Skip to main content
SpringerLink
Log in

Stochastic Dynamic Analysis of Large-Scale Nonlinear Structures

  • Conference paper
  • First Online:
Advances in Applied Nonlinear Dynamics, Vibration and Control -2021 (ICANDVC 2021)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 799))

  • 3164 Accesses

Abstract

Stochastic dynamic analysis of structures aims to explore the propagation of uncertainty in dynamic structures, referring to stochastic response and dynamic reliability analyses. For large-scale nonlinear structures, stochastic dynamic analysis is a challenging issue. In this study, a novel direct probability integral method (DPIM) is proposed to synchronously attack the problem of structural stochastic response and dynamic reliability analyses in an efficient and accurate way. The theoretical foundation of DPIM is the probability density integral equation (PDIE), an integral description of probability conservation, which decouples the evolution of probability density from the physical evolution of structure. Firstly, the PDIEs of static and dynamic structures are uniformly derived based on the principle of probability conservation, and then the equivalent differential equations are also highlighted. Then, the formula with Heaviside function for structural reliability estimation is advanced. Moreover, numerical procedures for structural stochastic responses, dynamic reliability and system reliability analyses based on DPIM are demonstrated. Finally, an example of 15-story hysteretic frame building illustrates the high efficiency and accuracy of DPIM for stochastic dynamic analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Goller B, Pradlwarter HJ, Schuëller GI (2013) Reliability assessment in structural dynamics. J Sound Vib 332:2488–2499

    Article  Google Scholar 

  2. Li J, Chen JB (2009) Stochastic dynamics of structures. Wiley, Singapore

    Book  Google Scholar 

  3. Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Courier Corporation, New York

    MATH  Google Scholar 

  4. Stefanou G, Savvas D, Papadrakakis M (2017) Stochastic finite element analysis of composite structures based on mesoscale random fields of material properties. Comput Methods Appl Mech Eng 326:319–337

    Article  MathSciNet  Google Scholar 

  5. Lin JH, Zhang YH, Zhao Y (2011) Pseudo excitation method and some recent developments. Procedia Eng 14:2453–2458

    Article  Google Scholar 

  6. Crandall SH (1963) Perturbation techniques for random vibration of nonlinear systems. J Acoust Soc Am 35:1700–1705

    Article  MathSciNet  Google Scholar 

  7. Roberts JB, Spanos PD (2003) Random vibration and statistical linearization. Dover, Mineola

    MATH  Google Scholar 

  8. Frank TD (2005) Nonlinear Fokker-Planck equations: fundamentals and applications. SSSYN. Springer, London. https://doi.org/10.1007/b137680

  9. To CWS (2005) A statistical nonlinearization technique for multi-degrees of freedom nonlinear systems under white noise excitations. J Sound Vib 286:69–79

    Article  MathSciNet  Google Scholar 

  10. Zhu WQ (1988) Stochastic averaging methods in random vibration. Appl Mech Rev 41:189–199

    Article  Google Scholar 

  11. Zhu WQ (2006) Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl Mech Rev 59:230–248

    Article  Google Scholar 

  12. Pichler L, Masud A, Bergman LA (2013) Numerical solution of the Fokker–Planck equation by finite difference and finite element methods—a comparative study. In: Papadrakakis M, Stefanou G, Papadopoulos V (eds) Computational methods in stochastic dynamics. COMPUTMETHODS, vol 26. Springer, Dordrecht, pp 69−85. https://doi.org/10.1007/978-94-007-5134-7_5

  13. Li KY, Gao W, Wu D, Song CM, Chen TC (2018) Spectral stochastic isogeometric analysis of linear elasticity. Comput Methods Appl Mech Eng 332:157–190

    Article  MathSciNet  Google Scholar 

  14. Wang WP, Chen GH, Yang DX, Kang Z (2019) Stochastic isogeometric analysis method for plate structures with random uncertainty. Comput Aided Geom Des 74:101772

    Article  MathSciNet  Google Scholar 

  15. Melchers RE, Beck AT (2018) Structural reliability analysis and prediction, 3rd edn. Wiley , London

    Google Scholar 

  16. Zhao YG, Ono T (2001) Moment methods for structural reliability. Struct Saf 23:47–75

    Article  Google Scholar 

  17. Wang C, Zhang H, Li QW (2019) Moment-based evaluation of structural reliability. Reliab Eng Syst Saf 181:38–45

    Article  Google Scholar 

  18. Goswami S, Ghosh S, Chakraborty S (2016) Reliability analysis of structures by iterative improved response surface method. Struct Saf 60:56–66

    Article  Google Scholar 

  19. Kim JH, Song JH (2020) Probability-adaptive kriging in n-Ball (PAK-Bn) for reliability analysis. Struct Saf 85:101924

    Article  Google Scholar 

  20. Hariri-Ardebili MA, Pourkamali-Anaraki F (2018) Support vector machine based reliability analysis of concrete dams. Soil Dyn Earthq Eng 104:276–295

    Article  Google Scholar 

  21. Song JH, Der Kiureghian A (2006) Joint first-passage probability and reliability of systems under stochastic excitation. J Eng Mech 132:65–77

    Article  Google Scholar 

  22. Vanvinckenroye H, Kougioumtzoglou IA, Denoël V (2019) Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique. Nonlinear Dyn 95:293–308

    Article  Google Scholar 

  23. Au SK, Beck JL (2001) First excursion probabilities for linear systems by very efficient importance sampling. Probab Eng Mech 16:193–207

    Article  Google Scholar 

  24. Ching J, Au SK, Beck JL (2005) Reliability estimation for dynamical systems subject to stochastic excitation using subset simulation with splitting. Comput Methods Appl Mech Eng 194:1557–1579

    Article  MathSciNet  Google Scholar 

  25. Valdebenito MA, Jensen HA, Labarca AA (2014) Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load. Comput Struct 138:36–48

    Article  Google Scholar 

  26. Li J (2016) Probability density evolution method: Background, significance and recent developments. Probab Eng Mech 44:111–117

    Article  Google Scholar 

  27. Chen JB, Li J (2007) The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Struct Saf 29:77–93

    Article  Google Scholar 

  28. Papadopoulos V, Kalogeris I (2016) A Galerkin-based formulation of the probability density evolution method for general stochastic finite element systems. Comput Mech 57(5):701–716. https://doi.org/10.1007/s00466-015-1256-9

    Article  MathSciNet  MATH  Google Scholar 

  29. Chen GH, Yang DX (2019) Direct probability integral method for stochastic response analysis of static and dynamic structural systems. Comput Methods Appl Mech Eng 357:112612

    Article  MathSciNet  Google Scholar 

  30. Lin YK (1976) Probabilistic theory of structural dynamics. Krieger Publishing Company, Malabar

    Google Scholar 

  31. Zhu WQ, Cai GQ (2017) Introduction to stochastic dynamics. Science Press, Beijing

    Google Scholar 

  32. Au C, Tam J (1999) Transforming variables using the Dirac generalized function. Am Stat 53:270–272

    MathSciNet  Google Scholar 

  33. Hoskins RF (2009) Delta functions: introduction to generalised functions. Elsevier, London

    Book  Google Scholar 

  34. Dick J, Kuo FY (2013) High-dimensional integration: the quasi-Monte Carlo way. Acta Numerica 22:133–288

    Article  MathSciNet  Google Scholar 

  35. Chen JB, Yang JY, Li J (2016) A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters. Struct Saf 59:20–31

    Article  Google Scholar 

  36. Liu ZJ, Liu W, Peng YB (2016) Random function based spectral representation of stationary and non-stationary stochastic processes. Probab Eng Mech 45:115–126

    Article  Google Scholar 

Download references

Acknowledgments

The supports of the National Natural Science Foundation of China (Grant Nos. 12032008, 11772079), and the China Postdoctoral Science Foundation (Grant No. 2019M661088) are much appreciated.

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Dalian University of Technology, Dalian, China

    Dixiong Yang & Guohai Chen

  2. School of Civil Engineering, Dalian University of Technology, Dalian, China

    Guohai Chen

Authors
  1. Dixiong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guohai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guohai Chen .

Editor information

Editors and Affiliations

  1. Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Xingjian Jing

  2. Shanghai University, Shanghai, Shanghai, China

    Hu Ding

  3. Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

    Jiqiang Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Yang, D., Chen, G. (2022). Stochastic Dynamic Analysis of Large-Scale Nonlinear Structures. In: Jing, X., Ding, H., Wang, J. (eds) Advances in Applied Nonlinear Dynamics, Vibration and Control -2021. ICANDVC 2021. Lecture Notes in Electrical Engineering, vol 799. Springer, Singapore. https://doi.org/10.1007/978-981-16-5912-6_63

Download citation

  • DOI: https://doi.org/10.1007/978-981-16-5912-6_63

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-16-5911-9

  • Online ISBN: 978-981-16-5912-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation