Keywords: dynamic analysis, fuzzy numbers, fuzzy set theory, direct integration method, ground motion accelerogram, seismic loading.

The design of reinforced concrete structures can be approached in different ways. The most common one with practicing engineers is the deterministic approach when the effect of various influences is estimated by the design engineer. This approach is suitable for companies with teams within which the experience is well established. Mostly it is the responsibility of an individual, who is well respected and whose subjective opinion is never questioned.

The other well-established approaches are based on the statistics, which in its nature is more rigorous as it tries to cover the objective information gathered over a period of time. This approach can be found at the later stages of design, mostly the reliability assessment. Of course, in order to achieve a reliable assessment, a considerably large amount of data needs to be gathered. Here, a close form of cooperation between the designer and the contractor is appreciated. But the real state of the matter is, this viewpoint is based on the situation in the Czech Republic, that if the designer and the contractor are not of the same body, each side tends to keep the detailed information to itself. This may transpire in the assessment of material parameters, when the contractor allows only a limited access to the data it possesses, which yields rather subjective estimation of material parameters, such as the modulus of elasticity, strengths, and especially their evolution over time.

Möller and Beer presented a concise approach to the reliability assessment, [1], which takes into account the effect of subjectivity quantified with help of the fuzzy set theory, known as the fuzzy randomness.

The work presented in this paper describes an approach to the very early stage of design, when any additional information related to the material properties, geometry, supports and loading can save time spent with unnecessary redesigning of a particular structural member or the entire structure. The proposed approach has been presented in an example of design of reinforced concrete structure resistant to seismic loading using the response spectrum method, [2]. In this paper, the approach to the seismic design using another numerical method, the direct step-by-step integration, [3,4], is presented. The quantities considered as fuzzy are the material parameters, while the ground motion accelerogram is taken as crisp at this stage. However, the material uncertainty transpires in the resulting displacements and accelerations of the structure. This effect and the way of viewing the results are explained with respect to a simple example of undamped free vibration. The method of the direct integration with a realistic crisp ground motion accelerogram is also explained.

An example of possible distribution of displacement obtained by the direct integration method is shown in Figure 1.

1

Möller, B. and Beer M., "Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics", Springer, 2004.

2

Štemberk, P. and Kruis, J., "Fuzzy Dynamic Structural Analysis of 2D Frames", Proc. of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing [CD-ROM]. Stirling: Civil-Comp Press Ltd, 2005. doi:10.4203/ccp.81.225

3

Bathe K.J., "Finite Element Procedures", Prentice-Hall, Inc.,1996.

4

Bittnar, Z. and Šejnoha, J., "Numerical Methods in Structural Mechanics", ASCE Press, 1996.

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