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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 16
CIVIL ENGINEERING COMPUTATIONS: TOOLS AND TECHNIQUES Edited by: B.H.V. Topping
Chapter 8
Structural Analysis and Optimal Design under Stochastic Uncertainty with Quadratic Cost Functions K. Marti
AeroSpace Engineering and Technology, Federal Armed Forces University Munich, Neubiberg/Munich, Germany K. Marti, "Structural Analysis and Optimal Design under Stochastic Uncertainty with Quadratic Cost Functions", in B.H.V. Topping, (Editor), "Civil Engineering Computations: Tools and Techniques", SaxeCoburg Publications, Stirlingshire, UK, Chapter 8, pp 173197, 2007. doi:10.4203/csets.16.8
Keywords: structural analysis, optimal plastic design, random model parameters, survival conditions of plasticity theory, state functions, robust decisions, quadratic cost functions, deterministic substitute problems, stochastic nonlinear programming.
Summary
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. [24]. 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 p_{s} 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 p_{s} of survival can be derived using linearisation methods. References
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