# Stochastic analysis of a stadium roof from deterministic wind tunnel measurements

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## Abstract

Dynamic analyses of structures under buffeting wind loads can be performed in a deterministic (Clough and Penzien, 1997) or stochastic (Preumont, 1994) context, both with a modal approach for computational efficiency reasons. In the first option, the forces are deterministically given, and the uncoupled modal equations of motion are solved either in the time domain with a stepby- step method, either in the frequency domain, with Fourier transformation. In the second option, the analysis relies on the determination of the Power Spectral Density (PSD) matrix of the structural response given that of the loading. The choice of one or another method usually depends on whether the loading is provided in the time or frequency domain and as a deterministic (a single time history) or stochastic manner. From a designer’s point of view, the wind loading can be defined using design codes (e.g. Eurocode, 2005) where analytical expressions of (i) the PSD of wind velocities (Davenport, Von Karman, etc) (ii) the coherence functions and (iii) the pressure coefficients are given to compute, finally, (iv) the PSD of the aerodynamic pressures. Design engineers are usually familiar with this probabilistic approach. Alternatively, the design may be conducted from aerodynamic pressures measured in a wind tunnel. This approach is more realistic than the aforementioned codified procedure since a number of phenomena as (a) the aerodynamic instabilities, (b) aerodynamic admittance (Scanlan and Jones, 1999), (c) site effects are taken into account. Pressures are thus given as unique (deterministic) time histories at each sensor. In a Finite Element context and a modal analysis, the generalized forces are computed from the measured pressures. With the firm wish to perform the analysis in a stochastic manner (for a number of good reasons mentioned next), we suggest to fit a probabilistic model to the measured data. Such a model could be fitted to the measured pressures right away, or any other subsequent quantity such as the generalized forces. The following discussion is about the most favorable quantity that has to be fitted and how to do it appropriately in view of typical measurement imperfections.

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