Keywords: fuzzy optimization, cable-stayed bridges.

In design and optimization problems material constants, loading and structure geometry are usually considered as given data, but in real world assumed values do not correspond with actual ones. All of this is accounted by safety factors, which amplify load magnitude, or reduce material strength, leading in general to over- conservative structures. As an alternative to safety factors one may try to describe the uncertain data and use this information during the optimization. Probabilistic description is nowadays common and very simple up to very sophisticated PDF can be used to describe uncertain parameters. However these procedures face serious difficulties when being implemented in engineering applications. This lead to nonprobabilistic description of uncertainty, in particular the fuzzy-set based analysis and instances of the worst case design also termed anti-optimization methods. The latter fixes bounds for the uncertain variables instead of defining probability functions needing less information than the probabilistic approach. The Two-Phase Method for fuzzy optimization of structures is based on the fuzzy-set method first proposed by Zadeh. In the first phase, the sequential fuzzy solution is obtained by using the Level Cuts Method, in which a fuzzy optimization problem is transformed into a series of ordinary optimization problems using different ?-level cuts in fuzzy constraints so as to determine a fuzzy optimization domain in the design space. In the second phase, the particular crisp solution is obtained by the Bound Search Method, in which having obtained the supremum and the infimum of the sequential fuzzy solution the particular optimum level is found using the bound search so as to provide a crisp optimization solution in the design space. This method is based on an alternative interpretation of the Belman-Zadeh optimality criteria which does not require the use of an artificial fuzzy objective.

Cable-stayed bridges are large and expensive structures that are now being widely used. The trend to increase the spans and the advent of innovative erection techniques make thus desirable the development of computational tools to assist on the preliminary design stage and/or erection control, which today relies mostly on the design staff expertise. The authors have been involved in the last few years in such a project, which led to the development of a programme dealing with the aspects of design which can be objectively expressed by a numerical merit measure, such as structural erection or maintenance cost.

The finite element based open code MODULEF was used as the basic tool for structural analysis, because code availability was a fundamental requirement for further developments. Out of the several element types included in the element library of the programme, only the FE required for two- and three-dimensional models of cable-stayed bridges were retained and adapted to specific needs. The analytic direct method was adopted for the purpose of sensitivity analysis, given the availability of the code, the discrete structural pattern and the large number of constraints under control.

The fundamental goal of the optimization process is to enhance the design in the sense of cost (or volume) reduction. This must be done with due attention paid on keeping the stresses, deflections and (eventually) frequencies within allowable limits. Cost and those constraints as weighted objectives whose simultaneous reduction is desirable. The solution of such problem provides an optimum design in the Pareto sense, depending on the relative priority (or weight) assigned to each objective. Since the simultaneous reduction of all goals is desirable, the minimax formulation looks particularly suitable for this purpose. This problem is discontinuous and non-differentiable and is therefore hard to solve. However, its solution is equivalent to that of an unconstrained convex scalar function, depending only on one control parameter, which may be solved by conventional quasi-Newton methods. This parameter must be steadily increased through the optimization process.

A numerical example with sizing design variables only and without erection stages consisting of a symmetric three-span cable-stayed bridge was solved. Monosymmetric I-shaped cross-sections are prescribed for the stiffening girders, while the pylons are made up of steel plates defining a rectangular hollow cross section. I-shaped transverse beams support the wearing surface. Three load cases, corresponding to live load on either side, central or whole span, were considered.

Fuzzy goals were generated involving Young's modulus, stress limits and loading. The influence of geometric design variables in the optimum cost is quite limited. Moreover both geometric and sizing design variables in steel cable stayed bridges are strictly controlled and their values are assumed to be deterministic.

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