Keywords: optimal design, truss, under uncertainty, random external loadings.

Using the first collapse-theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated [1].

Here, the design of trusses is considered, where the external load is considered to be stochastic.

In the first approach the recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP) [2]. In order to apply efficient numerical solution procedures (LP-solvers), approximate recourse problems based on discretization and the expected value problem are introduced [3,4].

In the second approach, based on the yield condition, a quadratic cost function will be introduced [5]. After the formulation of the stochastic optimization problem, the expected cost based optimization (ECBO) problem and the minimum expected cost (MEC) problem are formulated as representatives of appropriate substitute problems.

Subsequently, comparative numerical results using these methods are presented. Therefore, the three-bar truss is considered. The numerical results confirm the advantage of the presented stochastic opimization methods. Due to the different approaches, in some cases it is not possible to compare all methods in each detail. It turns out that a method which is superior to the others does not exist.

1

K. Marti, "Stochastic Optimisation Methods", Springer, Berlin, 2005.

2

P. Kall, S.W. Wallace, "Stochastic Programming", John Wiley & Sons, Chichester, 1994. doi:10.1007/BFb0035456

3

K. Marti, "Optimal design of trusses as a stochastic linear programming problem", In: A.S. Nowak, ed., "Reliability and optimization of structural systems", Ann Arbor, University of Michigan Press, Michigan, 231-239, 1998.

4

G. Stöckl, "Optimaler Entwurf elastoplastischer mechanischer Strukturen unter stochastischer Unsicherheit", Fortschritt-Berichte VDI, Reihe 18, Nr. 278, VDI-Verlag GmbH, Düsseldorf, 2003.

5

K. Marti, "Limit Load and Shakedown Analysis of Plastic Structures under Stochastic Uncertainty", To appear in: Computer Methods in Applied Mechanics and Engineering (CMAME), Elsevier, 2008. doi:10.1016/j.cma.2008.04.022

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