Safety assessment of structures in view of fuzzy randomness

Abstract

Structural safety can be realistically assessed only if the uncertainty in the structural parameters is appropriately taken into consideration and realistic computational models are applied. Uncertainty must be accounted for in its natural form. Stochastic models are not always capable of fulfilling this task without restrictions, as uncertainty may also be characterized by fuzzy randomness or fuzziness. On the basis of the theory of the fuzzy random variables the fuzzy probabilistic safety concept is introduced and formulated as the fuzzy first order reliability method (FFORM). This concept permits fuzziness, randomness and fuzzy randomness to be accounted for simultaneously. FFORM is illustrated by way of an example; hereby, the influence of the computational model is also demonstrated.

Introduction

Basic numerical investigations in structural engineering mainly concern the computation of structural responses and the assessment of structural safety. For that purposes appropriate structural models and suitably matched computational models must be applied in combination with reliable structural parameters close to reality.

This paper focuses on new methods for appropriately taking account of structural models and parameters with respect to their uncertainty. These models and parameters are usually established on the basis of plans, drawings, measurements, observations, experiences, expert knowledge, codes and standards and so on. In general, certain information and precise values do not exist. Uncertainty may result e.g. from human mistakes and errors in the manufacture, from the use and maintenance of constructions, from expert evaluations, and from a lack of information. Small samples, changing reproduction constraints and imprecise results of measurements are usual starting points for defining structural models and parameters.

These facts show that structural engineering is significantly characterized by uncertainty. In order to perform realistic structural analysis and proper safety assessment this uncertainty in both data and models must be appropriately taken into consideration. Computational models that are capable to numerically simulate the system behavior of the chosen structural model in this paper are considered to be sufficiently accurate, its uncertainty, e.g. resulting from weak formulations or numerical impreciseness, is not subject of the consideration.

In the following, some aspects of uncertainty are considered and available mathematical descriptions are discussed. Mathematical basics of the uncertainty characteristic fuzzy randomness are shortly described in Section 2 and explained more detailed in the appendix. Section 3 deals with quantification methods for fuzzy randomness and in Section 4 the developed fuzzy probabilistic safety concept is presented. The new safety analysis algorithm is demonstrated by way of an example in Section 5. In the conclusion some ideas for the further development of uncertainty concepts are provided.

When developing a concept for the safety assessment of structures considering uncertainty, the basic step is to define what uncertainty means. According to [1], uncertainty is the gradual assessment of the truth content of a postulation, e.g. in relation to the occurrence of a defined event. All non-deterministic parameters are characterized by uncertainty; these are referred to as uncertain parameters.

The classification and description of uncertainty may be carried out according to different criteria. Uncertainty is hereby classified in the manner shown in Fig. 1.

Whereas the type of the uncertainty indicates the cause of its manifestation, the characteristics of the uncertainty are described by the mathematical properties randomness, fuzziness and fuzzy randomness. The uncertainty characteristics depend on the type of uncertainty and the information content of the uncertain parameters.

Another classification distinguishes between data uncertainty and model uncertainty. This is linked to the definition of the model in any particular case. In the safety assessment the limit state surface is solely determined by the computational model. Model uncertainty in this case refers to uncertainty in the limit state surface; the uncertainty in the basic variables is referred to as data uncertainty.

Different methods are available for mathematically describing and quantifying uncertainty. These include e.g. probability theory [2], [3], interval algebra [4], convex modeling [5], fuzzy set theory [1], [6], the theory of fuzzy random variables [7], the subjective probability approach [8] and chaos theory [9].

Conventional methods for structural analysis and safety assessment only permit the simultaneous modeling of the uncertainty of structural parameters and structural models to a limited extent. The information content of the uncertainty of input parameters and models is often inadequately described and accounted for, or in some cases, ignored altogether. The possibilities available for taking uncertainty into account are limited.

Probabilistic concepts [10] presuppose sufficient information for determining stochastic input parameters, such as, e.g. expected values, variances, quantile values and probability distribution functions. The quality of the input information must be statistically assured by a sufficiently large set of sample elements. Probabilistic methods are only able to account for uncertainty with the characteristic randomness (stochastic uncertainty). Inaccuracies, unreliable data, or uncertainty which cannot be described or insufficiently described statistically can thus only be accounted for approximately. BAYES theorem serves as a suitable probabilistic method for processing subjective information. This conventional approach makes use of prior distributions on the basis of subjective probabilities. In contrast to the new approach described here, the BAYESian method generally yields crisp safety prognoses containing objective and subjective information. There is no way, however, of separating objectivity and subjectivity in these results. In view of the latter, probabilistic methods may only be applied to a limited extent.

Alternative methods based on fuzzy set theory and the theory of fuzzy random variables have only been applied in past few years. For both structural analysis and safety assessment, algorithms have been developed which take into account non-stochastic uncertainty [11]. In [12] a method is presented for the numerical simulation of the structural behavior of systems in which fuzzy structural parameters and fuzzy model parameters occur. This fuzzy structural analysis is based on α-level optimization. A possibilistic safety concept for assessing structural reliability using fuzzy variables is presented in [13]. Initial ideas relating to the fuzzy probabilistic method dealt with here are outlined in [14].

Uncertainty with the characteristic fuzzy randomness is described, quantified and processed on the basis of the theory of fuzzy random variables. This takes into account, as also in the case of fuzziness, both objective and subjective information. The theory of fuzzy random variables permits the modeling of uncertain structural parameters (which partly exhibit randomness) but which cannot be described using random variables without an element of doubt. The randomness is “disturbed” by a fuzziness component. The reasons for the existence of fuzzy randomness might be:

• Although samples are available for a structural parameter, these are only limited in number. No further information exists concerning the statistical properties of the universe.

• The statistical data material possesses informal uncertainty, i.e., the sample elements are of doubtful accuracy or were obtained under unknown or non-constant reproduction conditions.

• The available sample elements were generated under reproduction conditions which were non-constant but nevertheless known in detail.

In order to take account of uncertainty with the characteristic fuzzy randomness the method of fuzzy probabilistic safety assessment is developed. This comprehensive safety concept is formulated as a further development of introduced probabilistic approaches. This further development is demonstrated by way of the fuzzy first order reliability method (FFORM).