Safety Index, Stochastic Finite Elements and Expert Systems

  • F. Casciati
Part of the International Centre for Mechanical Sciences book series (CISM, volume 317)


The material presented in this Chapter has a two-fold objective. Firstly, it introduces those concepts in the Theory of Reliability which are useful in the Mechanics of Solids and Structures. The goal, here, is to link the general theoretical statements of Chapter 1 with the methods and concepts of the following two Chapters and, hence, with the subsequent specialistic applications.

Secondly, the present Chapter provides a short presentation of two research fields which are provoking a growing interest amongst practitioners, namely “Stochastic Finite Elements” (with their potentialities for dealing with complex mechanical systems) and “Expert Systems”. The application of the latter approach is mainly in monitoring and upgrading existing plants and buildings.


Expert System Failure Surface Basic Variable Structural Reliability Joint Probability Density 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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  1. [1]
    Casciati, F. and L. Faravelli: Fragility Analysis of Complex Structural Systems, Research Studies Press, Taunton, U.K. (1991)Google Scholar
  2. [2]
    Augusti, G., Baratta A. and F. Casciati: Probabilistic Methods in Structural Engineering, Chapman & Hall, London (1984)CrossRefzbMATHGoogle Scholar
  3. [3]
    Ditlevsen O. and O.H. Madsen: Proposal for a code from the direct use of reliability methods in structural design, JCSS, IABSE, Zurich, (1989), isbn 3–85748–059–2Google Scholar
  4. [4]
    Casciati, F., Grigoriu, M., Der Kiureghian A. and Y.K. Wen: Report of the work group on dynamics, 1st Draft, JCSS, (1990)Google Scholar
  5. [5]
    Shinozuka, M.: Stochastic Mechanics, Vol.I, Columbia Univ., (1987).zbMATHGoogle Scholar
  6. [6]
    Faravelli, L.: Finite elements analysis of stochastic nonlinear continua, in Liu W.K. and Beliytschko T. (Eds.), Computational Mechanics of Prob. and Reliab. Analysis, Elmepress, Lausanne (1989), 263–280Google Scholar
  7. [7]
    Casciati, F. and L. Faravelli: Uncertainty treatment in a vulnerability assistant expert system, in Expert Systems in Civil Engineering, IABSE Colloquium, Bergamo (1989), 97–106Google Scholar
  8. [8]
    Holloway, C.A.: Decision Making under Uncertainty Models and Choices, Prentice Hall, London (1979)Google Scholar
  9. [9]
    Casciati, F. and L. Faravelli: Structural reliability and structural design optimization, Proc. ICOSSAR 85, Kyoto (1985), III, 61–70.Google Scholar
  10. [10]
    Rjanitsyn, A.R.: Calcul a la Rupture et Pasticite des Construction (in French), Eyrolles, Paris (1959) (translation from 1954 Russian edition)Google Scholar
  11. [11]
    Cornell, C.A.: A probability based structural code, J. of the American Concrete Inst., 66, 12 (1969), 974–985Google Scholar
  12. [12]
    Hasofer, A.M. and N.C. Lind: An exact and invariant first order reliability format, J.of Eng.Mech., ASCE, 100, EM1 (1974) 111–121Google Scholar
  13. [13]
    Veneziano, D.: Contributions to Second-Moment Reliability Theory, Research Rep. R74–33, Massachussets Institute of Technology (1974)Google Scholar
  14. [14]
    Rackwitz, R.: Reliability analysis of structural components and systems, in Thoft Christensen P.(ed.), Reliability Theory and its Application in Structural and Soil Mechanics, M.Nijhoff W. Publ. (1983), 171–214Google Scholar
  15. [15]
    Shinozuka, M.: Basic analysis of structural safety, J. of Struct. Div., ASCE, 109 (3) (1983), 721–740CrossRefGoogle Scholar
  16. [16]
    Schittkowskj, K.: Computational Mathematical Programming, Series F, Computer and Systems, Sciences, Springer-Verlag (1985)Google Scholar
  17. [17]
    Schittkowskj, K.: NLPQL: a Fortran subroutine solving constrained nonlinear programming problems, Annals of Operations Research, 5, 6 (1985), 485–500Google Scholar
  18. [18]
    Rackwitz R. and B. Fiessler: Structural reliability under combined random sequences, Computer & Structures, 9 (1978), 489–494.CrossRefzbMATHGoogle Scholar
  19. [19]
    Breitung, K.: Asymptotic approximation for multinormal integrals, J. of Eng. Mech., ASCE, 110, 3 (1984), 357–366CrossRefGoogle Scholar
  20. [20]
    Tvedt, L.: On the Probability Content of a Parabolic Failure Set in a Space of Independent Standard Normally Distributed Random Variables, Section on Structural Reliability, A/S Veritas Research, Hovik, Norway (1985)Google Scholar
  21. [21]
    Hohenbichler, M. and R. Rackwitz: Nonnormal dependent vectors in structural safety, J. of Eng. Mech., ASCE, 107, 6 (1981), 1227–1238Google Scholar
  22. [22]
    Grigoriu, M.: Methods for approximate reliability analysis, Struct. Safety, 1 (1982), 155–165Google Scholar
  23. [23]
    Winterstein, S. and P. Bjerager: The use of higher moments in reliability estimation, Proc. ICASP 5, Vancouver, Vol. 2 (1987), 1027–1036Google Scholar
  24. [24]
    Ditlevsen, O., and H.O. Madsen: Probabilistic modelling of man-made load processes and their individual and combined effects, Proc. ICOSSAR 81 (1981), 103–104Google Scholar
  25. [25]
    Tichy, M.: The science of structural action, Prof. 4th ICASP (1983), 295–321Google Scholar
  26. [26]
    Casciati, F.: Load combination, in Lucia A.C. (ed.) Advances in Structural Reliability, D.Reidel Publ. Comp. (1987), 17–18Google Scholar
  27. [27]
    Ang, A.H.S., and W.H. Tang: Probability Concepts in Engineering Planning and Design, John Wiley & Sons (1983).Google Scholar
  28. [28]
    Wen, Y.K.: Structural Load Modelling & Combination from Performance & Safety Evaluation, Elsevier (1990)Google Scholar
  29. [29]
    Larrabee, R.D. and C.A. Cornell: Combination of various load processes, J. of Struct. Div., Asce, 107 (1) (1981), 223–239Google Scholar
  30. [30]
    Grigoriu, M.: Load combination analysis by translation processes, DIALOG 6–82, Proc. Euromech 155, Lyngby (1982), 165–183Google Scholar
  31. [31]
    Turkstra, C.J.: Theory and Structural Design Decision, Solid Mechanics Study, 2, Univ. of Waterloo (1972)Google Scholar
  32. [32]
    Ferry-Borges, J. and M. Castanheta: Structural Safety, Lab. Nacional de Eng. Civil, Lisbon (1972)Google Scholar
  33. [33]
    Wen, Y.K.: Statistical combination of loads, J. of Struct. Div., ASCE, 103 (5) (1977), 1079–1093Google Scholar
  34. [34]
    Casciati, F. and L. Faravelli: Load combination by partial safety factors, Nuclear Eng. and Design, 75, (1982), 439–452CrossRefGoogle Scholar
  35. [35]
    Faravelli, L.: A proposal on load combination for level I formats, Engineering Structures, 4 (1982), 197–206CrossRefGoogle Scholar
  36. [36]
    Der Kiureghian, A.: Reliability analysis, under stochastic loads, J. Struct. Div., ASCE, 106 (2) (1980), 414–429Google Scholar
  37. [37]
    Grigoriu, M.: Crossing of vector processes, in M.Grigoriu(ed) Risk, Structural Eng. and Human Error, Univ of Waterloo, Waterloo Press Waterloo, Ontario, Canada (1984), 89–112Google Scholar
  38. [38]
    Veneziano, D., Grigoriu, M. and C.A. Cornell: Vector process models for system reliability, J. of Eng. Mech. Div., ASCE, Vol. 103 (1977), 441–460Google Scholar
  39. [39]
    Lin, Y.K.: Probabilistic Theory of Structural Dynamics, Robert E. Krieger Publ. Co., Huntington, NY (1976)Google Scholar
  40. (40]
    Melsa, J.L. and A.P. Sage, An Introduction of Probability and Stochastic Processes, Prentice Hall, Inc., New Jersey (1973)Google Scholar
  41. [41]
    Roberts, J.B., and P.D. Spanos: Random Vibration and Statistical Linearization, John Wiley & Sons (1990)Google Scholar
  42. [42]
    Grigoriu, M.: Reliability of Daniels systems subject to Gaussian load processes, Symp. on Stochastic Struct. Dyn., Univ. of Illinois at Urbana Champaign (1988)Google Scholar
  43. [43]
    Grigoriu, M.: Reliability analysis of uncertain linear primarily secondary dynamic systems, Rep. N. 89–5, School of Civil and Environmental Engineering, Cornell Univ. (1989)Google Scholar
  44. [44]
    Wen, Y.K.: Methods of random vibration of inelastic structures, Applied Mech. Rev., 42 (2) (1989), 39–52Google Scholar
  45. [45]
    Wen, Y.K.: Equivalent linearization for hysteretic systems under random excitation, J. of Appl. Mech., Vol. 47 (1) (1980), 150–154CrossRefzbMATHGoogle Scholar
  46. [46]
    Spanos, P.D.: Stochastic linearization in structural dynamics, Applied Mechanics Rev., 34 (1) (1981), 1–6MathSciNetGoogle Scholar
  47. [47]
    Haber, T.T. and M.N. Noori: Random vibration of pinching, hysteretic systems, J. of Eng.Mech. Div., ASCE, 110 (7) (1984), 1036–1049CrossRefGoogle Scholar
  48. [48]
    Casciati, F. and L. Faravelli: Reliability assessment of nonlinear random frames, Nuclear Eng. & Design, 90 (1985), 341–356CrossRefGoogle Scholar
  49. [49]
    Park, Y.J., Wen Y.K. and A.H.S., Ang: Two-dimensional random vibration of hysteretic structures, J. of Earthquake Eng. & Struct. Dyn., 14 (1989), 543–557CrossRefGoogle Scholar
  50. [50]
    Casciati, F. and L. Faravelli: Stochastic equivalent linearization for 3-D frames, J. of Eng. Mech., ASCE, Vol. 114, (10) (1988), 1760–1771CrossRefGoogle Scholar
  51. [51]
    Casciati, F. and L. Faravelli: Hysteretic 3-dimensional frames under stochastic excitation, Res Mechanica, 26 (1989), 193–213Google Scholar
  52. [52]
    Casciati, F.: Stochastic dynamics of hysteretic media, Structural Safety, 6 (1989), 259–269CrossRefGoogle Scholar
  53. [53]
    Wen, Y.K. and H.-C. Chen: On fast integration for time-variant structural reliability, Prob. Eng. Mech., 3 (1987), 156–162Google Scholar
  54. [54]
    Cambou, D.: Applications of first-order uncertainty analysis in the finite element method in linear elasticity, Proc. ICASP 2, Aachen, (1975), 67–87Google Scholar
  55. [55]
    Esteva, L.: Second-moment probabilistic analysis of statistically loaded non-linear Structures, Proc. ICASP 2, Aachen (1975), 115–130Google Scholar
  56. [56]
    Handa, K. and K. Anderson: Application of finite element methods in the statistical analysis of structures, Proc. ICOSSAR 81, Trondheim (1981), 409–417Google Scholar
  57. (57]
    Hisada T. and S. Nakagiri: Stochastic finite element method developed for structural safety and reliability, Proc. ICOSSAR 81, Trondheim (1981), 395–408Google Scholar
  58. [58]
    Lawrence, M.: Basis random variables in finite element analysis, Int. J. of Num. Meth. in Eng., 24 (1987), 1849–1863MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    Righetti, G. and K. Harrop-Williams: Finite element analysis of random soil media, J. of Geotechnical Eng., 114 (1) (1988), 59–75CrossRefGoogle Scholar
  60. [60]
    Der Kiureghian A. and L.P. Liu: Finite element reliability methods, Lecture Notes for Structural Reliability: Methods & Applications, Berkeley (1989)Google Scholar
  61. [61]
    Der Kiureghian A.: Numerical methods in structural reliability, Proc. 4th ICASP, Florence (1985), 769–784Google Scholar
  62. [62]
    Nakagiri, S.: Stochastic Finite Element Method: An Introduction, (in Japanese), Baifiukan (1985)Google Scholar
  63. [63]
    Liu, W.K., et al.: Transient probabilistic systems, Computer Methods in Applied Mechanics and Engineering, 67 (1988), 27–54CrossRefzbMATHGoogle Scholar
  64. [64]
    Nakagiri, S.: Fluctuation of structural response, why and how, JSME Int. J., 30 (261) (1987), 369–374Google Scholar
  65. (65]
    Liu, W.K., Belytschko T. and A. Mani: Random field finite elements, Int. J. for Num. Meth. in Eng., 23 (1986), 1831–1845MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    Liu, W.K. and T., Belitschko (eds.): Computational Mechanics of Proabilistic and Reliability Analysis, Elmepress Int., Lausanne (1989)Google Scholar
  67. [67]
    Szentivanyi, B.: Computer solution of stochastically linear bar structure, a discrete time simulation approach, Zeszyty Naukowe Politechniki Poznanskiej, 26 (1981), 107–108Google Scholar
  68. [68]
    Vanmarke E. et al.: Random fields and stochastic finite elements, Structural Safety, 3 (1986), 143–166CrossRefGoogle Scholar
  69. [69]
    Faravelli, L. and D., Bigi: Stochastic finite elements for crash problems, Struct. Safety, 8 (1990), 113–130Google Scholar
  70. [70]
    Faravelli, L.: Response-surface approach for reliability analysis, J. of Eng.Mech., 115, 12 (1989), 2763–2781CrossRefGoogle Scholar
  71. [71]
    Olivi, L.: Response surface methodology in risk analysis, in Synthesis and Analysis Methods for Safety and Reliability Studies, Plenum Press (1980)Google Scholar
  72. [72]
    Petersen, R.G.: Design and Analysis of Experiments, M.Decker Inc, New York (1985)zbMATHGoogle Scholar
  73. [73]
    Faravelli, L.: Response variables correlation in stocahstic finite element analysis, Meccanica, 22, 2 (1988), 102–106MathSciNetCrossRefGoogle Scholar
  74. [74]
    Faravelli, L.: Stochastic finite elements by response surface techniques, in Computational Probabilistic Methods, ASME-AMD, Vol. 93 (1988), 197–203Google Scholar
  75. [75]
    Veneziano, D., Casciati, F. and L. Faravelli: Method of seismic fragility of complicated systems, 2nd CSNI Meeting on Prob. Meth. in Seismic Risk Assessment for Nuclear Power Plants, Livermore (1983), 67–88Google Scholar
  76. [76]
    Liu, W.K. et al: Applications of probabilistic finite element methods in elastic plastic dynamics, J. of Eng. For Industry (ASME), 109 (1987), 1–8CrossRefGoogle Scholar
  77. [77]
    Cohen, P.R.: Heuristic Reasoning about Uncertainty: an Artificial Intelligence Approach, Pitmans, Boston (1985)Google Scholar
  78. [78]
    Zadeh, L.A.: The role of fuzzy logic in the management of uncertainty in expert Systems, Fuzzy Sets and Systems, 11 (1968), 199–228MathSciNetGoogle Scholar
  79. [79]
    Dempster, A.P.: A generalization of bayesian inference, J. of the Royal Statistical Society B, 30 (1968), 205–247MathSciNetzbMATHGoogle Scholar
  80. [80]
    Shafer, G.: A Mathematical Theory of Evidence, Princeton Univ. Press, Princeton (1986)Google Scholar
  81. [81]
    Lindley, D.V.: The probabilistic approach to the treatment of uncertainty in artificial intelligence and expert systems, Stat. Science, 3 (1987), 17–24MathSciNetGoogle Scholar
  82. [82]
    Spiegelhalter, D.J.: Probabilistic reasoning in predictive expert systems, in Kanal L.N. and Lemmer J.F. (eds.), Uncertainty in Artificial Intelligence Elsevier Sc. Publ. (1986), 47–68Google Scholar
  83. [83]
    Pearl, J.: Fusion propagation and structuring in belief networks, Artificial Intelligence, 28 (1986), 9–15CrossRefGoogle Scholar
  84. [84]
    Andreassen S., Woldbye M., Falck., B. and S K. Andersen, MUNIN–A causal probabilistic network for interpretation of electromyographic findings, Proc. 10th Int. Joint Conf. on Artificial Intelligence, Milan (1987), 366–372Google Scholar
  85. [85]
    Lauritzen, S.L. and D.J. Spiegelhalter: Local computations with probabilities on graphical structures and their application to expert systems, J. Royal Statistical Society B, 50 (2) (1988), 157224Google Scholar
  86. [86]
    Gherardini, P.: Implementazione object-oriented di un modello di calcolo per reti causali (in Italian), Proc. XXXV Sc. Meeting of Italian Statistical Society, Padova (1990)Google Scholar
  87. [87]
    Maher, M.L., (ed.): Expert Systems for Civil Engineers: Technology and Application, American Society of Civil Engineering, ASCE, (1987).Google Scholar
  88. [88]
    The Building Surveyor (Off. J. of The Australian Institute of Building Surveyors), 7 (8) (1988)Google Scholar
  89. [89]
    Campbell, A. and S. Fitzgeral: The Deciding Factor, User’s Manual, Software Publ. Co. (1985)Google Scholar
  90. [90]
    INSIGHT, Knowledge System, Level Five Research, Merbourne Beach, Florida (1985)Google Scholar
  91. [91]
    Miyasato, G., Dong, W.M., Levitt, R.E., Boissonade A.C. and H.C. Shah: Seismic risk analysis system, Proc. Symp. Expert Systems in Civil Engineering, Seattle (1986) 121–132Google Scholar
  92. [92]
    Dong, W., Wong, F., Chiang, W., Kim, J.U. and H.C. Shah: An integrated system for seismic vulnerability and risk for engineering facilities, in Nelson J.K.(ed) Computer Utilization Structural Engineering, ASCE (1989), 408–417Google Scholar
  93. (93]
    Casciati, F. and L. Faravelli: L’impiego di ingegneria sismica (in Italian), Proc. 3rd Conf. Earthquake Eng. Italy, Rome (1987), 199–210Google Scholar
  94. [94]
    Casciati, F. and L. Faravelli: Individuazione meccanica dei solidi suscettibili di inquadramento esperti, (in Italian), Proc. 9th Conf. AIMETA (Ital Assoc. of Theoretical and Applied Mechanics ), Bari (1988), 553–556Google Scholar
  95. [95]
    Casciati, F. and L., Faravelli, Seismic vulnerability via knowledge based expert systems, Brebbia C.A. (ed.), Structural Repair and Maintenance of Historical Buildings, Computational Mechanics Publ., Southampton (1989), 299–307Google Scholar
  96. [96]
    Fenves, S.J., Ibarra-Arraya, E., Bielak, J. and C.H. Thewalt: Seismic resistance of existing buildings, in Nelson J.K.(ed.), Computer Utilization in Struct. Eng. (1989), 428–458Google Scholar
  97. [97]
    Subramani, M., Zaghw, A., and C.H., Conley: A KBES for seismic design of buildings, in Nelson J.,K. ( Ed. ), Computer Utilization in Struct. Eng. (1989), 342–351Google Scholar
  98. [98]
    Yao, J.T.P.: Safety and Reliability of Existing Structures, Pitman Publishing Ltd (1985)Google Scholar
  99. [99]
    Benedetti, D. and V. Petrini: Sulla vulnearbilita’ sismica di edifici in muratura: proposta di un metodo di valutazione (in Italian). Industria Costruzioni, 18 (1984), 66–74Google Scholar
  100. [100]
    Kafka P.: The Chernobil accident: a challenge of PSA as the tool for the prediction of event scenarios beyond the design basis and for safety improvements?, Trans. of 9th SMiRT, Vol. M (1987), 3–10Google Scholar
  101. [101]
    Vrouwenvelder, A.: Probabilistic Model Code, Assessment of Existing Structures, Report BI-88–010, IBBC, TNO, Delft (1988)Google Scholar
  102. [102]
    GNDT-CNR, Istruzioni per la compilazione della scheda di rilevamento esposizione e vulnerabilita’ sismica degli edifici (in Italian), (1986)Google Scholar
  103. [103]
    Benedetti, D., Benzoni, G. and M.A., Parisi: Seismic vulnerability and risk evaluation of old urban nuclei, Earthquake Eng. & Struct. Dynamics, 16 (1988), 183–201CrossRefGoogle Scholar
  104. [104]
    NEXPERT Object, Neuron Data Inc., Palo Alto (1987)Google Scholar

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© Springer-Verlag Wien 1991

Authors and Affiliations

  • F. Casciati
    • 1
  1. 1.University of PaviaPaviaItaly

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