Keywords: probabilistic design, assessment, failure probability, reliability, neural networks, splines, adaptive meta-model.

Reliability and safety concepts become widely spread as a new basis for structural evaluation in design and assessment of civil and structural engineering. Although this field of research is relatively young, different techniques have been proposed and optimised. Most of these techniques work very well on component level. Challenging is the extension of these techniques from elements, components and simple structures to complex systems. Simulation based concepts are very attractive to be used because of their inherent simplicity. Although, in stepping towards system reliability, some difficulties appear. One of the major disadvantages is the large number of simulations required to obtain sufficiently accurate results. Because of the complexity of large structures, with the possible need of non-linear finite element methods to calculate the outcome of the limit state function, the processing time might become unrealistic.

To meet this disadvantage, a methodology is searched for that minimises the number of calls to the finite element method or limit state function, that obtains a sufficient accuracy for the resulting failure probability and remains applicable for a high number of (random) variables. In here, the use of artificial intelligence might be an outcome. Recently, reliability analysis based on simulation methods (Monte Carlo or Directional Sampling) in combination with an Adaptive Response Surface were developed for this purpose [1]. The main idea is that the response consisting of a complex function of input variables is approximated by a simple function of the input variables. As the Response Surface is capable of handling the complex structural behaviour, the reliability analysis can be performed on the Response Surface, instead of using the original problem. The applicability has been demonstrated extensively.

Up to now, a low order polynomial is used for the response surface. Because of the simplicity of the response surface, this is the main gain in computation time. The number of calls to the real limit state function depends on the quality of the response surface used. Whenever the response surface is able to capture the real structural behaviour well, only a limited number of calls to the real limit state function is required. In case the difference in outcome between the response surface and real limit state function is larger, this results in an increased number of calls to the real limit state function.

Ideally, no functional form is preset. A `universal estimator' should be used. To increase the efficiency of the methodology, the low order polynomials that are used for the Response Surface, are extended using Neural Networks and Splines. Both are a more universal estimator and only require simple mathematics to calculate the outcome. A neural network architecture of two-layers, where the first layer is sigmoid and the second layer is linear, can be trained to approximate any function (with a finite number) of discontinuities arbitrarily well. Splines, or piecewise continuous cubic polynomials have similar advantages.

Both techniques are presented. The overall behaviour of the technique is illustrated referring to 15 benchmark examples. These rather academic examples were developed to compare different reliability methods with respect to criteria such as: multiple critical points, noisy boundaries, unions and intersections, space dimension, probability level, strong curvatures of the limit state function and limit state functions having no roots in the axis' direction. Two of the examples are treated in detail. The detailed analysis allows for mutual comparison between the different simulation techniques (Monte Carlo and Directional Sampling) and the used meta model (non, low order polynomial Response Surface, Splines, Neural Networks).

The technique is illustrated on two civil engineering applications. The first example treats the buckling of a ring-stiffened cylinder. The failure probability related to the overall buckling of a ring-stiffened cylinder submitted to an external hydrostatic pressure is dealt with. The second example focuses on the assessment of the overall stability of a Romanesque city wall of Leuven (B). The analysis focuses on the present safety of the city wall, regarding the uncertainties in load, geometry and resistance. The mutual efficiency of the different reliability algorithms is discussed.

1

L. Schueremans, "Probabilistic evaluation of structural unreinforced masonry", Ph. D. Thesis, Department of Civil Engineering, KULeuven, 2001.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £82 +P&P)