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Simulation in Time-Invariant and Time-Variant Reliability Problems

  • R. E. Melchers
Part of the Lecture Notes in Engineering book series (LNENG, volume 76)

Abstract

As a class of techniques useful in the solution of structural reliability problem formulations, simulation methods have attracted considerable interest as viable alternatives to numerical integration or First Order Second Moment or First Order Reliability (FOSM/FOR) methods. Further, the various methods which have been proposed owe a considerable debt to the better understanding of the mathematical nature of structural reliability problems engendered by FOSM and FOR methods. This theme is explored herein, as part of an overview of simulation methods for so-called “time-invariant” and “time-variant” structural reliability problems.

The strategies which are reviewed include importance sampling, adaptive, or search-based sampling, multiple limit state problems, and conditional expectation in rectangular co-ordinate systems. Directional sampling in the hyper-polar co-ordinate space also is described briefly, together with its extension for determination of outcrossing rates for continuous and discrete processes. Throughout the emphasis is on the ideas inherent in the procedures and their possible development rather than on a detailed historical perspective.

Keywords

Failure Probability Sampling Density Importance Sampling Directional Simulation Limit State Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    ANG, G.L., ANG, A. H.—S. and TANG, W.H., (1989), Kernel Method in Importance Sampling Density Estimation, (in) Proceedings 5th International Conference on Structural Safety and Reliability, San Francisco, (Ed.) Ang, A. H.—S., Shinozuka, M. and Schuëller, G.I., ASCE, 1193–1200.Google Scholar
  2. 2.
    ANG, G.L., ANG, A. H.—S. and TANG, W.H., (1991), Multi—Dimensional Kernel Method in Importance Sampling, (in) Proceedings 6th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Mexico City, (Ed.) Esteva, L. and Ruiz, S.E., CERRA, 289–295.Google Scholar
  3. 3.
    AUGUST, G., BARATTA, A. and CASCIATI, F., (1984), Probabilistic Methods in Structural Engineering, Chapman Hall, London.CrossRefGoogle Scholar
  4. 4.
    AYYUB, B.M. and CHIA, C.Y., (1984), Generalised Conditional Expectation for Structural Reliability Assessment, Structural Safety, (to appear).Google Scholar
  5. 5.
    AYYUB, B.M. and HALDER, A., (1984), Practical Structural Reliability Techniques, Journal of Structural Engineering, ASCE, 110 (8), 1707–1724.CrossRefGoogle Scholar
  6. 6.
    AYYUB, B.M. and LAI, K.L., (1989), Structural Reliability Assessment Using Latin Hyper—Cube Sampling, (in) Proceedings 5th International Conference on Structural Safety and Reliability, San Francisco, (Ed.) Ang, A. H.—S., Shinozuka, M. and Schuëller, G.I., ASCE, 1177–1184.Google Scholar
  7. 7.
    AYYUB, B.M. and LAI, K.L., (1989), Selective Sampling in Simulation—Based Reliability Assessment, Int. Jnl. Pres. Ves. and Piping, (to appear).Google Scholar
  8. 8.
    BJERAGER, P., (1988), Probability Integration by Directional Simulation, Journal of Engineering Mechanics, ASCE, 114 (8), 1285–1302.Google Scholar
  9. 9.
    BJERAGER, P., (1990), On Computational Methods for Structural Reliability Analysis, Structural Safety, 9 (2), 79–96.CrossRefGoogle Scholar
  10. 10.
    BJERAGER, P. and KRENK, S., (1989), Parametric Sensitivity in First Order Reliability Analysis, Journal of Engineering Mechanics, ASCE, 115 (7), 1577–1582.Google Scholar
  11. 11.
    BOLOTIN, V.V., (1981), Wahrscheinlichkeitsmethoden zur Berechnung von Konstruktionen, VEB Verlag fuer Bauwesen, Berlin, DDR.Google Scholar
  12. 12.
    BREITUNG, K., (1990), Probability Approximations by Log Likelihood Maximisation, Journal of Engineering Mechanics, ASCE, 117 (3), 457–477.Google Scholar
  13. 13.
    BREITUNG, K. and RACKWITZ, R., (1982), Nonlinear Combination of Load Processes, Journal of Structural Mechanics, 10 (2), 145–166.CrossRefGoogle Scholar
  14. 14.
    BUCHER, C.G., (1988), Adaptive Sampling — An Iterative Fast Monte Carlo Procedure, Structural Safety, 5 (2), 119–126.CrossRefGoogle Scholar
  15. 15.
    BUCHER, C.G., CHEN, Y.M. and SCHUELLER, G.I., (1988), Time Variant Reliability Analysis Utilising Response Surface Approach, Proceedings 2nd IFIP Conference on Reliability and Optimisation of Structural Systems, (Ed.) Thoft-Christensen, P., Springer Verlag, 1–14.Google Scholar
  16. 16.
    CHAN, H.Y., (1991), System Reliability Analysis With Time-Dependent Loads and Resistances, PhD Thesis, The University of Newcastle, Australia, March.Google Scholar
  17. 17.
    CHEN, Y.M., (1989), Reliability of Structural Systems Subjected to Time Variant Loads, Z.angew. Math. Mech., 69, T64 - T66.Google Scholar
  18. 18.
    CHEN, Y.M., SCHUELLER, G.I. and BOURGUND, U., (1988), Reliability of Large Structural Systems Under Time Varying Loads, (in) Proceedings 5th ASCE Speciality Conference on Probabilistic Methods in Civil Engineering, ( Ed.) Spanos, P.D., ASCE, 420–423.Google Scholar
  19. 19.
    COROTIS, R.B. and NAFDAY, A.M., (1989), Structural Systems Reliability Using Linear Programming and Simulation, Journal of Structural Engineering, ASCE, 115 (10), 2435–2447.CrossRefGoogle Scholar
  20. 20.
    CSENKI, A., (1989), An Improved Monte Carlo Method in Structural Reliability, Reliability Engineering and System Safety, 24, 275–292.CrossRefGoogle Scholar
  21. 21.
    DEAK, I., (1980), Three Digit Accurate Multiple Normal Probabilities, Numerische Mathematik, 35, 369–380.CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    DITLEVSEN, O., (1983), Gaussian Outcrossings from Safe Convex Polyhedrons, Journal of Engineering Mechanics, ASCE, 109 (1), 127–148.Google Scholar
  23. 23.
    DITLEVSEN, O. and BJERAGER, P., (1989), Plastic Reliability Analysis by Directional Simulation, Journal of Engineering Mechanics, ASCE, 115 (6), 1347–1362.Google Scholar
  24. 24.
    DITLEVSEN, O., HASOFER, A.M., BJERAGER, P. and OLESEN, R., (1987), Directional Simulation in Gaussian Processes, DCAMM Report No. 359, Technical University of Denmark, (See also Prob. Engg. Mech., 3 (4), 207–217, 1988).Google Scholar
  25. 25.
    DITLEVSEN, O., MELCHERS, R.E. and GLUVER, H., (1990), General Multi-Dimensional Probability Integration by Directional Simulation, Computers and Structures, 36 (2), 355–368.CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    DITLEVSEN, O., OLESEN, R. and HASOFER, A.M., (1985), Load Combination by Deâk Simulation, Proceedings SMiRT8, Brussels.Google Scholar
  27. 27.
    DITLEVSEN, O., OLESEN, R. and MOHR, G., (1987), Solution of a Class of Load Combination Problems by a Directional Simulation, Structural Safety, 4 (2), 95–109.CrossRefGoogle Scholar
  28. 28.
    DUPOIS, D.J. and MAES, M.A., (1991), A Computationally Effective Self-Adjusting Technique for Determining Failure Probabilities, (in) Proceedings 6th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Mexico City, (Ed.) Esteva, L. and Ruiz, S.E., CERRA, 240–249.Google Scholar
  29. 29.
    FU, G. and MOSES, F., (1987), A Sampling Distribution for System Reliability Assessment, Proceedings 1st IFIP Working Conference on Reliability and Optimisation of Structural Systems, Aalborg, Denmark, ( Ed.) Thoft—Christensen, P., 141–160.Google Scholar
  30. 30.
    FUJITA, M., SCHALL, G. and RACKWITZ, R., (1987), Time—Variant Component Reliabilities by FORM—SORM and Updating by Importance Sampling, (in) Lind, N.C., (Ed.), Proceedings 5th International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Vancouver, Canada, 520–527.Google Scholar
  31. 31.
    HARBITZ, A., (1983), Efficient and Accurate Probability of Failure Calculation By Use of the Importance Sampling Technique, (in) Proceedings 4th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Florence, (Ed.) Augusti, A., Boni, A. and Vannucchi, G., Pitagara, Editrice, Bologna, 825–836.Google Scholar
  32. 32.
    HARBITZ, A., (1986), An Efficient Sampling Method for Probability of Failure Calculation, Structural Safety, 3 (2), 109–115.CrossRefGoogle Scholar
  33. 33.
    HASOFER, A.M., DITLEVSEN, O. and OLESEN, R., (1987), Vector Outcrossing Probabilities by Monte Carlo, DCAMM Report No. 349, Technical University of DenmarkGoogle Scholar
  34. 34.
    HOHENBICHLER, M. and RACKWITZ, R., (1986), Asymptotic Crossing Rate of Gaussian Vector Processes Into Intersection of Failure Domains, Prob. Engrg. Mech., 1 (3), 177–179.CrossRefGoogle Scholar
  35. 35.
    HOHENBICHLER, M. and RACKWITZ, R., (1986), Sensitivities and Importance Measures in Structural Reliability, Civil Engineering Systems, 3 (4), 203–209.CrossRefGoogle Scholar
  36. 36.
    HOHENBICHLER, M. and RACKWITZ, R., (1988), Improvement of Second—Order Reliability Estimates by Importance Sampling, Journal of Engineering Mechanics, ASCE, 114 (12), 2195–2199.Google Scholar
  37. 37.
    IBRAHIM, Y. and RAHMAN, S., (1991), Reliability Analysis of Uncertain Dynamic Systems Using Importance Sampling, (in) Proceedings 6th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Mexico City, (Ed.) Esteva, L. and Ruiz, S.E., CERRA, 305–312, (See also: Ibrahim, Y., (1991), Observations on Applications of Importance Sampling in Structural Reliability Analysis, Structural Safety, 9(4), 269–282).Google Scholar
  38. 38.
    KARAMCHANDANI, A., (1990), New Methods in Systems Reliability, PhD Thesis, Stanford University.Google Scholar
  39. 39.
    KARAMCHANDANI, A., BJERAGER, P. and CORNELL, C.A., (1989), Adaptive Importance Sampling, (in) Proceedings 5th International Conference on Structural Safety and Reliability, San Francisco, (Ed.) Ang, A. H.—S., Shinozuka, M. and Schuëller, G.I., ASCE, 855–862.Google Scholar
  40. 40.
    KARAMCHANDANI, A. and CORNELL, A.C., (1989), Sensitivity of Failure Probability Estimates That Are Obtained Using Simulation, Structural Safety, (to appear).Google Scholar
  41. 41.
    KHAN, H., (1956), Use of Different Monte Carlo Sampling Techniques, (in) Meyer, H.A., (Ed.), Proceedings of Symposium on Monte Carlo Methods, John Wiley and Sons, New York, 149–190.Google Scholar
  42. 42.
    LEIRA, B.J., (1991), Importance Sampling Distributions for Gaussian Vector Processes, (in) Proceedings 6th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Mexico City, (Ed.) Esteva, L. and Ruiz, S.E., CERRA, 297–304.Google Scholar
  43. 43.
    LIN, H.Z. and DER KIUREGHIAN, A., (1987), Second-Order System Reliability Using Directional Simulation, (in) Proceedings 5th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Vancouver, (Ed.) Lind, N.C., Institute for Risk Research, University of Waterloo, Canada.Google Scholar
  44. 44.
    LIN, T.S. and COROTIS, R.B., (1985), Reliability of Ductile Systems With Random Strengths, Journal of Structural Engineering, ASCE, 111 (6), 1306–1325.CrossRefGoogle Scholar
  45. 45.
    MADSEN, H.O., (1988), Omission Sensitivity Factors, Structural Safety, 5 (1), 35–46.CrossRefGoogle Scholar
  46. 46.
    MADSEN, H.O., KRENK, S. and LIND, N.C., (1986), Methods of Structural Safety, Prentice Hall Inc., Englewood Cliffs, New Jersey.Google Scholar
  47. 47.
    MADSEN, H.O. and ZADEH, M., (1987), Reliability of Plates Under Combined Loading, Proc. Marine Struct. Rel. Symp., SNAME, Arlington, Virginia, 185–191.Google Scholar
  48. 48.
    MELCHERS, R.E., (1984), Efficient Monte Carlo Probability Integration, Research Report No. 7, Department of Civil Engineering, Monash University, Australia.Google Scholar
  49. 49.
    MELCHERS, R.E., (1987), Structural Reliability: Analysis and Prediction, Ellis Horwood/Wiley, Chichester, United Kingdom.Google Scholar
  50. 50.
    MELCHERS, R.E., (1989), Importance Sampling in Structural Systems, Structural Safety, 6 (1), 3–10.CrossRefGoogle Scholar
  51. 51.
    MELCHERS, R.E., (1989), Improved Importance Sampling for Structural System Reliability Calculation, (in) Proceedings 5th International Conference on Structural Safety and Reliability, San Francisco, (Ed.) Ang, A. H.-S., Shinozuka, M. and Schuëller, G.I., ASCE, 1185–1192.Google Scholar
  52. 52.
    MELCHERS, R.E., (), Load Space Formulation for Structural Reliability Calculation, Journal of Engineering Mechanics, ASCE, (to appear), (See also Research Report No. 044.11. 1989, Department of Civil Engineering and Surveying, The University of Newcastle).Google Scholar
  53. 53.
    MELCHERS, R.E., (1989), Structural Reliability Assessment for Major Structures, Proceedings of Institution of Civil Engineers, Part 2, 87 (3), 343–356.Google Scholar
  54. 54.
    MELCHERS, R.E., (1989), Discussion to: Bucher, C.G., Structural Safety, 6 (1), 65–66.CrossRefGoogle Scholar
  55. 55.
    MELCHERS, R.E., (1990), Radial Importance Sampling for Structural Reliability, Journal of Engineering Mechanics, ASCE, 116 (1), 189–203.Google Scholar
  56. 56.
    MELCHERS, R.E., (1990), Search Based Importance Sampling, Structural Safety, 9(2), 117–128, (See also Research Report No. 030.07. 1988, Department of Civil Engineering and Surveying, The University of Newcastle ).Google Scholar
  57. 57.
    MELCHERS, R.E., (1990), Directional Simulation for Time-Dependent Reliability Problems, Proceedings 3rd IFIP Conference on Reliability and Optimisation of Structural Systems, (Ed.), Der Kiureghian, A. and Thoft-Christensen, P., Springer Verlag, 261–272.Google Scholar
  58. 58.
    OUYPORNPRASERT, W., (1989), Efficient Computational Methods for Structural Reliability Analysis Based on Conditional Importance Sampling Functions, Z.angew. Math. Mech., 69 (4), T69 - T71.Google Scholar
  59. 59.
    OUYPORNPRASERT, W., BUCHER, C.G. and SCHUELLER, G.I., (1989), On the Application of Conditional Integration in Structural Reliability Analysis, (in) Proceedings 5th International Conference on Structural Safety and Reliability, San Francisco, (Ed.) Ang, A. H.-S., Shinozuka, M. and Schuëller, G.I., ASCE, 1683–1689.Google Scholar
  60. 60.
    RACKWITZ, R., (1985), Reliability of Systems Under Renewal Pulse Loading, Journal of Engineering Mechanics, ASCE, 111 (9), 1175–1184.Google Scholar
  61. 61.
    RACKWITZ, R., (1985), Personal Communication.Google Scholar
  62. 62.
    RACKWITZ, R., (1987), Discussion to: Harbitz, A., An Effective Sampling Method for Probability of Failure Calculation, Structural Safety, 4 (4), 313–314.CrossRefGoogle Scholar
  63. 63.
    RUBENSTEIN, R.Y., (1981), Simulation and the Monte Carlo Method, John Wiley, New York.CrossRefGoogle Scholar
  64. 64.
    SCHREIDER, Y.A., (Ed.), (1966), The Monte Carlo Method, Pergamon, Oxford, United Kingdom.Google Scholar
  65. 65.
    SCHUELLER, G.I., BUCHER, C.G., BOURGUND, U. and OUYPORNPRASERT, W., (1989), On Efficient Computational Schemes to Calculate Failure Probabilities, Probabilistic Engineering Mechanics, 4(1), 10–18, (See also: Lin, Y.K. and Schuëller, G.I., ( 1987 ), ( Eds.), Stochastic Structural Mechanics, Springer Verlag, Berlin ).Google Scholar
  66. 66.
    SCHUELLER, G.I. and STIX, R., (1987), A Critical Appraisal of Methods to Determine Failure Probabilities, Structural Safety, 4 (4), 293–309.CrossRefGoogle Scholar
  67. 67.
    SCHWARZ, R.F., (1980), Beitrag zur Bestimmung der Zuverlassigheit nicht linearer Strukturen under Berucksichtigung kombinierter Stochastescher Einwirkungen, Doctoral Thesis, Technical University, Munich.Google Scholar
  68. 68.
    SHINOZUKA, M., (1983), Basic Analysis of Structural Safety, Journal of Structural Engineering, ASCE, 109 (3), 721–740.CrossRefGoogle Scholar
  69. 69.
    SHINOZUKA, M., (1987), Stochastic Fields and Their Digital Simulation, (in) Stochastic Methods in Structural Dynamics, (Ed.) Schuëller, G.I. and Shinozuka, M., Martinus Nijhoff.Google Scholar
  70. 70.
    SHIRAKI, W., (1991), Probabilistic Evaluation of Elastic and Inelastic Earthquake Response Spectra Using an Efficient Simulation Technique, (in) Proceedings 6th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Mexico City, (Ed.) Esteva, L. and Ruiz, S.E., CERRA, 415–422.Google Scholar
  71. 71.
    STIX, R. and SCHUËLLER, G.I., (1982/1986), Problemstellung bei der Berechnung der Versagenswahrscheinlichkeit, Internal Working Report No. 3, Institute of Engineering Mechanics, University of Innsbruck, Austria.Google Scholar
  72. 72.
    STROUD, A.H., (1971), Approximate Calculation of Multiple Integrals, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  73. 73.
    THOFT-CHRISTENSEN, P. and BAKER, M.J., (1982), Structural Reliability Theory and Its Applications, Springer Verlag, Berlin.CrossRefGoogle Scholar
  74. 74.
    VENEZIANO, D., GRIGORIU, M. and CORNELL, C.A., (1977), Vector Process Models for System Reliability, Journal Engg. Mech. Divn., ASCE, 103 (EM3), 441–460.Google Scholar
  75. 75.
    VROUWENVELDER, A., (1983), Monte Carlo Importance Sampling - Application to Structural Reliability Analysis, TNO-IBBC, Report No. B-83–529/62.6. 0402, Rijswijk, The Netherlands.Google Scholar
  76. 76.
    WEN, Y.K. and CHEN, H.C., (1987), On Fast Integration for Time Variant Structural Reliability, Prob. Engrg. Mech., 2 (3), 156–162.CrossRefGoogle Scholar
  77. 77.
    WEN, Y.K. and CHEN, H.C., (1987), Reliability of Structural Systems Under Time Varying Loads, (in) Proceedings 5th International Conference on Applications of Statistics and Probability Theory in Civil Engineering, Vancouver, (Ed.) Lind, N.C., Institute for Risk Research, University of Waterloo, Canada, 366–373.Google Scholar
  78. 78.
    WEN, Y.K. and CHEN, H.C., (1989), System Reliability Under Time Varying Loads: I, Journal of Engineering Mechanics, ASCE, 115 (4), 808–823.Google Scholar
  79. 79.
    SHAO, S.F. and MUROTSU, Y., (1991), Reliability Evaluation of Methods for Systems With Complex Limit States, Proceedings 4th IFIP Conference on Reliability and Optimisation of Structural Systems, Munich, 11–13 September, (Ed.) Rackwitz, R. and Thoft-Christensen, P., Springer-Verlag.Google Scholar

Copyright information

© International Federation for Information Processing, Geneva, Switzerland 1992

Authors and Affiliations

  • R. E. Melchers
    • 1
  1. 1.Department of Civil Engineering and SurveyingThe University of NewcastleAustralia

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