[en] Abstract The analysis of large-scale structures subject to transient random loads, coherent in space and time, is a classic problem encountered in earthquake and wind engineering. The simulation-based framework is usually seen as the most convenient approach for both linear and nonlinear dynamics. However, the generation of statistically consistent samples of an excitation field remains a heavy computational task. In light of this, perturbation techniques are applied to develop and improve evolutionary spectral analysis. Advantageously performed in a standard modal basis, this evolutionary spectral analysis for linear structures requires the computation of the modal impulse response matrix. However, this matrix has no general closed-form expression in the presence of modal coupling. We propose therefore to model it by an asymptotic approximation, obtained by the inverse Fourier transform of an asymptotic expansion of the modal transfer matrix of the structure. This latter expansion considers the modal coupling as a perturbation of a main decoupled system. This strategy leads to an expansion known in a closed-form. Finally, the semi-group property allows the use of an efficient recurrence relation to approximate the modal evolutionary transfer matrix, i.e. the evolutionary extension of the transfer matrix. The asymptotic expansion-based method and the recurrence relation are then applied to nonlinear transient dynamics by using Gaussian equivalent linearization. This extension is formalized by a multiple timescales approach, allowing to consider a linearized structure, namely a time variant system, as piecewise linear time invariant depending on a statistical timescale. The proposed developments are finally illustrated on realistic civil engineering applications.
Disciplines :
Civil engineering
Author, co-author :
Canor, Thomas
Caracoglia, Luca
Denoël, Vincent ; Université de Liège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Language :
English
Title :
Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization
scite shows how a scientific paper has been cited by providing the context of the citation, a classification describing whether it supports, mentions, or contrasts the cited claim, and a label indicating in which section the citation was made.
Bibliography
[1] Ates, S., Seismic behaviour of isolated multi-span continuous bridge to nonstationary random seismic excitation. Nonlinear Dyn. 67:1 (2012), 263–282.
[2] Bartlett, M., An Introduction to Stochastic Processes. 1st ed., 1956, Cambridge University Press, Cambridge.
[3] Cai, G.Q., Suzuki, Y., Response of systems under non-Gaussian random excitations. Nonlinear Dyn. 45:1–2 (2006), 95–108.
[4] T. Canor, New perspectives on probabilistic methods for nonlinear transient dynamics in civil engineering (Ph.D. thesis), Université de Liège, Liège, 〈 http://hdl.handle.net/2268/166823〉, 2014.
[5] Canor, T., Blaise, N., Denoël, V., Efficient uncoupled stochastic analysis with non-proportional damping. J. Sound Vib. 331 (2012), 5283–5291.
[6] Canor, T., Blaise, N., Denoël, V., An asymptotic expansion-based method for a spectral approach in equivalent statistical linearization. Probab. Eng. Mech., 2014 〈 http://hdl.handle.net/10.1016/j.probengmech.2014.08.003〉.
[7] Caughey, T., Response of a nonlinear string to random loading. J. Appl. Mech. (ASME) 26 (1959), 341–344.
[8] Caughey, T., Response of van der Pol's oscillator to random loading. J. Appl. Mech. (ASME) 26 (1959), 345–348.
[9] Chen, L., Letchford, C., A deterministic-stochastic hybrid model of downbursts and its impact on a cantilevered structure. Eng. Struct. 26:5 (2004), 619–629.
[10] Chen, L., Letchford, C., Numerical simulation of extreme winds from thunderstorm downbursts. J. Wind Eng. Ind. Aerodyn. 9 (2007), 977–990.
[11] Chen, X., Analysis of along wind tall building response to transient nonstationary winds. J. Struct. Eng. 134:5 (2008), 782–791.
[12] Clough, R.W., Penzien, J., Dynamics of Structures. 2nd ed., 1993, McGraw-Hill, New York.
[13] Conte, J.P., Peng, B.F., An explicit closed-form solution for linear systems subjected to nonstationary random excitation. Probab. Eng. Mech. 11:1 (1996), 37–50.
[14] Cunha, A., The role of the stochastic equivalent linearization method in the analysis of the non-linear seismic response of building structures. Earthq. Eng. Struct. Dyn. 23:8 (1994), 837–857.
[16] Denoël, V., Degée, H., Asymptotic expansion of slightly coupled modal dynamic transfer functions. J. Sound Vib. 328 (2009), 1–8.
[17] Denoël, V., Detournay, E., Multiple scales solution for a beam with a small bending stiffness. J. Eng. Mech. 136:1 (2010), 69–77.
[18] Der Kiureghian, A., A coherency model for spatially varying ground motion. Earthq. Eng. Struct. Dyn. 25:1 (1996), 99–111.
[19] Di Paola, M., Digital simulation of wind field velocity. J. Wind Eng. Ind. Aerodyn. 74–76:0 (1998), 91–109.
[20] Dyrbye, C, Hansen, S, Wind Loads on Structures. 1st ed., 1997, John Wiley and Sons, New York.
[21] Emam, S.A., Nayfeh, A.H., On the nonlinear dynamics of a buckled beam subjected to a primary-resonance excitation. Nonlinear Dyn. 35:1 (2004), 1–17.
[22] Esmaeilzadeh Seylabi, E., Jahankhah, H., Ali Ghannad, M., Equivalent linearization of non-linear soil–structure systems. Earthq. Eng. Struct. Dyn. 41:13 (2012), 1775–1792.
[23] Eurocode1-1991-1-4 (1991) Eurocode 1: Actions on Structures Part 1–4: General Actions: Wind Actions, CEN.
[24] Gani, F., Légeron, F., Relationship between specified ductility and strength demand reduction for single degree-of-freedom systems under extreme wind events. J. Wind Eng. Ind. Aerodyn. 109:0 (2012), 31–45.
[25] Grigoriu, M., Stochastic Calculus. Applications in Science and Engineering. 1st ed., 2002, Birkhauser, Boston.
[26] Guha, T.K., Sharma, R.N., Richards, P.J., Internal pressure in a building with multiple dominant openings in a single wall: comparison with the single opening situation. J. Wind Eng. Ind. Aerodyn. 107–108:0 (2012), 244–255.
[27] Gupta, I.D., Trifunac, M.D., A note on the nonstationarity of seismic response of structures. Eng. Struct. 22:11 (2000), 1567–1577.
[28] Hammond, J.K., On the response of single and multidegree of freedom systems to non-stationary random excitations. J. Sound Vib. 7 (1968), 393–416.
[29] Hammond, J.K., Evolutionary spectra in random vibrations. J. R. Stat. Soc. Ser. B (Methodol.) 35:2 (1973), 167–188.
[30] Harichandran, R.S., Vanmarcke, E.H., Stochastic variation of earthquake ground motion in space and time. J. Eng. Mech. 112:2 (1986), 154–174.
[31] Hespanha, J., Linear Systems Theory. 2009, Princeton University Press, Oxford.
[32] Hinch, E., Perturbation Methods, vol. 1, 1991, Cambridge University Press, Cambridge.
[33] Holmes, J.D., Oliver, S.E., An empirical model of a downburst. Eng. Struct. 22:9 (2000), 1167–1172.
[35] M. Kahan, Approches stochastiques pour le calcul des ponts aux séismes (Ph.D. thesis), Ecole Nationale des Ponts et Chaussées, Paris, 1996.
[36] Kougioumtzoglou, I., Spanos, P., Nonlinear MDOF system stochastic response determination via a dimension reduction approach. Comput. Struct. 126:0 (2013), 135–148.
[37] Kreyszig, E., Advanced Engineering Mathematics. 9th ed., 2006, John Wiley and Sons, Inc., New York.
[38] Lin, J., Zhang, W., Williams, F.W., Pseudo-excitation algorithm for nonstationary random seismic responses. Eng. Struct. 16:4 (1994), 270–276.
[39] Lin, Y., Cai, G., Advanced theory and applications. Probabilistic Structural Dynamics, 2nd ed., 2004, McGraw-Hill, New York.
[40] Luongo, A., Paolone, A., On the reconstitution problem in the multiple time-scale method. Nonlinear Dyn. 19:2 (1999), 135–158.
[41] Morzfeld, M., Ajavakom, N., Ma, F., Diagonal dominance of damping and the decoupling approximation in linear vibratory systems. J. Sound Vib. 320 (2009), 406–420.
[42] Nason, G.P., Rv, Sachs., Kroisandt, G., Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 62:2 (2000), 271–292.
[43] Nayfeh, A.H., Pure and applied mathematics. Perturbation Methods, 1973, J. Wiley & Sons, New York.
[44] Nguyen, H., Manuel, L., Jonkman, J., Veers, P., Simulation of thunderstorm downbursts and associated wind turbine loads. J. Sol. Energy Eng.—Trans. ASME 135:2 (2013), 014–021.
[45] Perotti, F., Structural response to non-stationary multiple-support random excitation. Earthq. Eng. Struct. Dyn. 19 (1990), 513–527.
[46] Pradlwarter, H., Schuëller, G., Schenk, C., A computational procedure to estimate the stochastic dynamic response of large non-linear fe-models. Comput. Methods Appl. Mech. Eng. 192 (2003), 777–801.
[47] Preumont, A., Random Vibration and Spectral Analysis. 1994, Kluwer Academic Publishers, Dordrecht, The Netherlands.
[48] Priestley, M.B., Evolutionary spectra and non-stationary processes. J. R. Stat. Soc. Ser. B (Methodol.) 27:2 (1965), 204–237.
[49] Priestley, M.B., Power spectral analysis of non-stationary random processes. J. Sound Vib. 6 (1967), 86–97.
[51] Rayleigh, J., The Theory of Sound, vol. 1, 1945, Dover Publication, New York.
[52] Roberts, J, Spanos, P, Random Vibration and Statistical Linearization. 1st ed., 1999, Dover Pub. Inc., Mineola, NY.
[53] Rong, H.W., Meng, G., Xu, W., Fang, T., Response statistics of three-degree-of-freedom nonlinear system to narrow-band random excitation. Nonlinear Dyn. 32:1 (2003), 93–107.
[54] Schenk, C.A., Pradlwarter, H.J., Schuëller, G.I., On the dynamic stochastic response of fe models. Probab. Eng. Mech. 19:1–2 (2004), 161–170.
[55] Schenk, C.A., Pradlwarter, H.J., Schuëller, G.I., Non-stationary response of large, non-linear finite element systems under stochastic loading. Comput. Struct. 83:14 (2005), 1086–1102.
[56] Sengupta, A., Haan, F., Sarkar, P., Balaramudu, V., Transient loads on buildings in microburst and tornado winds. J. Wind Eng. Ind. Aerodyn. 96:10–11 (2008), 2173–2187.
[57] Shaska, K., Ibrahim, R.A., Gibson, R.F., Influence of excitation amplitude on the characteristics of nonlinear butyl rubber isolators. Nonlinear Dyn. 47:1–3 (2007), 83–104.
[58] Shinozuka, M., Sato, Y., Simulation of nonstationary random processes. J. Eng. Mech. Div. 93:1 (1967), 11–40.
[59] Shinozuka, M., Deodatis, G., Zhang, R., Papageorgiou, A., Modeling, synthetics and engineering applications of strong earthquake wave motion. Soil Dyn. Earthq. Eng. 18:3 (1999), 209–228.
[60] Smyth, A., Masri, S., Nonstationary response of nonlinear systems using equivalent linearization with a compact analytical form of the excitation process. Probab. Eng. Mech. 17:1 (2002), 97–108.
[61] Solari, G., Piccardo, G., Probabilistic 3-D turbulence modeling for gust buffeting of structures. Probab. Eng. Mech. 16 (2001), 73–86.
[62] Spanos, P.D., Kougioumtzoglou, I.A., Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination. Probab. Eng. Mech. 27:1 (2012), 57–68.
[63] Taflanidis, A.A., Jia, G., A simulation-based framework for risk assessment and probabilistic sensitivity analysis of base-isolated structures. Earthq. Eng. Struct. Dyn. 40:14 (2011), 1629–1651.
[64] To, C., The stochastic central difference method in structural dynamics. Comput. Struct. 23 (1986), 813–818.
[65] To, C., A stochastic version of the newmark family of algorithms for discretized dynamic systems. Comput. Struct. 44:3 (1992), 667–673.
[66] Tootkaboni, M., Graham-Brady, L., Stochastic direct integration schemes for dynamic systems subjected to random excitations. Probab. Eng. Mech. 25:2 (2010), 163–171.
[67] Tubaldi, E., Kougioumtzoglou, I.A., Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation. Earthq. Eng. Struct. Dyn. 44:1 (2015), 121–138.
[68] Vicroy, D.D., Assessment of microburst models for downdraft estimation. J. Airc. 29 (1992), 1043–1048.
[70] G. Wood, K. Kwok, An empirically derived estimate for the mean velocity profile of a thunderstorm downburst, in: Seventh AWES Workshop, Auckland, New Zealand, 1998.
[71] Xu, Y., Samali, B., Kwok, K., Control of along-wind response of structures by mass and liquid dampers. J. Eng. Mech. 118 (1992), 20–39.
[72] Zerva, A., Zerva, V., Spatial variation of seismic ground motions: an overview. Appl. Mech. Rev. (ASME) 55:3 (2002), 271–297.
[73] Zhang, Y., Hu, H., Sarkar, P., Modeling of microburst outflows using impinging jet and cooling source approaches and their comparison. Eng. Struct. 56 (2013), 779–793.
Similar publications
Sorry the service is unavailable at the moment. Please try again later.
This website uses cookies to improve user experience. Read more
Save & Close
Accept all
Decline all
Show detailsHide details
Cookie declaration
About cookies
Strictly necessary
Performance
Strictly necessary cookies allow core website functionality such as user login and account management. The website cannot be used properly without strictly necessary cookies.
This cookie is used by Cookie-Script.com service to remember visitor cookie consent preferences. It is necessary for Cookie-Script.com cookie banner to work properly.
Performance cookies are used to see how visitors use the website, eg. analytics cookies. Those cookies cannot be used to directly identify a certain visitor.
Used to store the attribution information, the referrer initially used to visit the website
Cookies are small text files that are placed on your computer by websites that you visit. Websites use cookies to help users navigate efficiently and perform certain functions. Cookies that are required for the website to operate properly are allowed to be set without your permission. All other cookies need to be approved before they can be set in the browser.
You can change your consent to cookie usage at any time on our Privacy Policy page.
