Finite Strip Modeling for Optimal Design of Folded Plate Structures

1046 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054finite strips. In particular, by assuming the thicknessof each plate as linearly varying, the thickness valuesat the nodal lines completely define the thickness distributionover the whole structure.– The structural topology depends on the mutualarrangement of the strips. In this sense, several alternativestructural topologies, for example, identifiedthrough an integer variable, needto be consideredduring the design process. This can be done bybuilding a family of derived topologies by the eliminationof one or more folded plates from a given basictopology.– The prestressing system is representedby a set ofpost-tensionedcables at one or both of their endsandit is fully definedby the intensity of the prestressingforces andby the longitudinal profile of thecables. In this study, the curvilinear profile of thecables is described by means of Bézier’s curves [10],which are polynomial curves of degree n 2 definedby the position of nþ1 control points.Basedon the above mentionedcriteria, in the proposedformulation,the components of the design vectorx can be identified as quantities belonging to one ofthe following classes:(a) Geometry: coordinates of the nodal lines andthickness of the finite elements;(b) Topology: integer value which identifies the structuralconfiguration;(c) Prestressing system: intensity of the prestressingforces andcoordinates of the control points whichdefine the cables profile.2.2. Definition of the behavioral design constraintsThe dimensions and the components of the vectorsg(x) and h(x) are clearly depending on the particulardesign problem which has to be solved. They representsome restrictions on the system behavior or performance,expressedas a function of the design variablesboth in explicit andimplicit way. In particular, thestructural performance at the serviceability stageusually drives the design process for the class of structuresconsidered here. Therefore, since the structuralresponse under the serviceability loads can be effectivelymodeled in the linear elastic range, the behavioraldesign constraints gðxÞ 0 and hðxÞ ¼0assumedin the proposedformulation deal with boththe static andkinematic fields evaluatedby means of alinear elastic analysis.With respect to the static field, the stress states ¼ sðxÞ must not leadto mechanical crisis relatedforexample to the local rupture of the materials. For thespecial class of structures investigatedhere, the totalstress fieldis generally given by the superposition of atransversal behavior, mainly regulatedby the bendingstresses at the local level, andof a longitudinal behavior,mainly regulatedby the membrane stress fieldat theglobal level. For this reason, the failure conditions onboth the membrane s a ¼ s a ðxÞ ¼½n x n y n xy Š T andbending s b ¼ s b ðxÞ ¼½m x m y m xy Š T stress fields,referredto the directions of orthotropy for non-isotropicstructures, are assumed to be independent between themanddefinedby the following design constraints:23n x n xn x n þ xn y n yn y n þ yn 2 xy ðn þ x n x Þðn þ y n y ÞgðxÞ¼ g n 2 xy ðn x þn x Þðn y þn y ÞaðxÞ¼ 0 ð2Þg b ðxÞm x m xm x m þ xm y m ym y m þ y64m 2 xy ðm þ x m x Þðm þ 7y m y Þ5m 2 xy ðm x þm x Þðm y þm y Þwhere the non-negative quantities n x ; n þ x ; n y ; nþ y andm x ; m þ x ; m y ; mþ y represent the limit values for theelementary stress states of mono-axial tension/compressionandsimple shear for the membrane andthebending stress fields respectively. In particular, the constraintsg a ðxÞ 0 and g b ðxÞ 0 are described in thestress space, by two limit surfaces, Iðs a Þ¼0andIðs b Þ¼0for the membrane andbending stress field, respectively,each of them defined by a couple of cones as shown inFig. 4 [11]. When the hypothesis of independent failureconditions cannot be assumed, a limit surface IðsÞ ¼0taking into account the interaction of the membrane andbending stress fields can also be effectively derived froma linear interpolation of the limit surfaces Iðs a Þ¼0 andIðs b Þ¼0. Clearly, the constraints gðx; r; yÞ 0 must beverifiedfor each loading condition r andin each point ofthe structures, or along the longitudinal coordinate y ofeach finite strip. In order to reduce the problem toalgebraic form, the constraints are verifiedonly in afinite number of transversal cross-sections.From the kinematic point of view, the displacementfield d ¼ dðxÞ shouldnot leadto loss of form. For thispurpose, it is introduced a suitable measure of deformationrepresentedby a local deformability indexd ¼ dðx;r;yÞ of the cross-sections:kdðx;r;yÞd 0 ðx;r;y;aÞkdðx;r;yÞ ¼ minð3Þpap kdðx;r;yÞkandby a global deformability index d ¼ dðx;rÞ of the