Rapid uncertainty quantification for non-linear and stochastic wind excited structures: a metamodeling approach

Abstract

The application of performance-based design (PBD) requires the modeling of the dynamic response of the system beyond the elastic limit. If probabilistic PBD is considered, this implies the need to propagate uncertainties through non-linear dynamic systems. This paper investigates the possibility of using advanced metamodeling techniques in order to define a computationally tractable approach for propagating uncertainty through a class of multi-degree-of-freedom non-linear dynamic systems subject to multivariate stochastic wind excitation. To this end, a scheme is introduced that is based on combining model order reduction with a recently introduced metamodeling approach that has been seen to be particularly effective in describing the dynamic response of uncertain non-linear systems of low dimensions. A case study consisting in a 40-story moment resisting frame subject to multivariate stochastic wind excitation and an array of non-linear viscous dampers is presented to illustrate the potential of the scheme.

Appendix 1: Target wind spectrum

The target power spectral density (PSD) function of the fluctuating wind velocity, $v_{z_j}(t)$, can be taken as [32]:

$$\begin{aligned} S_{v_{z_j}}(\omega ) = v_*^2 \frac{50z_j}{\pi {\bar{v}}_{z_j}} \frac{1}{\left[ 1+50 \left( \frac{\omega z_j}{2\pi {\bar{v}}_{z_j}}\right) \right] ^{5/3}} \end{aligned}$$

(19)

where $v_*$ is the shear velocity given by:

$$\begin{aligned} v_* = {\bar{v}}_{10} \beta \frac{k_a}{\text {ln}\left( \frac{10}{z_0}\right) } \end{aligned}$$

(20)

where ${\bar{v}}_{10}$ is the mean wind velocity at 10 m, β = 0.65, while $k_a = 0.4$ is the Von Kármán’s constant. The cross power spectral density can then be defined as:

$$\begin{aligned} S_{v_{z_j}v_{z_k}}(\omega ) = \sqrt{S_{v_{z_j}}(\omega )S_{v_{z_k}}(\omega )} \gamma _{jk}(\omega ), \quad j \ne k \end{aligned}$$

(21)

where $\gamma _{jk}$ is the coherence function between $v_{v_{z_j}}(t)$ and $v_{v_{z_j}}(t)$ that can be modeled as [33]:

$$ \gamma _{jk}({\varDelta } z,\omega ) = \text {exp} \left[ -\frac{\omega }{2\pi } \frac{C_z {\varDelta } z}{\frac{1}{2}({\bar{v}}_{z_j}+{\bar{v}}_{z_k})} \right] $$

(22)

where ${\varDelta } z= | z_j - z_k |$ is the height difference, while $C_z$ is a constant that can be set equal to 10 for design purposes [33].