Abstract
Yield stresses, allowable stresses, moment capacities (plastic moments), external loadings, manufacturing errors, etc., are not fixed quantities in practice, but must be modelled as random variables with a certain joint probability distribution. In reliability-oriented structural optimization the violation of the random behavioural constraints are evaluated by means of the corresponding probabilityp s of survival. Hence, the approximative computation ofp s and its sensitivities is of utmost importance. After the consideration of lower bounds ofp s based on a selection of certain redundants in the vector of internal forces/bending moments, and the consideration of upper bounds ofp s based on an optimizational representation of the yield or safety constraints by a pair of dual linear programs, a conical representation ofp s is introduced based on a coneY o of admissible pairs of external loads/strength increaments. Approximations ofp s can be constructed then by replacing the (finitely generated) coneY o by more simple ones, e.g. spherical or ellipsoidal cones. For the direct numerical computation of sensitivities ofp s and its bounds or approximations by using e.g. sampling methods or asymptotic expansion techniques based on Laplace integral representation of multiple integrals, exact differentiation formulae — of arbitrary order — forp s and its bounds or approximations with respect to deterministic input or design variables are obtained by applying the transformation method/stochastic completion techniques; the derivatives ofp s are represented again by certain expectations or multiple integrals.
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References
Arnbjerg-Nielsen, T. 1991:Rigid-ideal plastic model as a reliability analysis tool for ductil structures. Ph.D. Dissertation, Technical University of Denmark, Lyngby
Augusti, G.; Baratta, A.; Casciati, F. 1984:Probabilistic methods in structural engineering. London: Chapman and Hall
Bjerager, P. 1990: On computation methods for structural reliability analysis.Struct. Safety 9, 79–96
Bjerager, P.; Krenk, S. 1987: Sensitity measures in structural reliability analysis. In: Thoft-Christensen, P (ed.):Reliability and optimization of structural systems.Lecture Notes in Engineering 33, pp. 459–470. Berlin, Heidelberg, New York: Springer
Bjerager, P.; Krenk, S. 1989: Parameter sensitivity in first order reliability theory.J. Eng. Mech. Div. 115, 1577–1582
Breitung, K. 1991: Parameter sensitivity of failure probabilities. In: Der Kiureghian, A.; Thoft-Christensen, P. (eds.):Reliability and optimization of structural system '90, Lecture Notes in Engineering 61, pp. 43–51. Berlin, Heidelberg, New York: Springer
Cornell, C.A. 1967: Bounds on the reliability of structural systems.J. Struct. Div., ASCE 93, 171–200
Ditlevsen, O. 1979: Narrow reliability bounds for structural systems.J. Struct. Mech. 7, 453–472
Ditlevsen, O.; Bjerager, P. 1984: Reliability of highly redundant plastic structures.J. Eng. Mech. 110, 671–693
Frangopol, D.M. 1985: Sensitivity of reliability-based optimum design.J. Struct. Div. 111, 1703–1721
Galambos, J. 1977: Bonferroni inequalities.The Annals of Probability 5, 577–581
Haftka, R.T.; Gürdal, Z.; Kamat, M.P. 1990:Elements of structural optimization. Dordrecht-Boston-London: Kluwer
Hodge, P.G. 1959:Plastic analysis of structures. New York: McGraw-Hill
Kirsch, U. 1993:Structural optimization. Berlin, Heidelberg, New York: Springer
Kounias, E.G. 1968: Bounds for the probability of a union, with applications.The Annals of Mathematical Statistics 39, 2154–2158
Lawo, M. 1987:Optimierung im konstruktiven Ingenieurbau. Braunschweig: Vieweg
Lawo, M.; Thierauf, G. 1980:Stabtragwerke, Matrizenmethoden der Statik und Dynamik. Teil I: Statik. Brauschweig: F. Vieweg
Marti, K. 1990: Stochastic optimization methods in structural mechanics.ZAMM 70, T742-T745
Marti, K. 1994: Approximations and derivatives of probability functions. In: Anastassiou, G.; Rachev, S. (eds.)Approximation, probability, and related fields, pp. 367–377. New York: Plenum Press
Marti, K. 1995a: Differentiation of probability functions: the transformation method.Comp. Math. Appl. 30, 361–382
Marti, K. 1995b: Differentiation of probability functions arising in structural reliability. In: Rackwitz, R. (ed.)Reliability and optimization of structural systems, pp. 201–208. London: Chapman and Hall
Marti, K. 1995c: Computation of probability functions and its derivatives by means of orthogonal functions series expansions. In: Marti, K.; Kall, P. (eds.):Stoch. Prog.: numerical techniques and engineering applications.LNEMS 423, pp. 22–53. Berlin, Heidelberg, New York: Springer
Marti, K. 1996: Differentiation formulas for probability functions: the transformation method.Math. Prog., Series B 75, 201–220
Nafday, A.M.; Corotis, R.B. 1987: Failure mode enumeration for system reliability assessment by optimization algorithms. In: Thoft-Christensen, P. (ed.)Reliability and optimization of structural systems.Lecture Notes in Engineering.33, pp. 297–306. Berlin, Heidelberg, New York: Springer
Neal, B.G. 1965:The plastic methods of structural analysis. London: Chapman and Hall
Rackwitz, R.; Cuntze, R. 1987: Formulations of reliability-oriented optimization.Eng. Opt. 11, 69–76
Rozvany, G.I.N.; Bendsøe, M.P.; Kirsch, U. 1995: Layout optimization of structures.Appl. Mech. Reviews 48, 41–118
Simoes, L.M.C. 1990: Stochastically dominant modes of frames by mathematical programming.J. Struct. Eng. 116, 1040–1060
Spillers, W.R. 1972:Automated structural analysis: an introduction. New York, Toronto, Oxford: Pergamon
Tin-Loi, F. 1990: On the optimal plastic synthesis of frames.Eng. Opt. 16, 91–108
Tin-Loi, F. 1995: Plastic limit analysis of plane frames and grids using GAMS.Comp. & Struct. 54, 15–25
Zimmermann, J.J.; Corotis, R.B.; Ellis, J.H. 1991: Stochastic programs for identifying significant collaps models in structural systems. In: Der Kiureghian, A.; Thoft-Christensen, P. (eds.)Reliability and optimization of structural systems '90, pp. 359–365. Berlin, Heidelberg, New York: Springer
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Marti, K. Approximation and derivatives of probabilities of survival in structural analysis and design. Structural Optimization 13, 230–243 (1997). https://doi.org/10.1007/BF01197451
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DOI: https://doi.org/10.1007/BF01197451