TOOLS AND TECHNIQUES
Structural Analysis and Optimal Design under Stochastic Uncertainty with Quadratic Cost Functions
Aero-Space Engineering and Technology, Federal Armed Forces University Munich, Neubiberg/Munich, Germany
Problems from plastic analysis and optimal plastic design are based on the convex, linear or linearised yield/strength condition and the linear equilibrium equation for the stress (state) vector. In practice one has to take into account stochastic variations of the vector a=a(omega) of model parameters (e.g. yield stresses, plastic capacities, external load factors, cost factors, etc.), see e.g. [2-4]. Hence, in order to get robust optimal load factors x, robust optimal designs x, resp., i.e., maximum load factors, optimal designs insensitive with respect to variations of the vector of model parameters a, the basic plastic analysis or optimal plastic design problem with random parameters has to be replaced by an appropriate deterministic substitute problem, cf. [1]. As a basic tool in the analysis and optimal design of mechanical structures under uncertainty, the state function s* = s*(a,x) of the underlying structure is introduced. Depending on the survival conditions of plasticity theory, by means of the state function the survival/failure of the structure can be described by the condition s*<=(>)0. Interpreting the state function s* as the basic cost function, several relations to other cost functions, especially quadratic cost functions, are shown. Bounds for the probability of survival ps are obtained then by means of the Tschebyscheff inequality.
In order to obtain robust optimal decisions x*, a direct approach is proposed here based on the primary costs (weight, volume, costs of construction, costs for missing carrying capacity, etc.) and the recourse costs (e.g. costs for repair, compensation for weakness within the structure, damage, failure, etc.), where the above mentioned quadratic recourse cost criterion is used. The minimum recourse costs can be determined then by solving an optimisation problem having a quadratic objective function and linear constraints. For each vector a=a(omega) of model parameters and each design vector x one obtains an explicit representation of the "best" internal load distribution F*. Moreover, the expected recourse costs can be determined explicitly, where this function can be represented by means of a generalised "stiffness matrix" related to the given plastic analysis or optimal plastic design problem. Hence, corresponding to an elastic approach, the expected recourse function can be interpreted here as a generalised expected "compliance function", involving a generalised "stiffness matrix". Minimising the expected primary costs subject to constraints for the expected recourse costs (generalised "compliance"), or minimising the expected total primary and recourse costs, explicit finite dimensional parameter optimisation problems are obtained as deterministic substitute problems for finding robust optimal design x*, maximal load factors, respectively. The analytical properties of the resulting nonlinear programming problem are discussed, and applications, such as limit load/shakedown analysis problems, are considered. Furthermore, based on the expected "compliance function", explicit upper and lower bounds for the probability ps of survival can be derived using linearisation methods.
- [1]
- K. Marti, "Stochastic Optimization Methods", Springer-Verlag, Berlin-Heidelberg-New York, 2005.
- [2]
- K. Marti, I. Kaymaz, "Reliability analysis for elastoplastic mechanical structures under stochastic uncertainty", ZAMM, Vol. 86, No. 5, 358-384, 2006.
- [3]
- I. Kaymaz, K. Marti, "Reliability-based design optimization for elastoplastic mechanical structures", "Computer & Structures (CAS)", Vol. 85, Issue 10, 615-625, 2007.
- [4]
- K. Marti, "Computation of probabilities of survival/failure of technical, economic systems/structures by means of piecewise linearization of the performance function", to appear 2007 in "Structural and Multidisciplinary Optimization (SMO)".