#
Uncertainty Quantification and Simulation of Wind-Tunnel-Informed Stochastic Wind Loads^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Simulation of Multivariate Stochastic Wind Loads

#### 2.2. Wind-Tunnel-Informed POD-Based SRM

## 3. Uncertainty Quantification

#### 3.1. Errors Induced by Wind Tunnel Data

#### 3.2. Errors Induced by the Model

## 4. Wind Tunnel Tests

#### 4.1. Experimental Setup

#### 4.2. Processing of the Wind Tunnel Data

## 5. Results

#### 5.1. Preamble

#### 5.2. Errors Induced by the Variability of Short-Duration Wind Tunnel Records

#### 5.3. Model and Truncation Errors

#### 5.3.1. Model Errors

#### 5.3.2. Truncation Errors

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Standardization Scheme of Force Coefficients

**Figure A1.**Example time series of the standardized force coefficients for $\beta ={0}^{\circ}$ and the SM layout.

## References

- Zeldin, B.A.; Spanos, P.D. Random Field Representation and Synthesis Using Wavelet Bases. J. Appl. Mech.
**1996**, 63, 946–952. [Google Scholar] [CrossRef] - Kitagawa, T.; Nomura, T. A wavelet-based method to generate artificial wind fluctuation data. J. Wind Eng. Ind. Aerodyn.
**2003**, 91, 943–964. [Google Scholar] [CrossRef] - Li, Y.; Kareem, A. Simulation of multivariate random processes: Hybrid DFT and digital filtering approach. J. Eng. Mech.
**1993**, 119, 1078–1098. [Google Scholar] [CrossRef] - Mignolet, M.P.; Spanos, P.D. Simulation of Homogeneous Two-Dimensional Random Fields: Part I—AR and ARMA Models. J. Appl. Mech.
**1992**, 59, S260–S269. [Google Scholar] [CrossRef] - Shinozuka, M.; Deodatis, G. Simulation of stochastic processes by spectral representation. Appl. Mech. Rev.
**1991**, 44, 191–204. [Google Scholar] [CrossRef] - Deodatis, G. Simulation of ergodic multivariate stochastic processes. J. Eng. Mech.
**1996**, 122, 778–787. [Google Scholar] [CrossRef] - Cheynet, E.; Daniotti, N.; Bogunović Jakobsen, J.; Snæbjörnsson, J.; Wang, J. Unfrozen skewed turbulence for wind loading on structures. Appl. Sci.
**2022**, 12, 9537. [Google Scholar] [CrossRef] - Huang, G.; Peng, L.; Kareem, A.; Song, C. Data-driven simulation of multivariate nonstationary winds: A hybrid multivariate empirical mode decomposition and spectral representation method. J. Wind Eng. Ind. Aerodyn.
**2020**, 197, 104073. [Google Scholar] [CrossRef] - López-Ibarra, A.; Pozos-Estrada, A.; Nava-González, R. Effect of Partially Correlated Wind Loading on the Response of Two-Way Asymmetric Systems: The Impact of Torsional Sensitivity and Nonlinear Effects. Appl. Sci.
**2023**, 13, 6421. [Google Scholar] [CrossRef] - Wang, L.; McCullough, M.; Kareem, A. A data-driven approach for simulation of full-scale downburst wind speeds. J. Wind Eng. Ind. Aerodyn.
**2013**, 123, 171–190. [Google Scholar] [CrossRef] - Shinozuka, M. Stochastic fields and their digital simulation. In Stochastic Methods in Structural Dynamics; Springer: Dordrecht, The Netherlands, 1987; pp. 93–133. [Google Scholar]
- Tamura, Y.; Suganuma, S.; Kikuchi, H.; Hibi, K. Proper orthogonal decomposition of random wind pressure field. J. Fluids Struct.
**1999**, 13, 1069–1095. [Google Scholar] [CrossRef] - Carassale, L.; Piccardo, G.; Solari, G. Double modal transformation and wind engineering applications. J. Eng. Mech.
**2001**, 127, 432–439. [Google Scholar] [CrossRef] - Chen, L.; Letchford, C.W. Simulation of multivariate stationary Gaussian stochastic processes: Hybrid spectral representation and proper orthogonal decomposition approach. J. Eng. Mech.
**2005**, 131, 801–808. [Google Scholar] [CrossRef] - Chen, X.; Kareem, A. Proper orthogonal decomposition-based modeling, analysis, and simulation of dynamic wind load effects on structures. J. Eng. Mech.
**2005**, 131, 325–339. [Google Scholar] [CrossRef] - Ouyang, Z.; Spence, S.M.J. A performance-based wind engineering framework for envelope systems of engineered buildings subject to directional wind and rain hazards. J. Struct. Eng.
**2020**, 146, 04020049. [Google Scholar] [CrossRef] - Hu, L.; Li, L.; Gu, M. Error assessment for spectral representation method in wind velocity field simulation. J. Eng. Mech.
**2010**, 136, 1090–1104. [Google Scholar] [CrossRef] - Tao, T.; Wang, H.; Hu, L.; Kareem, A. Error analysis of multivariate wind field simulated by interpolation-enhanced spectral representation method. J. Eng. Mech.
**2020**, 146, 04020049. [Google Scholar] [CrossRef] - Davenport, A. The response of six building shapes to turbulent wind. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**1971**, 269, 385–394. [Google Scholar] - Simiu, E. Wind spectra and dynamic alongwind response. J. Struct. Div.
**1974**, 100, 1897–1910. [Google Scholar] [CrossRef] - Melbourne, W. Comparison of measurements on the CAARC standard tall building model in simulated model wind flows. J. Wind Eng. Ind. Aerodyn.
**1980**, 6, 73–88. [Google Scholar] [CrossRef] - Solari, G. Analytical estimation of the alongwind response of structures. J. Wind Eng. Ind. Aerodyn.
**1983**, 14, 467–477. [Google Scholar] [CrossRef] - Kaimal, J.C.; Finnigan, J.J. Atmospheric Boundary Layer Flows: Their Structure and Measurement; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Gurley, K.; Kareem, A. Simulation of correlated non-Gaussian pressure fields. Meccanica
**1998**, 33, 309–317. [Google Scholar] [CrossRef] - Suksuwan, A.; Spence, S.M.J. Optimization of uncertain structures subject to stochastic wind loads under system-level first excursion constraints: A data-driven approach. Comput. Struct.
**2018**, 210, 58–68. [Google Scholar] [CrossRef] - Lin, N.; Letchford, C.; Tamura, Y.; Liang, B.; Nakamura, O. Characteristics of wind forces acting on tall buildings. J. Wind Eng. Ind. Aerodyn.
**2005**, 93, 217–242. [Google Scholar] [CrossRef] - Tamura, Y.; Kareem, A. Advanced Structural Wind Engineering; Springer: Tokyo, Japan, 2013; Volume 482. [Google Scholar]
- Spence, S.M.J.; Kareem, A. Data-enabled design and optimization (DEDOpt): Tall steel building frameworks. Comput. Struct.
**2013**, 129, 134–147. [Google Scholar] [CrossRef] - Gurley, K.R. Modelling and Simulation of Non-Gaussian Processes; University of Notre Dame: Notre Dame, IN, USA, 1997. [Google Scholar]
- Welch, P. The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust.
**1967**, 15, 70–73. [Google Scholar] [CrossRef] - Solomon, O.M., Jr. PSD Computations Using Welch’s Method; Technical Report; Sandia National Labs.: Albuquerque, NM, USA, 1991. [Google Scholar]
- Tao, T.; Wang, H.; Yao, C.; He, X.; Kareem, A. Efficacy of interpolation-enhanced schemes in random wind field simulation over long-span bridges. J. Bridge Eng.
**2018**, 23, 04017147. [Google Scholar] [CrossRef] - Catarelli, R.A.; Fernández-Cabán, P.L.; Phillips, B.M.; Bridge, J.A.; Masters, F.J.; Gurley, K.R.; Prevatt, D.O. Automation and new capabilities in the university of Florida NHERI Boundary Layer Wind Tunnel. Front. Built Environ.
**2020**, 6, 558151. [Google Scholar] [CrossRef] - Wu, Y.; Gao, Y.; Li, D. Error assessment of multivariate random processes simulated by a conditional-simulation method. J. Eng. Mech.
**2015**, 141, 04014155. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) illustration of the building model used in the wind tunnel tests, and configuration of the pressure taps, (

**b**) wind tunnel testing with SM setup, and (

**c**) wind tunnel testing with PM setup.

**Figure 2.**(

**a**) SM setup, (

**b**) PM setup, and (

**c**) coordinate system adopted in estimating the wind loads.

**Figure 4.**Comparison between smoothed PSDs of short-duration records, ${S}_{R}\left(f\right)$, and target PSD of $C{F}_{x,20}\left(t\right)$ at the 20th floor and wind direction of $\beta ={0}^{\circ}$.

**Figure 5.**Comparison between smoothed PSDs of short-duration records, ${S}_{R}\left(f\right)$, and target PSD of $C{F}_{y,20}\left(t\right)$ at the 20th floor and wind direction of $\beta ={0}^{\circ}$.

**Figure 6.**Comparison between smoothed PSDs of short-duration records, ${S}_{R}\left(f\right)$, and target PSD of $C{T}_{z,20}\left(t\right)$ at the 20th floor and wind direction of $\beta ={0}^{\circ}$.

**Figure 7.**Mean error in the variance, ${\mu}_{\epsilon}$, with standard deviation, ${\sigma}_{\epsilon}$, for each floor, for the SM layout and $\beta ={0}^{\circ}$.

**Figure 8.**(

**a**) mean errors in the variance for SM and PM layouts, and (

**b**) standard deviation of the errors in the variance for the SM and PM layouts.

**Figure 9.**Map of the target correlation coefficients for the SM setup and wind direction $\beta ={0}^{\circ}$.

**Figure 10.**Mean error in the correlation coefficients for the SM setup and wind direction $\beta ={0}^{\circ}$.

**Figure 11.**Standard deviation of the difference between target and typical correlation coefficients for the SM setup and wind direction $\beta ={0}^{\circ}$.

**Figure 12.**(

**a**) mean errors in the correlation coefficients for SM and PM layouts, and (

**b**) standard deviation of the errors in the correlation coefficients for the SM and PM layouts.

**Figure 13.**Histograms of $\epsilon $ associated with SM layout and $\beta ={0}^{\circ}$ of (

**a**) $C{F}_{x}$, (

**b**) $C{F}_{y}$, (

**c**) $C{T}_{z}$, and (

**d**) correlation coefficients between the components of $\epsilon $.

**Figure 15.**Expected values and range of the model error in the correlation coefficients for the SM and PM setups.

**Figure 16.**Comparison of the error in the variance due to the use of a single smoothed short-duration wind tunnel record (wt) and the simulation model (sim).

**Figure 17.**Comparison of the difference between the correlation coefficients due to the use of a single smoothed short-duration wind tunnel record (wt) and the simulation model (sim).

**Figure 18.**Mean error in the variance, ${\mu}_{\epsilon}$, for each force coefficient component, considering 10, 15, and 25 contributing modes, for SM layout and wind direction $\beta ={0}^{\circ}$.

**Figure 19.**Mean error in the correlation coefficients, considering 10, 15, and 25 modes, for SM layout and wind direction $\beta ={0}^{\circ}$.

**Figure 20.**(

**a**) error in the variance, $\epsilon $, and (

**b**) errors in the correlation coefficients, $\phi $, considering 10, 15, and 25 contributing modes.

**Figure 21.**Expected error in the variance from the truncation of modes for both the SM and PM setups, considering 1, 5, 10, 15, 20, 25, and 75 contributing modes.

**Figure 22.**Expected difference in the correlation coefficients from the truncation of modes for all wind directions and the SM and PM setups, considering 1, 5, 10, 15, 20, 25, and 75 contributing modes.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duarte, T.G.A.; Arunachalam, S.; Subgranon, A.; Spence, S.M.J.
Uncertainty Quantification and Simulation of Wind-Tunnel-Informed Stochastic Wind Loads. *Wind* **2023**, *3*, 375-393.
https://doi.org/10.3390/wind3030022

**AMA Style**

Duarte TGA, Arunachalam S, Subgranon A, Spence SMJ.
Uncertainty Quantification and Simulation of Wind-Tunnel-Informed Stochastic Wind Loads. *Wind*. 2023; 3(3):375-393.
https://doi.org/10.3390/wind3030022

**Chicago/Turabian Style**

Duarte, Thays G. A., Srinivasan Arunachalam, Arthriya Subgranon, and Seymour M. J. Spence.
2023. "Uncertainty Quantification and Simulation of Wind-Tunnel-Informed Stochastic Wind Loads" *Wind* 3, no. 3: 375-393.
https://doi.org/10.3390/wind3030022