributed to the fact that multiquadrics RBF produce greater accuracy than other basis functions and MLPs. The accuracy of multiquadrics RBF method is comparable with the analytical FORM method 2. It can also be concluded that there are at least two advantages of using RBF networks to model a performance function: (a) it is not necessary to know the underlying relationship between the input variables and the output, unlike the response surface modelling technique [3]; and (b) the RBF network approach generally outperforms MLP networks and most mathematical regression models in terms of the modelling accuracy.

The illustrative example, although simple, do show that the RBF based FORM is feasible for reliability analysis. Because of the robust approximation of RBF network, RBF based FORM own unique characteristics. The performance function cannot be confined to be linear or explicit. Future works may include extension of the proposed method to other reliability methods such as second order reliability method.

1

A. Haldar and S. Mahadevan. "Reliability assessment using stochastic element analysis". New York: John Wiley & Sons, 2000

2

J. Deng, D.S. Gu, X.B. Li, and Z.Q. Yue. "Structural reliability analysis for implicit performance function using artificial neural network". Structural Safety. 27(1),25-48, 2004. doi:10.1016/j.strusafe.2004.03.004

3

A.T.C. Goh and F.H. Kulhawy. "Neural network approach to model the limit state surface for reliability analysis". Canadian Geotechnical Journal, 40(6): 1235-1244, 2003. doi:10.1139/t03-056

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