PSI - Issue 8

G. Fargione et al. / Procedia Structural Integrity 8 (2018) 566–572 Author name / Structural Integrity Procedia 00 (2017) 000–000

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Instead, if the generic component C i depends on other system components, the corresponding constraints between components {CbC} i must be met. They can be of three types: • Constraints on geometric parameters G v and/or G F . In this case the constraint limits for G V and/or G F depend on geometric properties of other components. • Constraints on materials. In this case some constraints on material properties M T P R depend on the material properties of other components. • Functional constraints. In this case the component is related to another component at functional level. As a consequence, in equation type (2) and (3) the generalized function f 2 depend on properties of the other component also. In this last step of the procedure, the characterizations of individual components (Section 2.3) and the system (Section 2.4) converge in the formalization of the problem of optimal materials choice and components dimensioning, for a system constituted by the set {C} of n c components, that can be formulate as follows: define the geometric parameters {G} and the materials {M} of all the set of components {C}, that optimize (or balance) the set {PF} of type (1) objective functions, while respecting the whole set of type (2) or (3) constraints on requirements {R}, the constraints on geometric parameters {G V } and {G F } and on material properties {M T P R } for all the set of components {C}, and the set {CbC} of all constraints between components {C} in the whole system. As anticipated earlier, a further feature of the method is that searching for the solution, which must include the optimal distribution of materials between system components, and the sizing of components, is guided by applying an efficiency principle, which presupposes the choice of the material and the sizing of components must be calibrated on the real performance needs. This principle translates into a further type of performance equation, which will be referred to as the Efficiency Function EF, expressed as: [ ] F V T R f R,G ,G ,M P EF CR 3 1 = − = (4) This type of equation quantifies the deviation of the Constraint Ratio CR (3) as regards to the unit value. Recalling that CR is the ratio between the generalized resistance of the component and the generalized stress to which it is subjected, the Efficiency Function EF, if minimized, under the condition CR ≥ 1, ensures that the constraints on requirement of type (2) is respected in the most efficient manner possible, that is, so that the component resistance exceeds the stress the component is subjected to, minimizing the gap between the two factors. This means calibrating the choice of material and free geometric variables in order to ensure constraint on the requirement, avoiding to select materials of excessively and unnecessarily high performances over the requirement, and to exceed on overdimensioning. In the formulation of the optimal material selection and component sizing problem, the Efficiency Functions set {EF} must be put beside the other performance functions set {PF}. After completing the detailed formulation of the problem, it is necessary to search for the optimal solution, that is, to define the geometric parameters {G} and materials {M} for the whole set of components {C} constituting the system, to optimize (or obtain a balancing) of all objective functions {PF} and {EF}, while respecting all constraints (on requirements to be constrained, on geometric parameters, on materials, between components). Finding the optimal solution is a much more complicated task than the more complex the system is. The proposed formulation is suitable for the use of various types of tools for multi-objective optimization, such as those introduced by Ashby (2000). In all cases, the objective functions {PF} and {EF} can be collected in a single overall function, using simple multi-attribute analysis models, that allow to develop unified functions, such as weighted sum of objective functions also not homogeneous, by means of normalization procedure proposed by Farag (2002). The introduction of weight coefficients allows to qualify the overall function, depending on the distribution of the values to be attributed to these coefficients. In this way, for each potential solution, it is possible to calculate the value of the unified objective function in order to classify the potential solutions and find the optimal one. 2.5. Optimal materials choice and components dimensioning