Importance of modal cross-correlation on wind loaded structures

Abstract

In civil engineering applications the dynamic analysis of large structure is often performed in a modal space. This method is known to offer an interesting decrease of the number of degrees of freedom as well as a decomposition of the structure’s response in “uncoupled” components. Even if the response in each mode can be computed independently from the responses in the other ones, in the context of stochastic loading, the coherence of these modal responses must be accounted for in the determination of the structural response. These quantities are known as the modal cross-correlations. In this paper we will show that these crosscorrelations can be important (contrarily to what is sometimes thought) even in case of wellseparated natural frequencies. This will be illustrated on the analysis of the famous Viaduct of Millau during an erection stage. mode shapes (m<<n) are collected in the mode shape matrix [ ] Φ and { } η represents the vector of modal amplitudes (or generalized coordinates). This subspace reduction can be useful in a deterministic analysis since the equation of motion in each mode can be solved independently from the others. In stochastic analysis, this subspace projection is also interesting since it gives a diagonal transfer matrix (see Table 1). Table 1 summarizes the main stages of a stochastic analysis. At this stage, it is supposed that the power spectral density (PSD) matrix of the applied forces is known. In case of a wind loaded structure subjected to buffeting forces this quantity can be expressed as a function of the PSD of the wind velocity and of the aerodynamic characteristics of the structure (REF, REF). By preand post-multiplication by the mode shapes [ ] Φ , the PSD matrix of the generalized forces can be obtained (see Table 1, line 1). Then the equations motion can be solved very easily by preand post-multiplying by the diagonal transfer matrix (see Table 1, line 2). Each component of the PSD matrix of the generalized coordinates can be expressed as a function of the corresponding element of the PSD matrix of the generalized forces only. As a final step the PSD matrix of the structural displacements can be estimated by preand post-multiplying again by the mode shapes. The computation of this matrix is the main objective of a stochastic analysis. Table 1. Summary of the progress of a stochastic analysis. Stage Operation Projection of the forces in the modal space ( ) [ ] ( ) [ ][ ] * T F F S S ω ω ⎡ ⎤ = Φ Φ ⎣ ⎦ Resolution in the modal space ( ) [ ] ( ) [ ] * , T conj F S H S H η ω ω ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦ ⎣ ⎦ Come back to displacements of structure ( ) [ ] [ ] ( ) [ ] x S Sη ω ω ⎡ ⎤ = Φ Φ ⎣ ⎦ T,conj stands for transposed conjuguate. Note that [H] is a diagonal matrix. 3 STOCHASTIC ANALYSIS OF LARGE STRUCTURES The design of structures is based on extreme displacements and internal forces. In this context, “extreme values” means maximum values that can be expected on a certain life time or duration of observation. For Gaussian processes the extreme value of a quantity can be estimated by the product of its variance and a peak factor (REF REF). Even if a rigorous value for the peak factor can be computed, for reasons of simplicity and computation time, the peak factor is often estimated by simplified relations (REF REF). This indicates that the main objective of a stochastic analysis can be reached once the covariance matrices of the structural displacements and internal forces are computed. If structural displacements are considered, the extreme values, needed for the design, can thus be estimated thanks to the variances of the displacements. The computation of the elements of the PSD matrix of the structural displacements consists in the main objective of a stochastic analysis. For large structures, the estimation of the whole PSD matrix of the displacements is however too expensive in terms of computation time. Two solutions are thus generally considered: • to compute the diagonal elements of this matrix only; this can be also realized for the most important degrees of freedom only • to integrate the last expression of Table 1 along frequencies and to estimate directly the covariance matrix of the displacements as a function of the covariance matrix of the modal amplitudes : [ ] [ ] [ ] cov cov T x η ⎡ ⎤ = Φ Φ ⎣ ⎦ (1)