Skip to main content
Log in

An efficient coupling of FORM and Karhunen–Loève series expansion

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

The topic of this paper is the solution of reliability problems where failure is influenced by the spatial random fluctuations of loads and material properties. Homogeneous random fields are used to model this kind of uncertainty. The first step of the investigation is the random field discretization, which transforms a random field into a finite set of random variables. The second step is the reliability analysis, which is performed using the FORM in this paper. A parametric analysis of the reliability index is usually performed with respect to the random field discretization accuracy. This approach requires several independent reliability analyses. A new and efficient approach is proposed in this paper. The Karhunen–Loève series expansion is combined with the FEM for the discretization of the random fields. An efficient solution of the reliability problem is proposed to predict the reliability index as the discretization accuracy increases.

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

Access this article

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Lemaire M (2009) Structural reliability. Wiley, New York

    Book  Google Scholar 

  2. Liu PL, Der Kiureghian A (1991) Optimization algorithms for structural reliability. Struct Saf 9:161–177

    Article  Google Scholar 

  3. Allaix DL, Carbone VI (2011) An improvement of the response surface method. Struct Saf 33:165–172

    Article  Google Scholar 

  4. Allaix DL, Carbone VI, Mancini G (2011) Advanced response surface method for structural reliability. Paper presented at the 13th International Conference on Civil, Structural and Environmental Engineering Computing, Chania, Greece

  5. Basaga B, Bayraktar A, Kaymaz I (2012) An improved response surface method for reliability analysis of structures. Struct Eng Mech 42(2):175–189

    Article  Google Scholar 

  6. Bucher CG, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7:57–66

    Article  Google Scholar 

  7. Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19:3–19

    Article  Google Scholar 

  8. Rajashekhar MR, Ellingwood BR (1993) A new look at the response surface approach for reliability analysis. Struct Saf 12:205–220

    Article  Google Scholar 

  9. Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div ASCE 100:111–121

    Google Scholar 

  10. Li Y, Vrouwenvelder T, Wijnants GH, Walraven J (2004) Spatial variability of concrete deterioration and repair strategies. Struct Concr 3:121–129

    Article  Google Scholar 

  11. Defaux G, Pendola M, Sudret B (2006) Using spatial reliability in the probabilistic study of concrete structures: the example of a RC beam subject to carbonation inducing corrosion. J Phys 136:243–253

    Google Scholar 

  12. Stewart MG, Mullard JA (2007) Spatial time-dependent reliability analysis of corrosion damage and the timing of first repair for RC structures. Eng Struct 29:1457–1464

    Article  Google Scholar 

  13. Allaix DL, Carbone VI, Mancini G (2011) Random fields for the modeling of deteriorated structures. Paper presented at the 11th International Conference on Applications of Statistics and Probability in Civil Engineering, Zurich, Switzerland

  14. Vanmarcke E (1983) Random fields, analysis and synthesis. The MIT Press, Cambridge

    MATH  Google Scholar 

  15. Brenner C, Bucher C (1995) A contribution to the SFE-based reliability assessment of non linear structures under dynamic loading. Probab Eng Mech 107:449–463

    Google Scholar 

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

    Article  Google Scholar 

  17. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York

    Book  MATH  Google Scholar 

  18. Li CC, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech 119:1136–1154

    Article  Google Scholar 

  19. Mahadevan S, Haldar A (1991) Practical random field discretization in stochastic finite element analysis. Struct Saf 9:283–304

    Article  Google Scholar 

  20. Matthies HG, Brenner CE, Bucher CG, Guedes Soares C (1997) Uncertainties in probabilistic analysis of structures and solids-stochastic finite elements. Struct Saf 19:283–336

    Article  Google Scholar 

  21. Sudret B, Der Kiureghian A (2000) Stochastic finite elements and reliability: a state-of-the-art report. Technical Report UCB/SEMM-2000/08, University of California, Berkeley, USA

  22. Vanmarcke E, Grigoriu M (1983) Stochastic finite element analysis of simple beams. J Eng Mech 109:1203–1214

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Allaix DL, Carbone VI (2008) Adaptive discretization of 1D homogeneous random fields. Paper presented at the Joint ESREL and SRA-Europe Conference, Valencia, Spain

  25. Allaix DL, Carbone VI (2010) Numerical discretization of stationary random processes. Probab Eng Mech 25:332–347

    Article  Google Scholar 

  26. Liu PL, Liu KG (1993) Selection of random field mesh in finite element reliability analysis. J Eng Mech 119:667–679

    Article  Google Scholar 

  27. Maymon G (1994) Direct computation of the design point of a stochastic structure using a finite element code. Struct Saf 14:185–202

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Ngah MF, Young A (2007) Application of the spectral stochastic finite element method for performance prediction of composite structures. Compos Struct 78:447–456

    Article  Google Scholar 

  30. Sudret B, Der Kiureghian A (2002) Comparison of finite element reliability methods. Probab Eng Mech 17:337–348

    Article  Google Scholar 

  31. Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Method Appl Mech Eng 198:1031–1051

    Article  MATH  Google Scholar 

  32. Huang SP, Quek ST, Phoon KK (2001) Convergence of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. Int J Numer Methods Eng 52:1029–1043

    Article  MATH  Google Scholar 

  33. Li LB, Phoon KK, Quek ST (2007) Comparison between Karhunen–Loève expansion and translation-based simulation of non-Gaussian processes. Comput Struct 85:264–276

    Article  MathSciNet  Google Scholar 

  34. Phoon KK, Huang HW, Quek ST (2005) Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion. Probab Eng Mech 20:188–198

    Article  Google Scholar 

  35. Betz W, Papaioannou I, Straub D (2014) Numerical methods for the discretization of random fields by means of the Karhunen–Loeve expansion. Comput Methods Appl Mech Eng 271:109–129

    Article  MathSciNet  MATH  Google Scholar 

  36. Phoon KK, Huang SP, Quek ST (2002) Simulation of second order processes using Karhunen–Loève expansion. Comput Struct 80:1049–1060

    Article  MathSciNet  Google Scholar 

  37. Allaix DL, Carbone VI (2012) Development of a numerical tool for random field discretization. Adv Eng Softw 51:10–19

    Article  Google Scholar 

  38. Haldar A, Mahadevan S (2000) Probability, reliability, and statistical methods in engineering design. Wiley, New York

    Google Scholar 

  39. Koduru SD, Haukaas T (2010) Feasibility of FORM in finite element reliability analysis. Struct Saf 32:145–153

    Article  Google Scholar 

  40. Rackwitz R (2001) Reliability analysis—a review and some perspectives. Struct Saf 23:365–395

    Article  Google Scholar 

  41. Valdebenito MA, Pradlwarter HJ, Schuëller GI (2010) The role of the design point for calculating failure probabilities in view of dimensionality and structural nonlinearities. Struct Saf 51:101–111

    Article  Google Scholar 

  42. Waarts PH, Vrouwenvelder ACWM (1999) Stochastic finite analysis of steel structures. J Constr Steel Res 52:21–32

    Article  Google Scholar 

  43. Haukaas T, Scott MH (2006) Shape sensitivities in the reliability analysis of non-linear frame structures. Comput Struct 84:964–977

    Article  Google Scholar 

  44. Charmpis DC, Schuëller GI, Pellissetti MF (2007) The need for linking micromechanics of materials with stochastic finite elements: a challenge for materials science. Comput Mater Sci 41:27–37

    Article  Google Scholar 

  45. Deodatis G (1991) Weighted integral method. I: stochastic stiffness matrix. J Eng Mech 117:1851–1884

    Article  Google Scholar 

  46. Chowdhury RN, Xu DW (1992) Reliability index for slope stability assessment—two methods compared. Reliab Eng Syst Saf 37:99–108

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Diego Lorenzo Allaix.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allaix, D.L., Carbone, V.I. An efficient coupling of FORM and Karhunen–Loève series expansion. Engineering with Computers 32, 1–13 (2016). https://doi.org/10.1007/s00366-015-0394-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-015-0394-1

Keywords

Navigation