Keywords: dynamic analysis, fuzzy numbers, fuzzy set theory, response surface function, seismic loading, response spectrum.

Concrete, as a convenient building material, inherently involves uncertainty about its composition, which is difficult to be eliminated completely, however, this uncertainty can be assessed by statistical, fuzzy, or other suitable tools. In the case of concrete structures, such as frames made of reinforced concrete, it is costly to obtain a sufficient experimental data set which would yield the desired statistical characteristics of material parameters necessary for a reliable structural analysis which should cover the effect of uncertainty. Instead, the knowledge gained with practicing engineers can be included as fuzzy numbers in the material modelling.

For design purposes, one may wish to conduct a statistical analysis, using the statistical characteristics of several measured events. In the case of an earthquake, the measured data for each site of interest is not particularly dense, leaving the statistical characteristics with little relevance. On the other hand, the expected seismic load at a site can be alternatively expressed by the fuzzy sets [1], which take into account the scarcity of seismic stations and the information about local sub-soil composition.

In this paper, an approach to dynamic analysis based on the fuzzy set theory is presented as an alternative to the classical stochastic dynamic analyses. The material parameters of reinforced concrete are considered to be fuzzy quantities with a given distribution, i.e. fuzzy numbers with a desired shape of the membership function [2]. The dynamic analysis is performed with help of the fuzzy arithmetic on either -cuts or computation-efficient (L,R) numbers [3]. The result of such an analysis is in the form of fuzzy numbers which compared with the stochastic approach is less expensive in terms of computation time and still is provides the designer with an idea of distribution of the sought quantity. In order to further improve the computational efficiency, inspired by [4], the concept of a surface response function [5,6] is utilized.

This approach is demonstrated in an illustrative example of a 2D frame where the effect of uncertain material parameters transpires in corresponding distributions of natural modal shapes and natural frequencies of an analysed two-dimensional frame. The results of the natural vibration analysis can be readily utilized in the investigation of the response spectrum of a structure. A methodology for a possible application to seismic design is explained. The question of how many -cuts are necessary and how their number influences the result is also tackled.

An example of a mode shape computed with fuzzy numbers is shown in Figure 225.1.

1

Zadeh, L.A., "Fuzzy sets", Information Control, 8 (3), 338-352, 1965. doi:10.1016/S0019-9958(65)90241-X

2

Valliappan, S., Pham, T.D., "Construction the membership function of a fuzzy set with objective and subjective information", Microcomputers in Civil Engrg., 8, 75-82, 1993.

3

Kaufman, A., Gupta, M.M., "Introduction to Fuzzy Arithmetic: Theory and Application", Nostrand Reinhold Company, Inc., New York, 1985.

4

Akpan, U.O., Koko, T.S., Orisamolu, I.R., Gallant, B.K., "Practical fuzzy finite element analysis of structures", Finite Elements in analysis and Design, 38, 93-111, 2001. doi:10.1016/S0168-874X(01)00052-X

5

Bucher, C.G., Chen, Y.M., Schuëler, G.I., "Time Variant Reliability Analysis Utilizing Response Surface Approach", Reliability and Optimization of Structural Systems '88, Eds. P. Thoft-Christensen, 1988.

6

Rajashekhar, M.R., Ellingwood, B.R., "A new look at the response surface approach for reliability analysis", Structural Safety, 12, 205-220, 1993. doi:10.1016/0167-4730(93)90003-J

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