Skip to main content
SpringerLink
Log in

A new computational scheme for structural static stochastic analysis based on Karhunen–Loève expansion and modified perturbation stochastic finite element method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Due to uncertainties, deterministic analysis cannot sufficiently reflect the performance of structures. Stochastic analysis can consider the influence of multiple uncertainties factors and improve the confidence of the analysis results. A new stochastic computational scheme, which has the features of Karhunen–Loève (K–L) expansion and modified perturbation stochastic finite element method (MPSFEM), is proposed for the structures with low-level uncertainties, called KL-MPSM for short. The material parameters are regarded as random fields and discretized by K–L expansion. The random variables obtained are substituted into MPSFEM to get the estimates of the first two order moments (mean and variance) of the structural responses. JC method is introduced to compute the reliability indexes and structures failure probability by utilizing the second-order estimates. A deep beam and a plane frame structure are presented as numerical examples to demonstrate the feasibility of KL-MPSM, and some random filed properties are studied. The results show that KL-MPSM has good accuracy, efficiency, and advantages in programming. Therefore, KL-MPSM is well suited for static stochastic analysis of structures with low-level uncertainties.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Liu WK, Belytschko T, Mani A (1986) Random field finite elements. Int J Numer Methods Eng 23(10):1831–1845

  2. Zhou H, Li J (2020) Energy-based collapse assessment of concrete structures subjected to random damage evolutions. Probab Eng Mech 60:103019

    Google Scholar 

  3. Wriggers P, Moftah SO (2006) Mesoscale models for concrete: Homogenisation and damage behaviour. Finite Elem Anal Des 42(7):623–636

    Google Scholar 

  4. Jiang L, Liu X, Xiang P, Zhou W (2019) Train-bridge system dynamics analysis with uncertain parameters based on new point estimate method. Eng Struct 199:109454

    Google Scholar 

  5. Liu X, Xiang P, Jiang L, Lai Z, Zhou T, Chen Y (2020) Stochastic analysis of train-bridge system using the Karhunen–Loève expansion and the point estimate method. Int J Struct Stab Dyn 20(02):2050025

    MathSciNet  Google Scholar 

  6. Wang T, Zhou G, Wang J, Zhao X (2016) Stochastic analysis for the uncertain temperature field of tunnel in cold regions. Tumn Undergr Sp Tech 59:7–15

    Google Scholar 

  7. Li Z, Pasternak H (2019) Experimental and numerical investigations of statistical size effect in s235jr steel structural elements. Constr Build Mater 206:665–673

    Google Scholar 

  8. Wang X, Li Z, Wang H, Rong Q, Liang RY (2016) Probabilistic analysis of shield-driven tunnel in multiple strata considering stratigraphic uncertainty. Struct Saf 62:88–100

    Google Scholar 

  9. Wriggers P, Allix O, Weißenfels C (2000) Virtual design and validation. Springer, Berlin

    Google Scholar 

  10. Zein S, Laurent A, Dumas D (2019) Simulation of a gaussian random field over a 3d surface for the uncertainty quantification in the composite structures. Comput Mech 63:1083–1090

    MathSciNet  MATH  Google Scholar 

  11. Rauter N (2021) A computational modeling approach based on random fields for short fiber-reinforced composites with experimental verification by nanoindentation and tensile tests. Comput Mech 67(2):699–722

    MathSciNet  MATH  Google Scholar 

  12. Zakian P (2021) Stochastic finite cell method for structural mechanics. Comput Mech 68(1):185–210

    MathSciNet  MATH  Google Scholar 

  13. Der Kiureghian A, Ke J-B (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3(2):83–91

    Google Scholar 

  14. Vanmarcke EMSGI, Shinozuka M, Nakagiri S, Schueller GI, Grigoriu M (1986) Random fields and stochastic finite elements. Struct Saf 3(3–4):143–166

    Google Scholar 

  15. Li C-C, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech-ASCE 119(6):1136–1154

    Google Scholar 

  16. Zhang J, Ellingwood B (1994) Orthogonal series expansions of random fields in reliability analysis. J Eng Mech-ASCE 120(12):2660–2677

    Google Scholar 

  17. Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Courier Corporation, North Chelmsford

    MATH  Google Scholar 

  18. Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability: a state-of-the-art report. Department of Civil and Environmental Engineering, University of California, Berkeley

    Google Scholar 

  19. Zheng Z, Dai H (2017) Simulation of multi-dimensional random fields by Karhunen–Loève expansion. Comput Methods Appl Mech 324:221–247

    MathSciNet  MATH  Google Scholar 

  20. Phoon KK, Huang SP, Quek ST (2002) Implementation of Karhunen–Loève expansion for simulation using a wavelet-Galerkin scheme. Probab Eng Mech 17(3):293–303

    Google Scholar 

  21. Kim H, Shields MD (2015) Modeling strongly non-gaussian non-stationary stochastic processes using the iterative translation approximation method and Karhunen–Loève expansion. Comput Struct 161:31–42

    Google Scholar 

  22. Tong M-N, Zhao Y-G, Zhao Z (2021) Simulating strongly non-gaussian and non-stationary processes using Karhunen–Loève expansion and l-moments-based hermite polynomial model. Mech Syst Signal Process 160:107953

    Google Scholar 

  23. Zhang D, Zhiming L (2004) An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions. J Comput Phys 194(2):773–794

    MATH  Google Scholar 

  24. Montoya-Noguera S, Zhao T, Yue H, Wang Yu, Phoon K-K (2019) Simulation of non-stationary non-Gaussian random fields from sparse measurements using Bayesian compressive sampling and Karhunen–Loève expansion. Struct Saf 79:66–79

    Google Scholar 

  25. Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230(6):2345–2367

    MathSciNet  MATH  Google Scholar 

  26. Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2):293–309

    MathSciNet  MATH  Google Scholar 

  27. Kamiński M (2007) Generalized perturbation-based stochastic finite element method in elastostatics. Comput Struct 85(10):586–594

    Google Scholar 

  28. Kamiński M (2010) Potential problems with random parameters by the generalized perturbation-based stochastic finite element method. Comput Struct 88(7–8):437–445

    Google Scholar 

  29. Kamiński M (2009) Perturbation-based stochastic finite element method using polynomial response function for the elastic beams. Mech Res Commun 36(3):381–390

    MathSciNet  MATH  Google Scholar 

  30. Kamiński M, Carey GF (2005) Stochastic perturbation-based finite element approach to fluid flow problems. Int J Numer Methods Heat Fluid Flow 15(7):671–697

    MathSciNet  MATH  Google Scholar 

  31. Kamiński M, Solecka M (2013) Optimization of the truss-type structures using the generalized perturbation-based stochastic finite element method. Finite Elem Anal Des 63:69–79

    MathSciNet  MATH  Google Scholar 

  32. Kamiński M, Kazimierczak M (2014) 2D versus 3D probabilistic homogenization of the metallic fiber-reinforced composites by the perturbation-based stochastic finite element method. Compos Struct 108:1009–1018

    Google Scholar 

  33. Kamiński M, Hien TD (1999) Stochastic finite element modeling of transient heat transfer in layered composites. Int Commun Heat Mass 26(6):801–810

    Google Scholar 

  34. Kamiński M (2008) On stochastic finite element method for linear elastostatics by the taylor expansion. Struct Multidiscip Optim 35(3):213–223

    Google Scholar 

  35. Feng W, Gao Q, Xiao-Ming X, Zhong W-X (2015) A modified computational scheme for the stochastic perturbation finite element method. Lat Am J Solids Struct 12:2480–2505

    Google Scholar 

  36. Wu F, Zhong WX (2016) A modified stochastic perturbation method for stochastic hyperbolic heat conduction problems. Comput Methods Appl Mech Eng 305:739–758

    MathSciNet  MATH  Google Scholar 

  37. Liu X, Zhang Y, Li X, Xiang P, Xie S (2022) Running stability analysis for express wagon with modified stochastic perturbation method. In: Second international conference on rail transportation

  38. Liu X, Xiang P, Jiang L, Lai Z, Zhou T, Chen Y (2020) Stochastic analysis of train-bridge system using the Karhunen-Loéve expansion and the point estimate method. Int J Struct Stab Dyn 20(02):2050025

    MathSciNet  Google Scholar 

  39. Grigoriu M (2010) Probabilistic models for stochastic elliptic partial differential equations. J Comput Phys 229(22):8406–8429

    MathSciNet  MATH  Google Scholar 

  40. Rauter N (2021) A computational modeling approach based on random fields for short fiber-reinforced composites with experimental verification by nanoindentation and tensile tests. Comput Mech 67(2):699–722

    MathSciNet  MATH  Google Scholar 

  41. Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech 198(9–12):1031–1051

    MATH  Google Scholar 

  42. Liu Z, Yang M, Cheng J, Tan J (2021) A new stochastic isogeometric analysis method based on reduced basis vectors for engineering structures with random field uncertainties. Appl Math Model 89:966–990

    MathSciNet  MATH  Google Scholar 

  43. Trinh M-C, Mukhopadhyay T (2021) Semi-analytical atomic-level uncertainty quantification for the elastic properties of 2d materials. Mater Today Nano 15:100126

    Google Scholar 

  44. Rackwitz R, Flessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494

    MATH  Google Scholar 

  45. Choi KK, Kim N-H (2004) Structural sensitivity analysis and optimization 2: nonlinear systems and applications. Springer, Berlin

    Google Scholar 

  46. Hoang V-L, Dang HN, Jaspart J-P, Demonceau J-F (2015) An overview of the plastic-hinge analysis of 3d steel frames. Asia Pac J Comput Eng 2(1):1–34

    Google Scholar 

  47. Olsson A, Sandberg G, Dahlblom O (2003) On latin hypercube sampling for structural reliability analysis. Struct Saf 25(1):47–68

    Google Scholar 

  48. Soize C (2006) Non-gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput Methods Appl Mech 195(1–3):26–64

    MathSciNet  MATH  Google Scholar 

  49. Grigoriu M (2016) Microstructure models and material response by extreme value theory. SIAM-ASA J Uncertain Quantif 4(1):190–217

    MathSciNet  MATH  Google Scholar 

  50. Staber B, Guilleminot J (2017) Stochastic modeling and generation of random fields of elasticity tensors: a unified information-theoretic approach. CR Mecanique 345(6):399–416

    Google Scholar 

  51. Malyarenko A, Ostoja-Starzewski M (2017) A random field formulation of Hooke’s law in all elasticity classes. J Elast 127(2):269–302

    MathSciNet  MATH  Google Scholar 

  52. Guilleminot J (2020) Modeling non-gaussian random fields of material properties in multiscale mechanics of materials. In: Uncertainty quantification in multiscale materials modeling. Elsevier, pp 385–420

Download references

Acknowledgements

This work was funded by the National Natural Science Foundation of China (Grant No. 11972379), the Key R &D Program of Hunan Province (2020SK2060), Hunan Science Fund for Distinguished Young Scholars (2021JJ10061), and Open fund of State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410075, Hunan, China

    Zhanjun Shao & Ping Xiang

  2. School of Civil Engineering and Architecture, Guangxi University, Nanning, 530004, Guangxi, China

    Zhanjun Shao & Xiumei Li

  3. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, China

    Ping Xiang

Authors
  1. Zhanjun Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiumei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Xiang.

Ethics declarations

Conflict of interest

The authors declare they have no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shao, Z., Li, X. & Xiang, P. A new computational scheme for structural static stochastic analysis based on Karhunen–Loève expansion and modified perturbation stochastic finite element method. Comput Mech 71, 917–933 (2023). https://doi.org/10.1007/s00466-022-02259-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-022-02259-7

Keywords

Search

Navigation