Fuzzy reliability analysis of concrete structures
Introduction
From the engineering point of view, a structural problem can be considered as “uncertain” when some lack of knowledge exists about the theoretical model which describes the structural system and its behavior, either with respect to the model itself, or to the value of its significant parameters. The uncertainty which affects the model, or a part of it, could be avoided through direct tests. Such process is typical, for example, of aeronautical and mechanical engineering, where tests on prototypes are performed before the series production and contribute to improve and to validate the model. In civil engineering the realization of structural prototypes is very unusual, not only for economical reasons, but also because a prototype tested in a laboratory can never fully represent the actual structure built on site.
To overcome such uncertainties, structural engineers always based their choices on the experience accumulated in the course of time. The same experience also allowed them to draw generalizations. However, difficulties arise when designers need to transfer the experience of the past to nowadays problems, where both the design choices and the nature itself of the structures are different. In this sense, particular attention must also be paid to special structures which cannot be listed in the traditional building categories and, as such, are not part of the experience inheritance from the past. In addition, due in particular to the growing complexity of structural systems faced by nowadays designers, the uncertain parameters involved in design evaluations tend to be very numerous and highly interacting. As a consequence, a complete understanding of the sensitivity of the structural behavior with respect to such uncertainties is usually quite hard, and specific mathematical concepts and numerical methods are required for a reliable assessment of the structural safety [5].
Reliability-based concepts are nowadays widely accepted in structural design, even if it is well known that, before such concepts can be effectively implemented, the actual design problem often needs to be considerably simplified. This is mainly due to the two following reasons:
- (1)
In their simplest formulation, reliability-based procedures require the structural performance to be represented by explicit functional relationships among the load and the resistance variables. But, unfortunately, when the structural behavior is affected by several sources of non-linearity, as always happens for concrete structures, such relationships are generally available only in an implicit form.
- (2)
For structural systems with several components, a complete reliability analysis includes both component-level and system-level estimates. Depending on the number and on the arrangement of the components, system reliability evaluations can become very complicated and even practically impossible for large structural systems.
This paper proposes a theoretical approach and numerical procedures for the reliability assessment of reinforced and prestressed concrete structures, based on detailed and representative mechanical models, and able to handle implicit formulation of the performance relationships and to perform system-level evaluations even for large structural systems [3], [4], [9].
The uncertainties regarding the geometrical and mechanical properties involved in the structural problem, can be approached by a probabilistic or by a fuzzy formulation.
The probabilistic approach assumes the intrinsic stochastic variability of the random variables as known. In the practice of structural design, however, it is very frequent that a lack of information occurs about such randomness and this makes the fuzzy approach more meaningful for a consistent solution of the problem. Think for example to a beam imperfectly clamped at one end. This link is usually modeled through a rotational spring having uncertain stiffness. The translation of this problem in probabilistic terms is not simple, since no information are usually available about the random distribution of the stiffness value. Conversely, it appears more direct and reasonable to consider a band of situations between the hinged and the clamped ones, which defines a design domain large enough to include the actual one under investigation. Situations of weak structural coupling are very frequent in structural engineering, as for instance happens for structures built in subsequent phases, for large span cable supported bridges and for high rise buildings.
For these reasons, in the present study the uncertainties are modeled by using a fuzzy criterion in which the model is defined through bands of values, bounded between suitable minimum and maximum extremes. The reliability problem is formulated at the load level, with reference to several serviceability and ultimate limit states. For the critical interval associated to each limit state, the membership function of the safety factor is derived by solving a corresponding anti-optimization problem. The planning of this solution process is governed by a genetic algorithm, which generates the sampling values of the parameters involved by the material and geometrical non-linear structural analyses.
The effectiveness of the proposed approach and its capability to handle complex structural systems are shown by carrying out a reliability assessment of a prestressed concrete continuous beam and of a cable-stayed bridge.
Section snippets
Randomness vs fuzziness
The uncertainties associated to a physical phenomena may derive from several and different sources. In the common language, something is uncertain when it assumes random meanings or behaviors (randomness), or when it is not clearly established or described (vagueness), or when it may have more than one possible meaning or status (ambiguity), or, finally, when it is described on the basis of too limited amount of information (imprecision). At a closer examination, randomness, vagueness, ambiguity
Non-linear analysis of reinforced and prestressed concrete framed structures
The standard design of reinforced and prestressed concrete frames is usually based on a linear elastic analysis under several load combinations and on a subsequent set of non-linear cross-sectional verifications. This kind of approach is simple, but contains some intrinsic inconsistencies in particular with respect to the use of global safety factors which affect not only the safety measurements, but also the results of the analyses in terms of both displacements and internal stresses [14].
When
Basic limit states of failure
Based on the general concepts of reinforced and prestressed concrete design, structural performances should generally be described with reference to a specified set of limit states, with regards to both serviceability and ultimate conditions [9]: such limits separate desired states of the structure from undesired ones.
Splitting cracks and considerable creep effects may occur if the compression stresses σc in concrete are too high. Besides, excessive stresses either in reinforcing steel σs or in
Applications
In the following, the fuzzy reliability analysis is used to model the structural behavior a prestressed concrete continuous beam and to support the design decisions regarding a segmentally erected cable-stayed bridge.
The first application regards a test beam, experimented by Lin [16], and intends to show, for a relatively simple structure, the level of detail allowed by non-linear analysis, the good accordance between experimental and numerical results and the effects due to the fuzziness of
Conclusions
In this paper, a general methodology for the fuzzy reliability analysis of structural systems has been presented and specialized to the case of reinforced and prestressed concrete structures. The reliability problem is formulated at the load level and the membership function of the safety factor over the failure interval is derived for several limit states by solving the corresponding anti-optimization problems. Particular attention is paid to both the solution of the optimization process,
Acknowledgements
This paper is dedicated to the memory of Francesco Martinez y Cabrera, formerly professor of “Theory and Design of Bridge Structures” at the Technical University of Milan, who started us to a comprehensive vision of structures and life.
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