Finite Strip Modeling for Optimal Design of Folded Plate Structures

Engineering Structures 26 (2004) 1043–1054www.elsevier.com/locate/engstructFinite strip modeling for optimal design of prestressedfolded plate structuresAlessio Bergamini a , Fabio Biondini b,a Consultant Engineer, Via F. Turati, 23, 24057 Martinengo (BG), Italyb Department of Structural Engineering, Technical University of Milan, Piazza L. da Vinci, 32, 20133 Milan, ItalyReceived29 July 2003; receivedin revisedform 29 January 2004; accepted5 March 2004AbstractThe optimal limit state design of prestressed thin-walled folded plate structures under multiple loading conditions is presented.The design variables include (a) geometrical quantities, like the thickness and the dimensions of the structural members, (b) topologicalparameters, which define the location andthe connectivity of such elements, and(c) the characteristics of the eventual prestressingsystem, described by the prestressing forces and the cables profile. Besides the physical and technological limits on suchvariables, the design constraints account for given limit states on both the stress and the displacement states which define thestructural behavior. The objective of the design process is to find the structural layout which minimizes the structural volumeand/or the total prestressing force according to the mentioned side and behavioral constraints. The solution of the correspondingnon-linear optimization problem is achieved by using a numerical procedure based on the complex method. The structural analysesrequired during the optimization process are performed by using the finite strip method. Some applications to the optimaldesign of beams, vaults and box-girder bridges show the effectiveness of the proposed procedure.# 2004 Elsevier Ltd. All rights reserved.Keywords: Folded plate structures; Finite strip method; Prestressing; Structural optimization; Complex method1. IntroductionDesign, construction andmanagement of everyengineering system usually involve several technologicalandmanagerial decisions aimedto minimize therequiredeffort or maximize the desiredbenefit. Instructural design, the importance of this optimizationprocess is emphasizedwhen the scheme of the carryingmechanism follows the shape of the structure itself,becoming in this way a direct expression of its functionalrequirements. Clearly, the complexity of thedesign choices involved in this phase depends on thetypology of the considered structural system.There is a great deal of structures for which both thegeometry andthe material properties can be consideredconstants along a main direction, straight or curved,while, generally, only the loading distribution may vary. Corresponding author. Tel.: +39-02-2399-4394; fax: +39-02-2399-4220.E-mail address: biondini@stru.polimi.it (F. Biondini).In many cases, the performance of these structures isalso improvedby means of proper longitudinal prestressingsystems. Fig. 1 shows some of these structuraltypes, which refer to thin-walledbeams, cylindrical andprismatic shell roofs [1] and box-girder bridges [6,9,13].For these structures, the design process should lead todefine the optimal morphology of the transversal crosssection,which means its geometry, size, shape andtopology,as well as the layout of the prestressing system,described by the prestressing forces and the cables profile.In such context, the attention of this paper is focussedonthe optimal design of thin-walledprestressedstructures composedby foldedplates andsubjectedtomultiple loading conditions [3,4]. A proper modeling ofthese structures can be foundwithin the framework ofthe finite strip method(Fig. 2). As well known, thismethodis basedon the formulation of a special classof finite elements as long as the structure andinterconnectedalongthe nodal lines that constitute thesides of the strips themselves [5,7,8]. As for many other0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2004.03.005

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.

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

A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054 1047The corresponding behavior design constraint is thenformulatedas follows:g c ðxÞ ¼dðxÞ d max 0 ð6Þ2.3. Choice of the objective functionFig. 4. Limit surfaces (a) Iðs a Þ¼0 and(b) Iðs b Þ¼0, associatedtothe limit states on the membrane and bending stress field, respectively(n þ x ¼ n p; n x ¼ vn p ; n þ y ¼ ln p; n y ¼ l 0 n p ; m þ x ¼ m p; m x ¼ vm p ;m þ y ¼ lm p; m y ¼ l 0 m p ).structure:dðx;rÞ ¼ 1 ð Ldðx;r;yÞdyð4ÞL 0where d ¼ dðx;r;yÞ is the nodal displacement vector ofthe cross-section at the longitudinal location y for theloading condition r, d 0 ¼ d 0 ðx;r;y;aÞ is the same vectorassociatedto a rigidrotation a of the undeformedcross-section aroundits center of gravity in thedeformed configuration, the operator jj jj denotes theEuclidean vector norm, and L is the longitudinal lengthof the structure. In particular, the deformability indicesd give a measure of the deformability of both the crosssectionsandthe structure, respectively. Their valuesmay vary between 0 and1, which represent the limitingsituations of rigidandinfinitely deformable structuralconfiguration, respectively. Basedon these indices, thestructure is enforcedto maintain a sufficient rigidity byintroducing an indirect constraint on the maximumvalue of the global deformability index d ¼ dðxÞ:dðxÞ ¼maxdðx;rÞ d maxð5Þrwhere d max is a suitable upper boundof the index d(x).Several quantities able to represent the structuralperformance may be chosen as target requirements foroptimal design. A proper choice of the objective functiondepends on the nature of the problem and representsa strategic phase of the optimization process. Inthis paper, attention is focussedon both the materialvolume andthe prestressing force, consideredto be themain aspects which contribute to define the total costof the structure. The objective function is then chosenas a linear combination of the total volume V(x) ofthe structure andthe weightedprestressing forcePðxÞ ¼ P i b iP i ðxÞ as follows:f ðxÞ ¼w V f V ðxÞþw P f P ðxÞVðxÞ¼ w VVðx 0 Þ þ w PðxÞPð7ÞPðx 0 Þwhere b i is the weight associatedto the prestressingforce of the cable i ( P i b i ¼ 1:0), w V and w P are weightswhich express the relative importance of the dimensionlessobjective functions f V (x) and f P (x), respectively, andx 0 is a reference vector of initial design.3. Numerical solution of the optimization problemThe optimization problem previously formulatedissolvedby using a numerical technique derivedby theso-calledcomplex method[2,12] which is, as known, adeterministic non-linear mathematical programmingprocedure. The complex algorithm falls in the class ofdirect or zero-order methods, since it does not requirederivatives but only the evaluation of the involvedfunctions. Such characteristic makes the search processvery effective for the solution of the problems investigatedinthe present study, where the relationshipbetween the objective function andthe design variablesis generally only available in an implicit form. In thefollowing, some details about the numerical algorithmandthe relatedsolution process are given.3.1. The complex methodThe complex methodis basedon the generation of asequence of complexes. A complex is a geometrical figureconsisting of k nþ1 vertices, of the segmentsinterconnecting them andof the relatedpolygonalareas that enclose a region of the n-dimensional designspace. Clearly, only the non-degenerate complexes, i.e.those which enclose a non-zero volume, are feasible forthe development of the numerical procedure.

1048 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054Fig. 5.Basic steps of the complex method: (a) reflection; (b) expansion, and (c) contraction.The basic idea of the method consists in comparingthe values assumedby the objective function in correspondenceof the k nþ1 vertices of the complex and,in a secondtime, in moving the worst vertex x w in sucha way that some improvement is achieved(Fig. 5). Inparticular, after an initial complex is heuristicallydefined, the method moves towards an optimalsolution through several basic steps which start withthe reflection of the worst vertex x w , or the vertex associatedtothe largest value of the objective function,through the opposite face of the complex (Fig. 5(a)). Ifthe new vertex is again the worst, an improvement isobtainedthrough an expansion (Fig. 5(b)) or a contraction(Fig. 5(c)). On the contrary, a new vertex isselectedandthe procedure is repeateduntil an optimalsolution is found. In this way, instead of improving thebest result, the search strategy works by eliminating theworst one. Clearly, at each iteration, the basic stepsneed to be properly modified in order to enforce thepath of the search process into the feasible region ofthe design space. As a concluding remark, it is outlinedthat a proper selection of the number of verticesusually requires k ffi 2n. Moreover, the center of gravityof the initial complex can be assumedas a goodchoice for the reference vector of initial design x 0 .3.2. Convergence criteriaThe capability of the complex methodin exploring awide region of the design space—eventually emphasizedbyusing several restarts of the solution processwith different initial complexes—usually leads to avoidpoints of local minimum. The optimal solution foreach restart is considered to be reached when someconvergence criteria are satisfied. In particular, twoconditions must be contemporarily verified:(1) The dimensionless ratio e 1 between the maximumvalue of the distance of each vertex of the complexx i from its center of gravity x 0 andthe effectivedimensions of the admissible region is lower than aspecifiedtolerance e 1,max , or:e 1 ¼ max Dx 1 ðx i x 0 Þ e1;maxð8Þi¼1;...;kwhere23Dx 1 0 00 Dx 2 0Dx ¼ 674 5 ; with0 0 Dx nDx j ¼ x þ j x j ; j ¼ 1;2; ...;n ð9Þ(2) The dimensionless standard deviation e 2 of thevalues of the objective function f(x i ), computedatthe vertices of the complex i ¼ 1; 2; ...;k, withrespect to the reference value f(x 0 ) is lower than aspecifiedtolerance e 2,max , or:vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu1X k f ðx i Þ f ðx 0 Þ 2e 2 ¼ tk f ðxi¼10 Þvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu1X k f ðx i Þ 2¼ t 1 e 2;max ð10Þk f ðx 0 Þi¼13.3. Scaling of the design variablesThe complex methodhas been proven to be veryeffective in structural optimization. However, especiallydue to the different nature of the involved quantities, aproper scaling of the design variables should be adoptedin order to avoid numerical problems of ill-conditioning.To this aim, the design vector x is expressedas a functionof its physical limits x and x þ as follows:x ¼ x þ Yðx þ x Þ ð11Þand the non-zero dimensionless components of the diagonalmatrix Y which vary in the range [0;1], are assumedas new scaleddesign variables.4. ApplicationsAs already mentioned, in the optimal design of thinwalledfoldedplatesstructures for which both thegeometry andmaterial properties are constant along acoordinate direction, straight or curved, the choice ofthe structural morphology consists in defining both thegeometry andtopology of the cross-section perpendicularto such direction. In addition, the layout of theprestressing system, when it is present, must be alsodefined. Considering such a context, some design

A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054 1049applications are now presentedandbriefly describedinorder to highlight the effectiveness of the proposeddesign methodology. Such applications are alldeveloped with reference to the following assumption:– The behavior of the material, assumedto be isotropicandlinear elastic, is definedby the Youngmodulus E ¼ 30:0 GPa andthe Poisson coefficientm ¼ 0:15. The self-weight of the structure is definedby a weight density c ¼ 25:0 kN=m 3 . For the prestressingsystem, the instantaneous losses are evaluatedbyassuming a friction coefficient betweencables andducts l ¼ 0:2 anda non-intentional curvatureof the cables K 0 ¼ 0:005 rad/m.– The behavioral design constraints on the stress fieldare defined by the limit values n þ x ¼ n x ¼ nþ y ¼n y ¼ 12:0 t MN=m for the membrane forcesand m þ x ¼ m x ¼ mþ y ¼ m y ¼ 1:2 t2 MNm=m forthe bending moments, where t (m) denotes the thicknessof the corresponding plate. The minimum thicknesst min ¼ 0:1 m is assumedat both the ends of eachfolded plate.– The convergence of the search process is regulatedby the tolerance coefficients e 1;max ¼ e 2;max ¼ 0:01.4.1. Prestressed beamThe simply supportedbeam shown in Fig. 6(a), withlength L ¼ 30:0 m andcross-section having heightH ¼ 3:0 m and variable width, is considered. Besidesits self-weight, the beam is subjectedto a uniform loadq ¼ 100:0 kN=m, appliedat the top side of the crosssection,andto the action of a straight prestressingcable located near the bottom side. The beam is modeledbyusing 20 finite strip elements having the samegeometrical dimensions. The thickness distributionalong the cross-section which minimizes both the totalvolume andthe intensity of the prestressing force(w V ¼ w P ¼ 1:0), is searchedfor. The optimal solutionis characterizedby the shape of the cross-section shownin Fig. 6(b), with a total volume Vðx opt Þ¼1:087 m 3 =manda prestressing force Pðx opt Þ¼5:550 MN.4.2. Prestressed roof elementThe roof element shown in Fig. 7(a), with lengthL ¼ 20:0 m,widthB ¼ 2:5 m andheight H ¼ 1:0 m,isconsidered. A longitudinal straight cable, prestressedwith a force P ¼ 100:0 kN, is locatedwithin the bottomplate. Besides its self-weight and a uniform lineload Q 0 ¼ 3:0 kN/m appliedat the endof both thelateral wings, the roof element is subjectedto two alternativeloading conditions shown in Fig. 7(a) andcharacterizedbythe uniform loads q ¼ 2:0 kN=m 2 andv ¼ 1:0 kN=m 2 , respectively. The thickness distributionover the cross-section which minimizes the totalvolume is searchedfor. To this purpose, a constantthickness is assumedfor each plate, with the exceptionof the lateral wings, for which two alternative solutions,with andwithout a terminal bulb, are considered.The optimal solutions are characterizedby theshapes of the cross-section shown in Fig. 7(b), (c). Itclearly appears that a higher thickness of the endpartsof the wings can leadto a lower thickness of all theother plates (Fig. 7(c)). This latter optimal solution hasa total volume Vðx opt Þ¼0:538 m 3 =m.4.3. Barrel vault with edge beamsThe barrel vault shown in Fig. 8(a), with lengthL ¼ 30:0 m, width B ¼ 10:0 m andconstant thickness,is considered. The vault is supported by the two edgebeams having a rectangular cross-section with heighth ¼ 1:0 m. Besides its self-weight, the structure is subjectedto two alternative loading conditions shownin Fig. 8(a) andcharacterizedby the uniform loadsq ¼ 3:0 kN=m 2 and v ¼ 1:0 kN=m 2 , respectively. Theshape of the vault andthe thickness of both the vaultandthe edge beams which minimize the total volumeare searchedfor. To this purpose, the shape of thevault is described by a third degree polynomial curvedefined by the symmetrical location of the controlpoints B and C, as shown in Fig. 8(b). The optimal solution,that is characterizedby the shape of the vaultxFig. 6.Prestressed beam. (a) Main dimensions and load. (b) Optimal cross-sectional shape.

1050 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054Fig. 7. Roof element. (a) Main dimensions and loads. (b) Optimal thickness distributions, wings with constant thickness and (c) wings with aterminal bulb.andthe thickness distribution shown in Fig. 8(b), has atotal volume Vðx opt Þ¼0:753 m 3 =m.4.4. Bridge deck with prestressed girdersThe bridge deck shown in Fig. 9(a), with lengthL ¼ 40:0 m, width B ¼ 12:0 m andheight H ¼ 3:0 m,is considered. Besides its self-weight and the weight ofthe non-structural elements g p ¼ 4kN=m 2 , the structureis subjectedto the four alternative loading conditionsshown in Fig. 9(a), with q ¼ 8:0 kN=m 2 . Boththe girders are prestressed with a cable whose profile isdescribed by a third degree polynomial curve. Thethickness distribution over the cross-section and thecables profile which minimize both the total volumeandthe prestressing force (w V ¼ w P ¼ 1:0), are searchedfor.To this purpose, the width of the finite stripsis assumedto be variable. Moreover, two different constraintsfor the global deformability index d(x) are considered,respectively with (1) d max;1 ¼ 0:75 and(2) d max;2 ¼ 0:50. The optimal thickness distributionandthe cable layout are shown for both cases (1) and(2) in Fig. 9(b) and(c), respectively. The correspondingoptimal values of the total volume andof the prestressingforce are Vðx opt;1 Þ¼3:783 m 3 =m and Pðx opt;1 Þ¼7:079 MN for case (1), Vðx opt;2 Þ¼4:286 m 3 =m andPðx opt;2 Þ¼8:482 MN for case (2). It can be noticedthat solution (1), being associatedto a less restrictiveconstraint on the d-index, leads to lower values of volumeandprestressing with respect to solution (2).In order to better highlight the main aspects of thedesign process, Figs. 10–14 show the characteristicsof the optimal solution (2) andthe correspondingevolution of the search procedure. In particular, Fig. 10shows the distribution along the longitudinal coordinatey of the maximum values attainedover all theloading conditions by the components of the behavioraldesign vectors g a (x) and g b (x), relatedto theFig. 8.Barrel vault. (a) Main dimensions and loads; (b) Optimal shape and thickness distribution.

A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054 1051Fig. 9. Bridge deck with girders. (a) Main dimensions and loads. Optimal thickness distribution and cable location at the middle span for(b) d max;1 ¼ 0:75, and(c) d max;2 ¼ 0:50. (d) Optimal cable layout.Fig. 11. Bridge deck with girders—case (2). Global deformability indexd ¼ dðx;rÞ of the structure for each loading condition r ¼ 1; ...; 4.Fig. 10. Bridge deck with girders—case (2). Maximum design constraintvalues on (a) the membrane static field g a;j ðxÞ and(b) the bendingstatic field g b;j ðxÞ ðj ¼ 1; ...; 6Þ.Fig. 12. Bridge deck with girders—case (2). Distribution of the localdeformability index of the cross-sections d ¼ dðx;r;yÞ for each loadingcondition r ¼ 1; ...; 4.

1052 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054Fig. 13. Bridge deck with girders—case (2). Evolution of the tolerancecoefficients e 1 and e 2 for convergence criteria.membrane andthe bending stress state, respectively.From the inspection of such figures, which highlightthe more critical cross-sections with respect to eachlimit condition, it can be noticed that the design constraintsare nowhere violated. Fig. 11 refers insteadtothe deformability index d(x,r) which expresses, in anaverage sense, the kinematic displacement field. Alsoin this case, it appears that the upper limit d max;2 ¼0:50 is never violated. However, due to the averageFig. 14. Bridge deck with girders—case (2). Evolution of the objectivefunction f(x) andof its components f V (x) and f P (x) for the currentbest solution (w V ¼ w P ¼ 1:0; Vðx 0 Þ¼6:421 m 3 =m,Vðx opt Þ¼4:286 m 3 =m; Pðx 0 Þ¼16:654 kN, Pðx opt Þ¼8:482 kN).meaning of the global deformability index, some localviolations are expectedat the sectional level near thesupports for the more demanding loading conditions.This is shown in Fig. 12, which refers to the localdeformability index d ¼ dðx;r;yÞ computedfor eachcross-section.With reference to the path of the search process,Fig. 13 shows the evolution of the parameters e 1 and e 2Fig. 15. Box-girder bridge with internal diaphragms. (a) Main dimensions and loads. (b) Cross-sectional topologies. (c) Optimal shape, topologyandthickness distribution.

A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054 1053Fig. 16. Box-girder bridge with internal diaphragms. Distribution of the topologies at the vertices of the complex and evolution of the objectivefunction f ðxÞ DD f V ðxÞ for the current best solution.

1054 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054adopted to check the convergence of the iterativeprocedure. Finally, Fig. 14 shows the correspondingevolution of the objective function f(x) andof its componentsf V (x) and f P (x).4.5. Box-girder bridge deck with internal diaphragmsThe box-girder bridge deck shown in Fig. 15(a), withlength L ¼ 40:0 m, total width B ¼ 16:0 m andheightH ¼ 2:0 m, is considered. Besides its self-weight, theweight of the non-structural elements g p ¼ 4kN=m 2andthe service loadp ¼ 2:0 kN=m 2 , the structure issubjected to the four alternative loading conditionsshown in Fig. 15(a), with q ¼ 8kN=m 2 . The thicknessdistribution andboth the shape andtopology of thecross-section which minimize the total volume, are searchedfor.To this purpose, the topologies associatedtothe four alternative symmetrical cross-sections shownin Fig. 15(b), each of them derived from the first fundamentalone by eliminating or not one or more internaldiaphragms, are considered. The search processleads to an optimal solution which corresponds to thetwo-cellular box girder shown in Fig. 15(c), with a totalvolume Vðx opt Þ¼7:737 m 3 =m. It is worth noting thatthe high value of thickness of the bottom slab intension is due to the hypothesis of homogeneousmaterial andelastic behavior. The disposition ofproper longitudinal reinforcement, designed with referenceto the effective composite andnon-linear nature ofthe material, clearly allows a lower thickness of theconcrete slab.Finally, with reference to the path of the search process,Fig. 16 shows the distribution of the topologies atthe vertices of the complex andthe evolution of theobjective function f(x) for the current best solutionduring the iterative process.5. ConclusionsThe problem of finding the optimal shape, size andtopology of prestressed folded plate structures undermultiple loading conditions has been investigated. Thedesign problem has been formulated as a mathematicaloptimization problem, accounting for both static andkinematic constraints, andit has been solvedby usinga numerical algorithm basedon the complex method.The structural analyses needed for the optimizationprocess have been performedby using the finite stripmethod. A number of applications have shown theeffectiveness of the proposedprocedure.In particular, the design tools presented in this papercan usefully be appliedat the conceptual design stage,where the designer is usually interested in comparingbetween them a variety of alternative optimal solutions,derived, for example, by using different designmodels. These solutions, which the proposed procedureidentifies with wide generality, rationality and objectivity,can be then usedas a basis for a more detaileddesign of the reinforcement. Clearly, future developmentsare expectedin order to obtain a better controlof the design problem, especially on the definition ofthe design constraints, which should also account fortechnological aspects, as well as for aesthetical andadditional functional requirements.References[1] ASCE. Design of cylindrical concrete shell roofs. Manual ofengineering practice. New York (NY): ASCE; 1952, p. 31.[2] Belegundu AD, Chandrupatla TR. Optimization concepts andapplications in engineering. Upper Saddle River (NJ): Prentice-Hall; 1999.[3] Bergamini A, Biondini F. Optimisation of folded plate structures.Proceedings of the Second International Conference onAdvances in Structural Engineering and Mechanics (ASEM02),Seoul, Korea, August 21–23. 2002.[4] Biondini F, Bontempi F, Malerba PG. Ottimizzazione di formadi strutture a folded-plates. Proceedings of the 13th C.T.E. Conference,Pisa, Italy, November 9–10–11. 2000 [in Italian].[5] Cheung YK. Finite strip methodin structural analysis. Oxford:Pergamon Press; 1976.[6] Kristek V. Theory of box girder bridges. Prague: John WileyandSons; 1979.[7] Loo YC, Cusens A. The finite strip methodin bridgeengineering. London: E. & F.N. Spon; 1978.[8] Malerba PG, Toniolo G. Metodi di discretizzazionedell’analisi strutturale. Milan, Italy: Masson Editore; 1981[in Italian].[9] Martinez Y Cabrera F, Menni C. I ponti a cassone monocellularea profilo deformabile. Technical Report, 33. Istituto diScienza e Tecnica delle Costruzioni, Politecnico di Milano,Tamburini Editore, Milan, 1974 [in Italian].[10] Mortenson ME. Mathematics for computer graphicsapplications. New York (NY): Industrial Press; 1999.[11] Nielsen MP. Limit analysis andconcrete plasticity. Boca Raton(FL): CRC Press; 1999.[12] Rao SS. Engineering optimization—theory andpractice. NewYork (NY): John Wiley andSons; 1996.[13] Schlaich J, Scheef H. Concrete box-girder bridges. Structuralengineering documents, 1e. International Association for BridgeandStructural Engineering (IABSE); 1982.