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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 109

Dynamic Elastic Analysis of MDOF Structures with Random Strengths

S. Benfratello and F. Giambanco

Department of Structural and Geotechnical Engineering, Universita' degli Studi di Palermo, Italy

Full Bibliographic Reference for this paper
S. Benfratello, F. Giambanco, "Dynamic Elastic Analysis of MDOF Structures with Random Strengths", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 109, 2004. doi:10.4203/ccp.79.109
Keywords: uncertainties, probabilistic analysis, dynamic analysis, elastic analysis.

Summary
The numerical values of the constants appearing in constitutive laws show a random behaviour and the international standard ENV 13005 [1] established the criteria for the probabilistic characterization of such values. The structural response must be therefore considered as random variable whose characteristics have to be appropriately determined. In the framework of structural analysis a fundamental role is played by elastic limit analysis which provides the maximum load multiplier such that the structure remains everywhere in elastic field. Many important actions on structures such as wind, wave and earthquake, possess a strongly dynamic behaviour and the appropriate tools for structural dynamic analysis have to be adopted. Further, stress levels in the structure during transient phase can be much greater than the corresponding ones in stationary phase and therefore appropriate tools for performing limit analysis in the case of dynamic actions are of great interest. In the paper the dynamic elastic limit analysis of MDOF discrete or discretised structures subjected to analytical forcing function (i.e. unit step function or sinusoidal function) by assuming the yield stress of material as random variable, whose characteristics are known, is performed. In the paper the exact probability density function of the limit elastic load multiplier is determined as that of the minimum of random variables obtained as a linear transformation of the random yield stresses. The -th limit elastic multiplier is given as

   i = 1,2, ...,n (43)

being the number of degrees of freedom of the structure, are the -th element of the vectors (yield stress vector) and (actual stress vector), respectively. In matrix form equation (43) can be written as follows

(44)

As it can be easily seen, the coefficients of transformation (44) are time dependent. The exact probability density function of the limit elastic load multiplier is given by

(45)

Since the computational effort required by equation (45) can be very high a very simple approximated formula is also presented. This approximated formula represents an upper bound of the exact one and it is obtained from equation (45) simply by assuming the random variable as independent. Numerical application to the truss sketched in figure 1 confirms that the proposed approach allows to easily take into account the role played by the transient phase. In figure 2 the probability density function of the limit elastic multiplier is reported.
Figure 1: Redundant truss.

Figure 2: Probability density function of the limit elastic dynamic load multiplier.

References
1
UNI CEI ENV 13005:2000, Guide to the expression of uncertainty in measurement.

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