On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

Open access


Civil engineering is one of the many fields of occurrences of uncertain parameters. The present paper in an attempt to present and describe the most common methods used for inclusions of uncertain parameters. These methods can be applied in the area of civil engineering as well as for a larger domain. Definitions and short explanations of methods based on probability, interval analysis, fuzzy sets, and convex sets are presented. Selected advantages, disadvantages, and the most common fields of implementation are indicated.

An example of a cantilever beam presented in this paper shows the main differences between the methods. Results of the performed analysis indicate that the use of convex sets allows us to obtain an accuracy of results similar to stochastic models. At the same time, the computational speed characteristic for interval methods is maintained.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • 1. G. Alefeld G. Mayer “Interval analysis: Theory and applications” J Comput Appl Math 121(1):421–464 2000.

  • 2. T. M. Allen A. S. Nowak R. J. Bathurst “Calibration to determine load and resistance factors for geotechnical and structural design” Transportation Research E-Circular (E-C079) 2005.

  • 3. Y. Ben-Haim “A non-probabilistic measure of reliability of linear systems based on expansion of convex models” Struct Saf 17(2):91–109 1995.

  • 4. Y. Ben-Haim “Convex models of uncertainty: applications and implications” Erkenntnis 41(2):139–156 1994.

  • 5. Y. Ben-Haim I. Elishakoff “Convex models of uncertainty in applied mechanics” Elsevier 1990.

  • 6. J. M. Bernardo and A. F. Smith Bayesian theory 2001.

  • 7. P. Bocchini G. Deodatis “Critical review and latest developments of a class of simulation algorithms for strongly non-Gaussian random fields” Probabilist Eng Mech 23(4):393–407 2008.

  • 8. B. Cambou “Application of first-order uncertainty analysis in the finite element method in linear elasticity” Proc. 2nd Int. Conf. on Applications of Statistics and Probability in Soil and Structural Engrg 1975 67–87.

  • 9. S.-H. Chen X.-W. Yang “Interval finite element method for beam structures” Finite Elem Anal Des 34(1):75–88 2000.

  • 10. L. F. Contreras E. T. Brown M. Ruest “Bayesian data analysis to quantify the uncertainty of intact rock strength” Journal of Rock Mechanics and Geotechnical Engineering 10(1):11–31 2018.

  • 11. G. Corliss C. Foley R. B. Kearfott “Formulation for reliable analysis of structural frames” Reliable Computing 13(2):125–147 2007.

  • 12. W. De Mulder D. Moens D. Vandepitte “Modeling uncertainty in the context of finite element models with distance-based interpolation” Proceedings of the 1st International Symposium on Uncertainty Quantification and Stochastic Modeling 2012.

  • 13. O. Dessombz et al. “Analysis of mechanical systems using interval computations applied to finite element methods” J Sound Vib 239(5):949–968 2001.

  • 14. I. Ekeland “Mathematics and the Unexpected” The University of Chicago Press 1990.

  • 15. I. Elishakoff “Essay on uncertainties in elastic and viscoelastic structures: From A. M. Freudenthal’s criticisms to modern convex modeling” Comp Struct 56(6):871–895 1995.

  • 16. I. Elishakoff “Some Questions in Eigenvalue Problems in Engineering” Numerical Treatment of Eigenvalue Problems Vol. 5 / Numerische Behandlung von Eigenwertaufgaben Band 5: Workshop in Oberwolfach ed. by J. Albrecht et al. Basel: Birkhauser Basel pp. 71–107 1991.

  • 17. I. Elishakoff et al. “How to find the range of eigenvalues due to uncertain elastic modulus and mass density?” Whys and Hows in Uncertainty Modelling: Probability Fuzziness and Anti-Optimization ed. by I. Elishakoff Springer-Verlag Wien pp. 341–355 1999.

  • 18. EN 1990 “Eurocode - Basis of structural design.” European Committee for Standardization 1:1–116 2004.

  • 19. EN 1995-1-1 “Eurocode 5: Design of Timber Structures - Part 1-1: General - Common Rules and Rules for Buildings.” European Committee for Standardization 1:1–121 2008.

  • 20. R. G. Ghanem “Hybrid stochastic finite elements and generalized Monte Carlo simulation” J Appl Mech (ASME) 65(4):1004–1009 1998.

  • 21. R. G. Ghanem P. D. Spanos “Stochastic Finite Elements: A Spectral Approach” New York NY USA: Springer-Verlag New York Inc. 1991.

  • 22. W. Gilewski et al. “Truss Structures with Uncertain Parameters – Geometrical Interpretation of the Solution based on Properties of Convex Sets” Procedia Engineering 111:249–253 2015.

  • 23. W. Gilewski et al. “Truss structures with uncertain parameters – geometrical interpretation of the solution based on properties of convex sets” Theoretical Foundations of Civil Engineering Vol. 7: Structural Mechanics ed. by S. Jemioło and M. Gajewski Warszawa: Warsaw University of Technology Publisher House pp. 41–52 2016.

  • 24. T. Hickey Q. Ju M. H. Van Emden “Interval arithmetic: From principles to implementation” J ACM 48(5):1038–1068 2001.

  • 25. Y. H. Huang “Pavement Analysis and Design” Pearson Education Inc. 2012.

  • 26. Z. Kang Y. Luo A. Li “On non-probabilistic reliability-based design optimization of structures with uncertain-but-bounded parameters” Struct Saf 33(3):196–205 2011.

  • 27. R. B. Kearfott “Interval Computations: Introduction Uses and Resources” Euromath Bulletin 2(1):95–112 1996.

  • 28. N. D. Lagaros G. Stefanou M. Papadrakakis “An enhanced hybrid method for the simulation of highly skewed non-Gaussian stochastic fields” Comput Methods Appl Mech Eng 194(45):4824–4844 2005.

  • 29. M. Litwin M. Górecki “Assembly mistakes of steel structures (in Polish)” Budownictwo i Architektura 4:63–72 2009.

  • 30. W. K. Liu A. Mani T. Belytschko “Finite element methods in probabilistic mechanics” Probabilist Eng Mech 2(4):201–213 1987.

  • 31. J. E. Lundberg T. V. Galambos “Load and resistance factor design of composite columns” Struct Saf 18(2):169–177 1996.

  • 32. P. Mackiewicz “Fatigue life of asphalt mixtures used in pavements (in Polish)” Oficyna Wydawnicza Politechniki Wrocławskiej 2016.

  • 33. D. Moens D. Vandepitte “A survey of non-probabilistic uncertainty treatment in finite element analysis” Comput Methods Appl Mech Eng 194(12):1527–1555 2005.

  • 34. R. Moore W. Lodwick “Interval analysis and fuzzy set theory” Fuzzy Set Syst 135(1):5–9 2003.

  • 35. R. E. Moore “Interval arithmetic and automatic error analysis in digital computing” PhD thesis Stanford University 1962.

  • 36. R. L. Muhanna R. L. Mullen “Formulation of Fuzzy Finite-Element Methods for Solid Mechanics Problems” Computer-Aided Civ Inf 14(2):107–117 1999.

  • 37. R. L. Muhanna R. L. Mullen H. Zhang “Interval finite elements as a basis for generalized models of uncertainty in engineering mechanics” Reliable Computing 13(2):173–194 2007.

  • 38. R. L. Mullen R. L. Muhanna “Bounds of structural response for all possible loading combinations” J Struct Eng 125(1):98–106 1999.

  • 39. R. L. Mullen R. L. Muhanna “Structural analysis with fuzzy-based load uncertainty” Probabilistic Mechanics & Structural Reliability ASCE 310–313 1996.

  • 40. A. Neumaier A. Pownuk “Linear systems with large uncertainties with applications to truss structures” Reliable Computing 13(2):149–172 2007.

  • 41. J. Niczyj “Multi-criterion reliability optimization and technical assessment of bar structures on the background of fuzzy set theory (in Polish)” Polish Szczecin Univ. of Technology Publishers Nr 581 (3):3–229 2003.

  • 42. M. Papadrakakis V. Papadopoulos “Robust and eƥcient methods for stochastic finite element analysis using Monte Carlo simulation” Comput Methods Appl Mech Eng 134(3):325–340 1996.

  • 43. J. Pełczyński T. Rzeżuchowski J. Wąsowski “Description of united solution sets by in-equalities for truss structures” 4th ECCOMAS Young Investigators Conference (YIC 2017) 2017.

  • 44. K. Phoon H. Huang and S. Quek “Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion” Probabilist Eng Mech 20(2):188–198 2005.

  • 45. Z. Qiu “Comparison of static response of structures using convex models and interval analysis method” Int J Numer Meth Eng 56(12):1735–1753 2003.

  • 46. Z. Qiu “Convex models and interval analysis method to predict the effect of uncertain-but-bounded parameters on the buckling of composite structures” Comput Methods Appl Mech Eng 194(18):2175–2189 2005.

  • 47. Z. Qiu and I. Elishakoff “Anti-optimization technique – A generalization of interval analysis for nonprobabilistic treatment of uncertainty” Chaos Solitons and Fractals 12(9):1747–1759 2001.

  • 48. U Radoń “Reliability analysis of Mises truss” Arch Civ Mech Eng 11(3):723–738 2011.

  • 49. M. V. R. Rao R. L. Muhanna R. L. Mullen “Interval Finite Element Analysis of Thin Plates” 7th International Workshop on Reliable Engineering Computing 2016 111–130.

  • 50. S. S. Rao L. Chen “Numerical solution of fuzzy linear equations in engineering analysis” Int J Numer Meth Eng 43(3):391–408 1998.

  • 51. R. J. Ross ed. “Wood handbook – wood as an engineering material. General Technical Report FPL-GTR-190” U.S. Department of Agriculture Forest Service Forest Products Laboratory 2010.

  • 52. T. J. Ross “Fuzzy logic with engineering applications” John Wiley & Sons 2004.

  • 53. S. P. Shary “On optimal solution of interval linear equations” SIAM J Numer Anal 32(2):610–630 1995.

  • 54. M. Shinozuka G. Deodatis “Response Variability Of Stochastic Finite Element Sys-tems” J Eng Mech 114(3):499–519 1988.

  • 55. A Simoneau E Ng M. A. Elbestawi “Chip formation during microscale cutting of a medium carbon steel” Int J Mach Tool Manu 46(5):467–481 2006.

  • 56. J Skrzypczyk “Fuzzy methods in the analysis of uncertain systems” Zeszyty Naukowe. Budownictwo/Politechnika Śląska (86):183–196 1999.

  • 57. A. P. Smith J. Garloff H. Werkle “Verified solution for a simple truss structure with uncertain node locations” Proceedings of the 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering Weimar Germany 2009.

  • 58. E Sobczyńska W Wasilewski M Gregoriou-Szczepaniak “Issues of modeling of masonry structures: case study of tenement house at Szara street in Warsaw (in Polish)” Journal of Civil Engineering Environment and Architecture 63(4):607–615 2016.

  • 59. P. D. Spanos R. Ghanem “Stochastic finite element expansion for random media” J Eng Mech 115(5):1035–1053 1989.

  • 60. G. Stefanou “The stochastic finite element method: Past present and future” Comput Methods Appl Mech Eng 198(9–12):1031–1051 2009.

  • 61. P. Van Hentenryck L. Michel and Y. Deville “Numerical: a modeling language for global optimization” MIT Press 1997.

  • 62. E. Vanmarcke M. Grigoriu “Stochastic finite element analysis of simple beams” J Eng Mech 109(5):1203–1214 1983.

  • 63. P. Wojnarowski “Hydrocarbons reserves estimation for fields in early stage of development with application of interval analysis (in Polish)” Wiertnictwo Nafta Gaz 27:781–787 2010.

  • 64. N. Xiao et al. “Interval finite elements for uncertainty in frame structures” 11th International Conference on Structural Safety & Reliability pp. 16–20 2013.

  • 65. F. Yamazaki et al. “Neumann expansion for stochastic finite element analysis” J Eng Mech 114(8):1335–1354 1988.

  • 66. K.-V. Yuen “Bayesian methods for structural dynamics and civil engineering” John Wiley & Sons 2010.

  • 67. L. A. Zadeh “Fuzzy sets” Information and Control 8(3):338–353 1965.

  • 68. H. Zhang “Nondeterministic linear static finite element analysis: an interval approach” PhD thesis Georgia Institute of Technology 2005.

Journal information
Impact Factor

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.304
Source Normalized Impact per Paper (SNIP) 2018: 0.866

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 67 67 7
PDF Downloads 49 49 9