Keywords: optimal plastic design under stochastic uncertainty, FORM, explicit representation of the projection problem, explicit formulations of (RBDO) problems.

Problems from optimal plastic design are based on the convex or piecewise linear yield criterion and the linear equilibrium equation for the generic stress (state) vector . Having taken into account, in practice, stochastic variations of the vector of the model parameters, e.g. yield stresses, external loadings, cost coefficients, etc., the basic stochastic optimal plastic design problem is replaced - in order to get robust optimal designs - by a deterministic substitute problem with probabilistic constraints. Survival/failure of an elastic-plastic structure is described with the new limit state function based on the static limit theorem for elastic-plastic structures. This function can be represented explicitly by the minimum value function of a convex or linear program, related to the basic survival conditions, depending on the parameter vector and the design vector . Hence, the probability of survival can be represented [1] by . Using (FORM), the probability of survival is approximated then by the formula , where denotes the length of a so-called beta- or "design" point . In general, the computation of this projection of the origin 0 to the failure domain is very difficult. However, in the present case, due to the new limit state function , an explicit representation of the projection problem for the calculation of the design point is available. Consequently, also the necessary optimality conditions for the projection problem are available explicitly. For the basic reliability-based design optimization (RBDO) problem [2]

where is the initial cost function, denotes the domain of feasible designs , and is a given reliability level, a standard solution technique is the two-level method with two nested optimization problems. However, based on the above mentioned explicit representation of the projection problem, two explicit representations of the (RBDO) problem are obtained:

i)

Using the explicit necessary conditions of the projection problem:
Since the projection problem is given explicitly, also the corresponding necessary (sufficient) optimality conditions can be stated explicitly. Hence, in the present case, the optimality condition in the above (RBDO) problem is replaced by the necessary optimality conditions for derived from the projection problem. Numerical results show that somewhat higher initial -values must be chosen in some cases for guaranteeing the desired reliability level .

ii)

Relaxation methods:
In many cases the reliability condition can be represented by

    for all

where is a given set of dual variables , and is the projection of the origin in the parameter space onto a given (simple) surface in -space depending on , where this projection can be determined explicitly. Hence, a semi-infinite optimization problem is obtained in this case. This problem can be solved then by known approximation techniques reducing e.g. the problem to a finite dimensional parameter optimization problem again.

Methods for the derivation of explicit (RBDO) problems are shown, and some numerical examples are given.

1

K. Marti, "Plastic Sturcutral Analysis under Stochastic Uncertainty", MCMDS, 9, Issue 3, 303-325, 2003. doi:10.1076/mcmd.9.3.303.24149

2

N. Kushel and R. Rackwitz, "Two basic problems in reliability-based structural optimization", Mathematical Methods of Operational Research, 46, 309-333, 1997. doi:10.1007/BF01194859

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