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Expected Total Cost Minimum Design of Plane Frames by Means of Stochastic Linear Programming Methods

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Coping with Uncertainty

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 633))

Abstract

Yield stresses, allowable stresses, moment capacities (plastic moments with respect to compression, tension and rotation), applied loadings, cost factors, manufacturing errors, etc., are not given fixed quantities in structural analysis and optimal design problems, but must be modeled as random variables with a certain joint probability distribution. Problems from plastic analysis and optimal plastic design are based on the convex yield (feasibility) criterion and the linear equilibrium equation for the stress (state) vector.

After the formulation of the basic mechanical conditions including the relevant material strength parameters and load components as well as the involved design variables (as, e.g., sizing variables) for plane frames, several approximations are considered: (1) approximation of the convex yield (feasible) domain by means of convex polyhedrons (piecewise linearization of the yield domain); (2) treatment of symmetric and non symmetric yield stresses with respect to compression and tension; (3) approximation of the yield condition by using given reference capacities.

As a result, for the survival of plane frames a certain system of necessary and/or sufficient linear equalities and inequalities is obtained. Evaluating the recourse costs, i.e., the costs for violations of the survival condition by means of piecewise linear convex cost functions, a linear program is derived for the minimization of the total costs including weight-, volume- or more general initial (construction) costs. Appropriate cost factors are given. Considering then the expected total costs from construction as well as from possible structural damages or failures, a stochastic linear optimization problem (SLP) is obtained. Finally, discretization of the probability distribution of the random parameter vector yields a (large scale) linear program (LP) having a special structure. For LP’s of this type numerous, very efficient LP-solvers are available – also for the solution of very large scale problems.

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Authors and Affiliations

  1. Aero-Space Engineering and Technology, Federal Armed Forces University Munich, 85577, Neubiberg, Germany

    Kurt Marti

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  1. Kurt Marti
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Correspondence to Kurt Marti .

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Editors and Affiliations

  1. Aerospace Engineering and, Technology, Federal Armed Forces University Munich, Werner-Heisenberg-Weg 39, Neubiberg/München, 85577, Germany

    Kurt Marti

  2. Applied Systems Analysis (IIASA), International Institute for, Schloßplatz 1, Laxenburg, 2361, Austria

    Yuri Ermoliev

  3. Applied Systems Analysis (IIASA), International Institute for, Schloßplatz 1, Laxenburg, 2361, Austria

    Marek Makowski

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Marti, K. (2010). Expected Total Cost Minimum Design of Plane Frames by Means of Stochastic Linear Programming Methods. In: Marti, K., Ermoliev, Y., Makowski, M. (eds) Coping with Uncertainty. Lecture Notes in Economics and Mathematical Systems, vol 633. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03735-1_8

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