Abstract
Modern engineering has included the basic sciences and their accompanying mathematical theories among its primary tools. The theory of probability is one of the more recent entries into standard engineering practice in various technological disciplines. Probability and statistics serve useful functions in the solution of many engineering problems. However, not all technological manifestations of uncertainty are amenable to probabilistic representation. In this paper we identify the conceptual limitations of probabilistic and related theories as they occur in a wide range of engineering tasks. We discuss the structure and properties of an alternative, non-probabilistic, method — convex modelling — for quantitatively representing uncertain phenomena.
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J. Arbocz and J. G. Williams: 1977, ‘Imperfection Surveys on a 10 ft. Diameter Shell Structure’,Amer. Inst. of Aeronautics and Astronautics Journal 15, 949–956.
Z. Artstein: 1974, ‘On the Calculus of Closed Set-Valued Functions’,Indiana University Mathematics Journal 24, 433–441.
Z. Artstein and J. C. Hansen: 1985, ‘Convexification in Limit Laws of Random Sets in Banach Spaces’,Annals of Probability 13, 307–309.
R. J. Aumann: 1965, ‘Integrals of Set-Valued Functions’,Journal of Mathematical Analysis and Applications 12, 1–12.
Y. Ben-Haim: 1985,The Assay of Spatially Random Material, Kluwer Academic Publishers, Dordrecht, Holland.
Y. Ben-Haim: 1990, ‘Detecting Unknown Lateral Forces on a Bar by Vibration Measurement’,Journal of Sound and Vibration 140, 13–29.
Y. Ben-Haim: 1992, ‘Convex Models for Optimizing Diagnosis of Uncertain Slender Obstacles on Surfaces’,Journal of Sound and Vibration 152, 327–341.
Y. Ben-Haim: 1993, ‘Convex Models of Uncertainty in Radial Pulse Buckling of Shells’,ASME Journal of Applied Mechanics 60, 683–688.
Y. Ben-Haim: 1993, ‘Failure of an Axially Compressed Beam with Uncertain Initial Deflection of Bounded Strain Energy’,International Journal of Engineering Science 31, 989–1001.
Y. Ben-Haim: ‘A Non-Probabilistic Concept of Reliability’, to appear inStructural Safety.
Y. Ben-Haim and E. Elias: 1987, ‘Indirect Measurement of Surface Temperature and Heat Flux: Optimal Design Using Convexity Analysis,International Journal of Heat and Mass Transfer 30, 1673–1683.
Y. Ben-Haim and I. Elishakoff: 1989, ‘Non-Probabilistic Models of Uncertainty in the Non-Linear Buckling of Shells with General Imperfections: Theoretical Estimates of the Knockdown Factor’,ASME Journal of Applied Mechanics 56, 403–410.
Y. Ben-Haim and I. Elishakoff: 1990,Convex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam.
Y. Ben-Haim and I. Elishakoff: 1991, ‘Convex Models of Vehicle Response to Uncertain but Bounded Terrain’,ASME Journal of Applied Mechanics 58, 354–361.
Y. Ben-Haim and N. Shenhav: 1984, ‘The Measurement of Spatially Random Material’,SIAM Journal of Applied Mathematics 44, 1150–1163.
R. F. Drenick: 1968, ‘Functional Analysis of Effects of Earthquakes’,2nd Joint United States—Japan Seminar on Applied Stochastics, Washington, D.C., Sept. 19–24.
R. F. Drenick: 1970, ‘Model-Free Design of Aseismic Structures’,Journal of the Engineering Mechanics Division’, Proceedings of the ASCE 96, 483–493.
D. Dubois and H. Prade: 1988,Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York.
I. Elishakoff: 1983,Probabilistic Methods in the Theory of Structures, Wiley, New York.
I. Elishakoff and Y. Ben-Haim: 1990, ‘Dynamics of a Thin Shell Under Impact with Limited Deterministic Information on its Initial Imperfections’,International Journal of Structural Safety 8, 103–112.
I. Elishakoff and P. Colombi: 1993, ‘Combination of Probabilistic and Convex Models of Uncertainty when Scarce Knowledge is Present on Acoustic Excitation Parameters’,Computer Methods in Applied Mechanics and Engineering,104, 187–209.
I. Elishakoff, G. Q. Cai and J. H. Starnes, jr.: 1993, ‘Non-Linear Buckling of a Column with Initial Imperfection via Stochastic and Non-Stochastic Convex Models’,International Journal of Non-Linear Mechanics 27.
I. Ekeland: 1988,Mathematics and the Unexpected, University of Chicago Press, Chicago.
B. R. Gaines: 1978, ‘Fuzzy and Probability Uncertainty Logics’,Information and Control 38, 154–169.
D. Givoli and I. Elishakoff: 1992, ‘Stress Concentration at a Nearly Circular Hole with Uncertain Irregularities’,ASME Journal of Applied Mechanics 59, 65–71.
W. H. Hartt and N. K. Lin: 1986, ‘A Proposed Stress History for Fatigue Testing Applicable to Offshore Structures’,International Journal of Fatigue 8: 91–93.
S. W. Kirkpatrick and B. S. Holmes: 1989, ‘Effect of Initial Imperfections on Dynamic Buckling of Shells’,ASCE Journal of Engineering Mechanics 115, 1075–1093.
H. E. Lindberg: 1992, ‘An Evaluation of Convex Modelling for Multimode Dynamic Buckling’,ASME Journal of Applied Mechanics 59, 929–936.
H. E. Lindberg: 1992, ‘Convex Models for Uncertain Imperfection Control in Multimode Dynamic Buckling’,ASME Journal of Applied Mechanics 59, 937–945.
K. Murota and K. Ikeda: 1992, ‘On Random Imperfections for Structures of Regular-Polygonal Symmetry’,SIAM J. Appl. Math. 52, 1780–1803.
H. G. Natke and T. T. Soong: 1993, ‘Topological Structural Optimization Under Dynamic Loads’, in S. Hernandez and C. A. Brebbia (eds.),Optimization of Structural Systems and Applications, Proceedings of 3rd Intl. Conf. on Computer Aided Optimum Design of Structures, Elsevier Applied Science, London.
D. E. Newland: 1986, ‘General Linear Theory of Vehicle Response to Random Road Roughness’, in I. Elishakoff and R. H. Lyons (eds.),Random Vibrations — Status and Recent Developments, Elsevier Science Publishers, Amsterdam, pp. 303–326.
J. Pearl: 1988,Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufman Publishers, San Mateo.
F. C. Schweppe: 1968, ‘Recursive State Estimation: Unknown but Bounded Errors and System Inputs,IEEE Transactions on Automatic Control AC-13, 22–28.
F. C. Schweppe: 1973,Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs.
N. Shenhav and Y. Ben-Haim: 1984, ‘A General Method for Optimal Design of Nondestructive Assay Systems’,Nuclear Science and Engineering 88, 173–183.
M. Shinozuka: 1970, ‘Maximum Structural Response to Seismic Excitations’,Journal of the Engineering Mechanics Division’, Proceedings of the ASCE 96, 729–738.
K. Sobczyk and B. F. Spencer: 1992,Random Fatigue: From Data to Theory, Academic Press, Boston.
T. T. Soong: 1981,Probabilistic Modelling in Science and Engineering, Wiley, New York.
T. T. Soong: 1990,Active Structural Control: Theory and Practice, Wiley, New York.
S. M. Stigler: 1986,The History of Statistics: The Measurement of Uncertainty Before 1900, Belnap Press, Cambridge.
P. Suppes and M. Zanotti: 1977, ‘On Using Random Relations to Generate Upper and Lower Probabilities’,Synthese 36, 427–440.
A. Talmor, Y. Leichter, Y. Ben-Haim and A. Kushlevsky: 1990, ‘Adaptive Assay of Radioactive Pulmonary Aerosol with an External Detector’,International Journal of Applied Radiation and Isotopes, Part A,41, 989–993.
U. Vulkan and Y. Ben-Haim: 1989, ‘Global Optimization in the Adaptive Borehole Assay of Uranium’,International Journal of Applied Radiation and Isotopes, Part E: Nuclear Geophysics 3, 97–105.
H. S. Witsenhausen: 1968, ‘A Minimax Control Problem for Sampled Linear Systems’,IEEE Transactions on Automatic Control AC-13, 5–21.
H. S. Witsenhausen: 1968, ‘Sets of Possible States of a Linear System Given Perturbed Observations’,IEEE Transactions on Automatic Control AC-13, 556–558.
J. T. P. Yao: 1985,Safety and Reliability of Existing Structures, Pitman, Boston.
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Ben-Haim, Y. Convex models of uncertainty: Applications and implications. Erkenntnis 41, 139–156 (1994). https://doi.org/10.1007/BF01128824
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DOI: https://doi.org/10.1007/BF01128824