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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 77
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Paper 119

Fuzzy Cluster Design: A New Way for Structural Design

B. Möller, M. Beer and M. Liebscher

Institute of Structural Analysis, University of Technology, Dresden, Germany

Full Bibliographic Reference for this paper
, "Fuzzy Cluster Design: A New Way for Structural Design", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 119, 2003. doi:10.4203/ccp.77.119
Keywords: fuzzy numbers, fuzzy random variables, fuzzy structural analysis, fuzzy probabilistic safety assessment, fuzzy clustering, uncertain structural design, nonlinear structural design.

Summary
Structural behavior and 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 of data and models 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 fuzzy set theory a general method for fuzzy structural analysis has been developed. This method is formulated in terms of the $ \alpha$-level optimization combined with a modified evolution strategy. Every known analysis algorithm for the realistic simulation of load-bearing behavior may be applied in this fuzzy structural analysis in the sense of a deterministic fundamental solution. With the aid of fuzzy structural analysis fuzzy structural parameters are mapped onto fuzzy structural responses, like e.g. fuzzy internal forces or fuzzy displacements.

A fuzzy probabilistic safety concept has been introduced on the basis of the theory of fuzzy random variables. This concept permits fuzziness, randomness and fuzzy randomness to be accounted for simultaneously. It represents a comprehensive safety concept, which is formulated as a further development of introduced probabilistic approaches. This further development is described by way of the Fuzzy First Order Reliability Method (FFORM). The application of the fuzzy probabilistic safety concept leads to fuzzy safety prognoses, i.e. a fuzzy failure probability and a fuzzy reliability index are obtained.

By applying fuzzy structural analysis and fuzzy probabilistic safety assessment all uncertainty of structural parameters is appropriately mapped onto fuzzy structural responses and fuzzy safety prognoses. The developed algorithm of fuzzy structural analysis yields sets of discrete points in the space of the fuzzy result values. These points are evaluated by membership values. Additionally, the parameter coordinates in the space of the fuzzy input values and fuzzy model parameters which belong to these result points are known.

On the basis of these results an appropriate structural design has to be derived. Due to the uncertainty of structural parameters an uncertain structural design fits the reality in the best way. That means that modified fuzzy structural parameters and fuzzy parameters of fuzzy random variables are determined as being uncertain design values. By comparing requirements regarding structural responses and safety levels with the fuzzy results, permissible points in the space of the fuzzy input values are detected. Permissible uncertain design values are derived on the basis of a fuzzy cluster analysis algorithm in the space of the fuzzy input values.

The generated sets of modified fuzzy input values represent alternative structural design variants. Again they are introduced in fuzzy structural analysis and fuzzy probabilistic safety assessment to obtain the assigned fuzzy result values. After having checked the complete permissibility of the fuzzy results the design variants are compared according to value and uncertainty of the assigned structural responses and safety levels.

The values of structural responses and safety levels are computed by applying defuzzification methods. Thereby the centroid method and the defuzzification algorithms after Chen and Jain are proposed. The distance between the defuzzified result values and the initially stated permissible values serves as the first criterion to assess the fuzzy results.

The uncertainty of the fuzzy results is assessed on the basis of Shannon's entropy. This is used when searching for a preferably robust structural design as the second criterion. As the robustness cannot be measured in absolute terms a relative sensitivity measure is introduced. If the uncertainty of the fuzzy results takes low values in relation to the uncertainty of the fuzzy input values, the assigned structural design is then considered as being robust. This means that moderate fluctuations of input values within the considered uncertainty only lead to small fluctuations in structural response and safety levels.

In general the requirements stated above cannot be satisfied simultaneously, a multiple solution is obtained. Depending on the particular design problem concerned the criterions may be weighted by different factors assigned to different fuzzy results and combined to only one objective function. This yields a compromise solution that meets the requirements in the particular case at best.

The comprehensive application of the developed algorithms is demonstrated by way of an example. Hereby, a geometrically and physically nonlinear algorithm for the analysis of reinforced concrete structures is adopted.

In conclusion, the developed concept for uncertain structural design permits to design a structure on the basis of an arbitrary nonlinear fuzzy structural analysis and fuzzy probabilistic safety assessment and additionally takes account of uncertainty. For the first time this concept provides a capable tool for determining permissible uncertain structural parameters and for assessing fuzzy input and fuzzy result values.

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