Stochastic Structural Optimization with Quadratic Loss Functions

Structural Analysis and Optimal Structural Design under Stochastic Uncertainty using Quadratic Cost Functions are treated in this chapter: Problems from plastic analysis and optimal plastic design are based on the convex, linear or linearized 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.). Hence, in order to get robust optimal load factors x, robust optimal designs x, resp., the basic plastic analysis or optimal plastic design problem with random parameters has to be replaced by an appropriate deterministic substitute problem. As a basic tool in the analysis and optimal design of mechanical structures under uncertainty, a state function \(s^* =s^*(a,x)\) of the underlying structure is introduced. The survival of the structure can be described then by the condition \(s^* \le 0\) . Interpreting the state function \(s^*\) as a cost function, several relations \(s^*\) to other cost functions, especially quadratic cost functions, are derived. Bounds for the probability of survival \(p_s\) are obtained then by means of the Tschebyscheff inequality. In order to obtain robust optimal decisions \(x^*\) , i.e., maximum load factors, optimal designs insensitive with respect to variations of the model parameters \(a=a(\omega )\) , a direct approach is presented then 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 cost criterion is used. The minimum recourse costs can be determined then by solving an optimization 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 then an explicit representation of the best internal load distribution \(F^*\) . Moreover, also the expected recourse costs can be determined explicitly. The expected recourse function may be represented by means of a “generalized stiffness matrix”. Hence, corresponding to an elastic approach, the expected recourse function can be interpreted here as a generalized expected compliance function, which depends on a generalized “stiffness matrix”. Based on the minimization of the generalized compliance or the minimization of the expected total primary and recourse costs, explicit finite-dimensional parameter optimization problems are achieved for finding robust optimal design \(x^*\) or a maximal load factor \(x^*\) . The analytical properties of the resulting programming problem are discussed, and applications, such as limit load/shakedown analysis, are considered. Furthermore, based on the expected “compliance function”, explicit upper and lower bounds for the probability \(p_s\) of survival.