Finite Strip Modeling for Optimal Design of Folded Plate Structures

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1044 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054Fig. 1. Some examples of structures for which both the geometry and the material properties can be considered constants along a main direction.(a) Bridge decks with prestressed girders and box-girder bridges; (b) Prestressed prismatic and cylindrical shell roofs; (c) Prestressed thin-walledbeams.classical modeling techniques, eventual prestressing systemscan also be properly modeled by means of a set ofloads equivalent to the prestressing actions taking bothinstantaneous and time dependent losses into account.Basedon such kindof modeling, the choice of thedesign variables considers (a) geometrical quantities,like the thickness andthe dimensions of the finitestrips, (b) topological parameters, which define thelocation andthe connectivity of such elements, and(c)the characteristics of the eventual prestressing system,described by the prestressing forces and the cables profile.In particular, the design space of feasible structuralmorphologies is restrictedto satisfy physical andtechnologicallimits on the design variables, as well as givenlimit states on both the stress andthe displacementstates which define the structural behavior at the serviceabilitystage. In the selection of these design constraints,special attention is paidto the definition of aproper deformability index which accounts for theparticular nature of the structures investigatedin thepresent work. The optimal choice within this set offeasible solutions is basedon the definition of an objectivefunction to be minimizedwhich represents aweightedsum of the structural volume andthe totalprestressing force. The solution of the correspondingnon-linear optimization problem is achievedby using anumerical procedure based on the well-known complexmethod [2,12].Fig. 2. Reference systems and finite element model. (a) Finite strip element between the nodal lines i and j in the local reference system (x,y,z);(b) Finite strip modeling of the structure in the global reference system (X,Y,Z); (c) Thickness distribution of the cross-section.
A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054 1045In the following, the general criteria of the structuralmodeling and the basic formulation of the proposeddesign methodology are presented and explained. Specialattention is paidto the choice of the design variables,the definition of the design constraints and theformulation of the objective function. The effectivenessof the proposedmethodology is finally proven by someapplications to the optimal design of beams, vaults andbox-girder bridges.2. Formulation of the optimization problemThe problem of the optimum structural design consistsof finding a set of design variables which accountsfor assigneddesign constraints andoptimizes one ormore given target requirements. Therefore, from amathematical point of view, the purpose of the designprocess is to finda vector x which optimizes the valueof an objective function f(x), according to either sideconstraints with bounds x and x + , or inequalitygðxÞ 0 and/or equality hðxÞ ¼0 behavioral constraints.Since, without any loss of generality, minimizationproblems only may be considered, theprevious optimization problem is cast in the followingform:minff ðxÞjxx x x þ ; gðxÞ 0; hðxÞ ¼0g ð1Þwhich, in the general case, represents a non-linear programmingproblem.2.1. Choice of the design variablesIn structural design, the choice of the design variablesrepresents a crucial point which drives the wholedesign process. For the class of structures consideredhere, the design variables can be conveniently definedon the base of the finite strip model adopted for thestructural analyses. This means that the design modelandthe analysis model are strongly relatedbetweenthem, even if they express different points of view ofthe structural problem. Specifically, the analysis modelis defined as a set of folded plates interconnected toeach other in the longitudinal direction along one oftheir external nodal lines. Therefore, the design variableswhich define the design model can be chosen asdescribed in the following points (Fig. 3):– The structural shape is directly defined by thelocation of the nodal lines.– The structural size is relatedto the thickness of theFig. 3. Choice of the design variables. (a) Geometry: coordinates of the nodal lines and thickness of the finite elements; (b) Topology: structuralconfiguration; (c) Prestressing system: intensity of the prestressing forces andcoordinates of the control points of the cables profile.
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Finite Strip Modeling for Optimal Design of Folded Plate Structures

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