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

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## 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.

## Keywords

Metamodeling Reduced order modeling Uncertain dynamic systems Monte Carlo simulation Wind engineering Multi-degree-of-freedom systems## Notes

### Funding

This research effort was supported in part by the National Science Foundation (NSF) under Grant No. CMMI-1750339. This support is gratefully acknowledged.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Spence SMJ, Gioffrè M (2012) Large scale reliability-based design optimization of wind excited tall buildings. Probab Eng Mech 28:206–215CrossRefGoogle Scholar
- 2.Bernardini E, Spence SMJ, Kwon DK, Kareem A (2015) Performance-based design of high-rise buildings for occupant comfort. J Struct Eng. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001223 Google Scholar
- 3.Tabbuso P, Spence SMJ, Palizzolo L, Pirrotta A, Kareem A (2016) An efficient framework for the elasto-plastic reliability assessment of uncertain wind excited systems. Struct Saf 58:69–78CrossRefGoogle Scholar
- 4.Chuang WC, Spence SMJ (2017) A performance-based design framework for the integrated collapse and non-collapse assessment of wind excited buildings. Eng Struct 150:746–758CrossRefGoogle Scholar
- 5.Gioffrè M, Gusella V (2007) Peak response of a nonlinear beam. J Eng Mech 133(9):963–969CrossRefGoogle Scholar
- 6.Ciampoli M, Petrini F, Augusti G (2011) Performance-based wind engineering: towards a general procedure. Struct Saf 33(6):367–378CrossRefGoogle Scholar
- 7.Petrini F, Ciampoli M (2012) Performance-based wind design of tall buildings. Struct Infrastruct Eng 8(10):954–966Google Scholar
- 8.Caracoglia L (2014) A stochastic model for examining along-wind loading uncertainty and intervention costs due to wind-induced damage on tall buildings. Eng Struct 78:121–132CrossRefGoogle Scholar
- 9.Cui W, Caracoglia L (2015) Simulation and analysis of intervention costs due to wind-induced damage on tall buildings. Eng Struct 87:183–197CrossRefGoogle Scholar
- 10.Cui W, Caracoglia L (2017) Exploring hurricane wind speed along us atlantic coast in warming climate and effects on predictions of structural damage and intervention costs. Eng Struct 122:209–225CrossRefGoogle Scholar
- 11.Spiridonakos MD, Chatzi EN (2015) Metamodeling of dynamic nonlinear structural systems through polynomial chaos NARX models. Comput Struct 157:99–113CrossRefGoogle Scholar
- 12.Mai CV, Spiridonakos MD, Chatzi EN, Sudret B (2016) Surrogate modelling for stochastic dynamical systems by combining narx models and polynomial chaos expansions. Int J Uncertain Quantif 6:313–339MathSciNetCrossRefGoogle Scholar
- 13.Mai CV (2016) Polynomial chaos expansions for uncertain dynamical systems—applications in earthquake engineering. Ph.D. Thesis, ETH Zurich, SwitzerlandGoogle Scholar
- 14.Eman H, Pradlwarter HJ, Schuëller GI (2000) A computational procedure for the implementation of equivalent linearization in finite element analysis. Earthq Eng Struct Dyn 29:1–17CrossRefGoogle Scholar
- 15.Pradlwarter HJ, Schuëller GI, Schenk CA (2003) A computational procedure to estimate the stochastic dynamic response of large non-linear FE-models. Comput Methods Appl Mech Eng 192:777–801ADSCrossRefzbMATHGoogle Scholar
- 16.Schenk CA, Pradlwarter HJ, Schuëller GI (2004) On the dynamic stochastic response of FE models. Probab Eng Mech 19:161–170CrossRefGoogle Scholar
- 17.Jensen HA, Catalan AA (2007) On the effects of non-linear elements in the reliability based optimal design of stochastic dynamical systems. Int J Non-Linear Mech 42:802–816CrossRefzbMATHGoogle Scholar
- 18.Valdebenito MA, Schuëller GI (2011) Efficient strategies for reliability-based optimization involving non-linear, dynamical structures. Comput Struct 89:1797–1811CrossRefGoogle Scholar
- 19.Beck AT, Kougioumtzoglou IA, dos Santos KRM (2014) Optimal performance-based design of non-linear stochastic dynamical RC structures subject to stationary wind excitation. Eng Struct 78:145–153CrossRefGoogle Scholar
- 20.Mitseas IP, Kougioumtzoglou IA, Beer M (2016) An approximate stochastic dynamics approach for nonlinear structural system performance-based multi-objective optimum design. Struct Saf 60:67–76CrossRefGoogle Scholar
- 21.Wilson EL (2002) Three dimensional static and dynamic analysis of structures: a physical approach with emphasis on earthquake engineering. Computers and Structures Inc., BerkeleyGoogle Scholar
- 22.Diniz SMC, Sadek F, Simiu E (2004) Wind speed estimation uncertainties: effects of climatological and micrometeorological parameters. Probab Eng Mech 19(4):361–371CrossRefGoogle Scholar
- 23.Diniz SMC, Simiu E (2005) Probabilistic descriptions of wind effects and wind-load factors for database-assisted design. J Struct Eng 131(3):507–516CrossRefGoogle Scholar
- 24.Bashor R, Kijewski-Correa T, Kareem A (2005) On the wind-induced response of tall buildings: the effects of uncertainties in dynamic properties and human comfort thresholds. In: Proceedings of the \(10^{{\rm th}}\) Americas conference on wind engineeringGoogle Scholar
- 25.Billings SA (2013) Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains. Wiley, New YorkCrossRefzbMATHGoogle Scholar
- 26.Billings SA, Wei HL (2008) An adaptive orthogonal search algorithm for model subset selection and non-linear system identification. Int J Control 81(5):714–724MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Wei HL, Billings SA (2008) Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information. Int J Model Identif Control 3(4):341–356CrossRefGoogle Scholar
- 28.Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230:2345–2367ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 29.Lagarias JC, Reeds JA, Wright MH, Wright PE (1998) Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J Optim 9(1):112–147MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Symans MD, Constantinou MC (1998) Passive fluid viscous damping systems for seismic energy dissipation. J Earthq Technol 35(4):185–206Google Scholar
- 31.Deodatis G (1996) Simulation of ergodic multivariate stochastic processes. J Eng Mech 122:778–787CrossRefGoogle Scholar
- 32.Kaimal JC, Wyngaard JC, Izumi Y, Coteé OR (1972) Spectral characteristics of surface-layer turbulence. Q J R Meteorol Soc 98(417):563–589ADSCrossRefGoogle Scholar
- 33.Davenport GA (1967) The dependence of wind load upon meteorological parameters. In: Proceedings of the international research seminar on wind effects on building and structures. University of Toronto Press, pp 19–82Google Scholar