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*Correspondence to: S. Kravanja, Faculty of Civil Engineering, University of Maribor, Slovenia 17, SI-2000 Maribor,Slovenia. E-mail: stojan.kravanjauni-mb.siCCC 00295981/98/02032936$17.50Received 12 August 1996( 1998 John Wiley & Sons, Ltd.Revised 5 January 1998INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng. 43, 329364 (1998)THE MINLP OPTIMIZATION APPROACH TOSTRUCTURAL SYNTHESISPART III: SYNTHESIS OF ROLLER AND SLIDING HYDRAULICSTEEL GATE STRUCTURESS. KRAVANJA1,*, Z. KRAVANJA2 AND B. S. BEDENIK11niversity of Maribor, Faculty of Civil Engineering, Smetanova 17, SI-2000 Maribor, Slovenia2niversity of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, SI-2000 Maribor, SloveniaABSTRACTPart III of this three-part series of papers describes the synthesis of roller and sliding hydraulic steel gatestructures performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLPapproach enables the determination of the optimal number of gate structural elements (girders, plates),optimal gate geometry, optimal intermediate distances between structural elements and all continuous andstandard crossectional sizes. For this purpose, special logical constraints for topology alterations andinterconnection relations between the alternative and fixed structural elements are formulated. They havebeen embedded into a mathematical optimization model for roller and sliding steel gate structures GATOP.GATOP has been developed according to a special MINLP model formulation for mechanical superstruc-tures (MINLP-MS), introduced in Parts I and II. The model contains an economic objective function ofself-manufacturing and transportation costs of the gate. As the GATOP model is non-convex and highlynon-linear, it is solved by means of the Modified OA/ER algorithm accompanied by the Linked Two-PhaseMINLP Strategy, both implemented in the TOP computercode. An exampleof the synthesis is presented asa comparative design research work of the already erected roller gate, the so-called Intake Gate in Aswan IIin Egypt. The optimal result yields 29)4 per cent of net savings when compared to the actual costs of theerected gate. ( 1998 John Wiley & Sons, Ltd.KEY WORDS:structural synthesis; optimization; topology optimization; discrete variable optimization; Mixed-IntegerNon-linear Programming; MINLP; the Modified OA/ER algorithm; MINLP strategy; hydraulic gate; sliding gate; rollergate; Aswan1. INTRODUCTIONThis paper describes the Mixed-Integer Non-linear Programming (MINLP) approach to thesynthesis of roller and sliding gate structures, i.e. the simplest types among vertical-lift hydraulicsteel gates, see Figure 1. Roller and sliding gates are also regarded as the most frequentlyFigure 1. Vertical-lift hydraulic steel gate structuremanufactured types of hydraulic steel gates for headwater control. They are used to regulate thewater stream on hydro-electric plants, dams or spillways.As hydraulic steel gates are very special structures, only a few authors have discussed theiroptimization, e.g. Kravanja et al.,13 Jongeling and Kolkman4 as well as Almquist et al.5Particular interest was shown in the optimization not of these (roller and sliding gates) but ofsimilar structures. In such investigations, Vanderplaats and Weisshaar6 as well as Gurdal et al.7optimizedstiffenedlaminated compositepanels, Butler8 and Ringertz9 optimized stiffenedpanels,Farkas and Jarmai10 optimizedwelded rectangular cellular plates, Finckenoret al.11 treated skinstringer cylinders and Gendy et al.12 stiffened plates. Almost all authors used Non-linearProgramming (NLP) techniques. Gurdal et al.7 proposed the genetic algorithm, while Kravanjaet al.13 introduced MINLP algorithms and strategies to the simultaneous topology and para-meter optimization of the gate.In Parts I of this three-part series of papers, a general view of the MINLP approach to thesimultaneous topology and parameter optimization of structures is presented. Part II describesthe extension to the simultaneous standard dimension optimization. Based on the superstructureapproach defined in Parts I and II, the main objective of this paper (Part III) is the MINLPsynthesis of roller and sliding hydraulic steel gate structures, obtained at minimal gate costs andsubjected to defined design, material, stress, deflection and stability constraints. As the MINLPapproach enables simultaneous topology, parameter and standard dimension optimization,a number of gate structural elements (girders and plates), the gate global geometry, intermediatedistances between structural elements and all continuous and standard dimensions are obtainedsimultaneously.This last part of the three-part series of papers is divided into three main sections:1. Section 2 describes how different topology and standard dimension alternatives are postu-lated and how their interconnection relations are formulated by means of explicit logicalconstraints in order to perform topology and standard dimension alterations within theoptimization procedure.330S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Figure 2. Gate superstructure, constructed by three main gate element, each containing six horizontal and nine verticalgirders2. Section 3 represents a general MINLP optimization model for roller and sliding gatestructures GATOP.3. Finally, in Section 4, the proposed superstructure MINLP approach is applied to thesynthesis of an already erected roller gate, the so-called Intake Gate in Aswan II in Egypt.2. SUPERSTRUCTURE ALTERNATIVES AND THE MODELLING OFTHEIR DISCRETE DECISIONS2.1. Postulation of topology and standard dimension alternativesThe first step in the synthesis of the gate is the generation of an MINLP superstructure inwhich different topology/structure and standard dimension alternatives are embedded to beselected as the optimal result. The gate superstructure also contains the representation ofstructural elements which may construct each possible structure alternative as well as differentsets of discrete values, defined for each standard dimension alternative. As both the roller andslidinggates have almostthe same structure, it was reasonable to propose a superstructure,whichcould well be useful for both of them.2.1.1. opology alternativesThe gate superstructure typically includes a representation of main gate elements, where eachgate element is composed of various structural elements, such as horizontal girders, verticalgirders, stiffeners and plate elements of the skin plate, see Figure 2. The superstructure comprisesn main gate elements, n3N, each containing m horizontal girders, m3M, the (3#2v) number ofvertical girders through the entire gate, v3, and the corresponding (m!1)(2#2v) number ofskin-plate elements.THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III331( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)To each mth horizontal girder of the nth main gate element an extra binary variable yn,misassigned. The number of horizontal girders and corresponding plate elements of the skin plate,distributed over the nth main gate element, can therefore be determined simply by+myn,m. Notethat the proposed minimal number of identical vertical girders is 3 and that they can take onlyodd numbers. If a binary variable yvis assigned to each v, v3, the number of vertical girders canbe obtained by (3#2+vyv). In the same way an even number (2#2+vyv) of equal horizontalpartitions of the entire gate is proposed. In the case of vertical girders, we can see that thestructural elements can also be determined by suitable linear combinations of binary variables.Among the maximal number M.!9of horizontal girders per each main gate element, the upperand lower girders together with the minimal number M.*/of intermediate horizontal girders andtheadjoining skin-plateelements are treatedas fixedstructural elements,which are always presentin the optimization. All other (M.!9!M.*/) intermediate horizontal girders with the corres-ponding number of adjoined skin-plate elements are then regarded as alternative structuralelements, which may either disappear or be selected. Since only alternative structural elementsparticipate in the discrete optimization, the size of the discrete decisions is significantly reduced.Each possible combination between selected alternative structural elements and fixed structuralelements forms an extra gate topology/structure alternative.2.1.2. Standard dimension alternativesFour standard dimensions are additionally defined to represent the standard thicknesses ofsheet-iron plates: the thickness of the skin-plate tsnfor each nth main gate element, the flangethickness of the horizontal girder tfn, the web thickness of the outer horizontal girder t065wn,m, m1or mM, and the web thickness of the inner horizontal girder t*/wn,m, 1(m(M. Since thethickness tsnhas a common value for the entire skin-plate of the nth main gate element and thetfnare the same for all horizontal girders of the nth main gate element, i.e. they correspond tothe common standard design variables for the superstructure or its part d45,comtfrom the specialMINLP-MS model formulation in Part II. Similarly, the web thicknesses t065wn,m, which takea common value for both outer horizontal girders of the nth main gate element, and t*/wn,m, whichare the same for all the inner horizontal girders, correspond to the common standard designvariables d45,#0.a,cof the alternative structural elements. An extra set of discrete values of standarddimension alternatives and a special set of the same size of binary variables are introduced foreach mentioned standard dimension.Each standard dimension tsnshall be expressed within the given i standard dimension alterna-tives, i3I, standard dimension tfnby k alternatives, k3K, standard dimension t065wn,mby p alterna-tives, p3P, and standard dimension t*/wn,mby r alternatives, r3R. The vector of i binary variablesyn,iand the vector of i discrete values q/,*are assigned to the variable tsn, the vectors of k elementsyn,kand q/,to the variable tfn, the vectors of p elements yn,pand q/,1to the variable t065wn,mand thevectors of r elements yn,rand q/,3to the variable t*/wn,m. Consequently, the overall vector of binaryvariables assigned to the gate superstructure is yMyn,m, y1,v, yn,i, yn,k, yn,p, yn,rN.2.2. Modelling of discrete decisionsThe postulated gate superstructure of topology and standard dimension alternatives can beformulated as an MINLP problem using a special MINLP model formulation (MINLP-MS) forsimultaneous topology, parameter and standard dimension optimization of mechanical super-structures, described in Part II. As can be seen from the (MINLP-MS), the objective function is332S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.typically subjected to structural analysis and logical constraints. While structural analysisconstraints represent a mathematical model of a synthesized structure, logical constraints areused for the explicit modelling of logical decisions.Modelling of discrete decisions to determine topology alternatives is an objective of the highestimportance for the synthesis. In order to perform these decisions within the MINLP optimiza-tion, interconnection logical constraints Dy#R(d,p)r are proposed. While variables, theirboundsand mostof theconstraintsof theMINLP-MSmodel formulationare representedin PartII, interconnection logical constraints and the constraints defining topology alterations aredescribed in this section. The latter ones are derived from the following basic integer or mixed-integer logical constraints:(a) Multiple choice constraints for selecting among a set of units I:Select exactly M units:+i|IyiM(1)Select M units at the most:+i|Iyi)M(2)Select at least M units:+i|Iyi*M(3)(b) If then conditions:if unit k is selected then unit i must be selected: yk!yi)0(4)(c) Activation or deactivation of continuous variables, inequalities or equations:1. example to relate continuous variable x to the scalar value :x y(5)if y1Nx, if y0Nx02. an opposite relation:x(1!y)(6)if y1Nx0, if y0Nx3. example for the bounds of continuous variable x:xLOy)x)xUPy(7)if y1NxLO)x)xUP, if y0N0)x)0(d) Integer cuts constraint eliminates unnecessary integer combinations ykMykiDi1, . . . , mN3M0,1Nm, e.g. those found at previous MINLP iterations:+i|BIkyi! +i|NIkyi)DBIkD!1(8)whereBIkMiDyki1N,NIkMiDyki0NIn order to describe the modelling of discrete decisions, a general gate superstructure fromFigure2 is addressedin whichthe defined structuralelementsare typically horizontaland verticalTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III333( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Figure 3. Vertical cross-section through the gate: (a) gate superstructure with fixed ( ) and alternative (- - -) structuralelements; (b) minimal structure with only fixed structural elementsgirders. As the modelling of vertical girders is simplified and needs no special interconnectionlogical constraints, the modelling of discrete decisions regarding horizontal girders proved to bemore sophisticated.2.2.1. Modelling of topology alterationsLet us consider the vertical cross-section of the gate element superstructure with fixed andalternative horizontal girders, see Figure 3(a). The number of fixed and alternative girders andtheir locations in the superstructure can be described by the following logical constraints:M.*/) +m|Mym)M.!9(9)ym)ym1,m2, 3, . . . , M!1(10)yM.!91(11)Logical constraint (9) defines the minimal (M.*/) and maximal (M.!9) number of structuralelements (girders). While number M.*/represents the number of fixed structural elements, thedifference between the maximal and minimal number of elements (M.!9!M.*/) gives thenumber of alternative structural elements. Constraint (10) defines the direction of the removal ofalternative elements: from the top down the superstructure. From Figure 3 is evident that themostupperelementis the fixed one, whichis setby the constraint(11). It then furtherfollows from334S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.constraint (10) that all the rest fixed elements are located at the bottom of the superstructure.Hence, constraints (9)(11) represent the explicit model for topology alterations of horizontalgirders.2.2.2. Modelling of interconnection relationsInterconnection relations between alternative and fixed structural elements within the super-structure require special attention paid to the structural synthesis performed by the MINLPapproach. Interconnection relations either restore the connections between currently selected(existing) structural elements or cancel the relations between currently rejected (disappearing)structural elements. Since MINLP methods optimize the topology and parameters simulta-neously,it is necessaryto definethese interconnectionrelations in an expliciteequationalform, sothat they can enable interconnections and disconnections between the elements during theoptimization process. Special interconnection logical constraints for interconnection relationsbetween the alternative and fixed structural elements have been proposed.They will be embeddedinto the MINLP optimization model of the gate structure, enabling the latter to thus becomeself-sufficient for automatic topology and parameter optimization.The modelling of interconnection logical constraints, however, requires additional effort, sincemost element constraints include functions not only of their own variables but also of thevariables belonging to their adjoining structural elements. Such an example is, e.g. the constraintof the moment of inertia In,mof the mth horizontal girder of the nth main gate element (seeequation (23) in the following section), which includes the substituted expression (S6) of theskin-plate effective width bsn,mwith the heights (between girders and the sill) hm1and hm1of bothadjoining girders. The constraints of the mth intermediate horizontal girder are typically func-tions of three heights: hm1, hmand hm1, and two vertical distances between horizontal girders:dhm1and dhm. The distance dhmis simply defined by the constraintdhmhm1!hm,m2, 3, . . . , M!1(12)Theproblem arises if hm1is not defined when the adjoiningupper alternativegirderto the mthhorizontal girder does not exist. For example, let us consider the third girder in Figure 3(a) whichis the uppermost existing intermediate element. In order to define h4so as to fulfil the constraintsof girder 3, h4should temporarily become equal to the height of the uppermost fixed girderh6hM(Figure 3(b). The main idea is to set all heights of non-existing intermediate girders(girders 4 and 5 in Figure 3(a) to the value of h6by means of the logical constraintsdhm)d61).)ym,m2, 3, . . . , M!1(13)dhm*d-08,%9).)ym,m2, 3, . . . , M!1(14)Note, that constraints (13) and (14) restore the upper d61).and lower d-08,%9).bounds of thedistance dh.when the corresponding girder exists (ym1) and set it to zero, otherwise. When thedistance is zero, it follows from constraint (12) that hmbecomes equal to hM. In this way alldistancesand heightsare definedfor anygirder thatbecomesthe uppermostselectedintermediateone and re-establishes its connection to the uppermost fixed girder.As the uppermost selected intermediate girder is connected to the uppermost fixed girder (e.g.girder 3 to girder 6 in Figure 3(a), the latter should also, in the similar way, be connected to theformerone (girder 6 to girder3 in Figure3(a). Constraints for the uppermostfixed girderare thenjust functions of two heights: hMand hM1, and a distance dhM1. The problem arises if someTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III335( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)intermediategirdersdonot exist,e.g. girders 4 and 5 inFigure 3(a). Insuch cases,hM1should notbe considered. Instead, the height hs(h3in Figure 3(a) of the upper selected intermediate girdershould be defined and substituted for hM1. The vertical distance dhsof the uppermost selectedintermediate girder is then defined by the constraint:dhshM!hs(15)The selection of the height hsamong all hmcan be performed by the following logical constraints:hs*hm!h614(1!ym1)!h614(1!ym)!h614)ym1,m2, 3, . . . , M!2(16)hs)hm#h614(1!ym1)#h614(1!ym)#h614)ym1,m2, 3, . . . , M!2(17)hs*hM1!h614(1!yM1)(18)hs)hM1#h614(1!yM1)(19)Constraints (16) and (17) set hsto the height hmof that mth existing horizontal girder (ym1),which has the existing adjoining lower girder (ym11) and the disappearing adjoining uppergirder (ym10). However, for mM!1 the next upper girder always exists, since it is fixed,i.e. (yM1). In this case we need additionalconstraints, i.e. (18) and (19), which set hsto the heighthM1.3. THE MINLP OPTIMIZATION MODEL FOR ROLLER AND SLIDINGHYDRAULIC STEEL GATE STRUCTURESGATOPAn MINLP mathematical optimization model for roller and sliding hydraulic steel gate struc-tures GATOP (GATe OPtimization) has been developed. The model has proven efficient for thesynthesis of roller and sliding gates. As an interface for mathematical modelling and datainputs/outputs GAMS (General Algebraic Modelling System) by Brooke et al.,14 a high-levellanguage has been used. The first version of GATOP was developed to perform NLP optimiza-tion problems of fixed gate structures, while the dead weight of the gate structure was consideredin the objective function, see Reference 15. The new GATOP version is a much more general one:many alternative horizontal girders, vertical girders and plate elements of the skin-plate are nowsimultaneously represented in a composite form of the gate superstructure. Thus, the newGATOP model is formulated as an MINLP problem performing gate synthesis.In the process of the development of the GATOP model, the following assumptions andsimplifications have been considered:1. A simplified static system for roller and sliding gates is to be used. It includes independentsimply supported horizontal girders that are combined with independent clamped skin-plate elements. Such a static system is convenient for gates in which the horizontal girdersare much longer than their intermediate vertical distances. Vertical girders have the sameheight as horizontal ones.2. In the above case, the horizontal girders transmit almost all the water load, so that theparticipation of vertical girders can be neglected. Although the vertical girders are notanalysed, they are nevertheless considered as a geometrical and economic fact in theobjective function.3. Only the water load, i.e. the hydrostatic pressure on the skin plate, is taken into considera-tion, while the dead weight, friction and buoyancy are neglected.336S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.The MINLP model GATOP has been developed according to the special MINLP modelformulationfor mechanical superstructures(MINLP-MS),where logical constraints for topologyalterations and interconnection relations between structural elements, proposed in the previoussection, have been included. Thus, the superstructures economic objective function, structuralanalysis constraints and logical constraints, such as pure integer logical constraints, interconnec-tion logical constraints, logical constraints for continuous variables with bounds and relations aswell as logical constraints for standard design variables, have been defined. The model alsocontainstheinitializationsofvariables. Heuristicand MILP constraintsare additionallyincludedto perform the optimization faster and more efficiently. A list of symbols of the GATOP model isshown in Appendix I.3.1. Input dataAll necessary input data, such as geometry boundaries of the gate, design requirements for thegate, water opening, water load, material characteristics, stresses, deflections and stability allow-ances are put into the model, see constants (input data) in the list of symbols in Appendix I andFigure 4. The topology boundaries of the gate superstructure are included as the maximal andminimal number of girders and skin-plate elements.3.2. Objective functionThe economic objective function is defined by equation (20). The objective of the optimizationis to minimize the objective variable cost which represents self-manufacturing costs of the gate,such as costs of material: c.!5%3)o)4&70-, the sheet-iron cutting costs: c#65)4A, welding costs:c8%-$)4&8%-$, anti-corrosion resistant painting costs: c!#31)S4&!#31as well as transportation costs:c53!/41)o)4&70-. The optimal result with an optimal gate topology (optimal gate structure) andoptimal parameters (dimensions) will be obtained as the trade-offof the mentioned costs. Notethat index n is used for main gate elements, m for horizontal girders and v for vertical girders. Seealso the list of symbols in Appendix I:min:cost(c.!5%3#c53!/41)o)4&70-#c#65)4A#c8%-$)4&8%-$#c!#31)S4&!#31(20)where:4&70-+n|NM(L4#2b*4)tsn#(2b4&#(dg!1)b7&)tfn#(2t4#(dg!1)t78)bwn#(2b*4#2b4&#L4#(dg!1)b7&)d#03)hgn#(L4!4b4&/3!b7&(dg!1) +m|MMbfn,m(tfin,m#d#03)N#L4) +m|MMbwin,m(twn,m#d#03)NN(S1)4A+n|NM2L4#4b*4#4b4&#(m1#4d)h/#(m2#2d)b8/!2dg+m|MMbfn,mN#(m3#2d) +m|MMb8*/,.N#(4L4!8b4&/3) +m|MMyn,mNN(S2)THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III337( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Figure 4. Roller gate338S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.4&8%-$+n|NM(m4#4d)h/#m5+m|MMb8*/,.N#4L4+m|MMyn,mNN(S3)S4&!#31+n|NM(m6#2dg)bwn#2dg#4b4b*4#2L4!2)hgn# (2L4!8b4&/3#(2!2dg)b7&) +m|MMbfn,mN#2L4+m|MMbwin,mNN(S4)wheremi: sliding gate:roller gate:m16m114m22m26m30m34m44m412m58m516m62m663.3. Structural analysis constraintsStructural analysis non-linear and linear constraints (21)(79) define the design, stress, deflec-tion and stability constraints for the main gate elements, i.e. girders and skin-plate elements, aswell as for the entire gate.The non-linear and linear parameter constraints (21)(23) define the cross-section area charac-teristics, i.e. the coefficients of the effective width of the skin plate for the horizontal girders upperflange,ln,m, eccentricitydistances from the outer edgeof the horizontal girder flange to the gravitycentre of the girder section, en,m, and the moment of inertia of the horizontal girder, In,m. Thecoefficients for effective width, ln,m, are defined by Figure 2 of DIN 19704:16 (l1); here they areproposed to be calculated by constraints (21). The next constraints (24)(28) express normal andshear stresses, acting in the section of horizontal girders. The combinations between these stressesare not decisive, because p.!9acts in the middle of the girder and q.!9acts at both girder ends.Constraints (29)(31) explain the local buckling constraints for the plate sections of girder webs.Constraints (29) are activated when the water load acts on the outer side of the skin plate andconstraints (30) are actuated when water acts on the inner side of the skin plate. The platebuckling reduction coefficient s, dependent on the plate slenderness ratio jP, is defined byEUROCODE-3:17 s0)6/Jj2P!0)13. In this paper, the buckling coefficient s is determinedby the equation s1/(!0)22703#1)75365j1). Lateral buckling of horizontal girders is verifiedby constraints (32) in the case of compressed girder flanges when water acts on the inner side ofthe skin plate. All stability constraints (29)(32) are defined according to EUROCODE-3. Thehorizontalgirder deflectionis expressed by means of the inequality constraint (33). All constraintsof the lower and upper horizontal girders are defined in the same manner as intermediate girders,with the exception that different coefficients of the effective width ln,mare used here, seeconstraints (34) and (35).THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III339( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Figure 5. Stresses in the girder and skin plate elementCoefficients k3n,mand k4n,mof normal stresses, which act in the skin-plate elements are defined byDIN 1970416 in Table II (clamped plate). In the model, they are defined by constraints (36), (37),(44) and (45), whose functions were also approximated by the least-square technique. It should benoted that coefficients ln,m, k3n,mand k4n,mare also given elsewhere, see e.g. Reference 18. Con-straints (38)(41) and (46)(49) explain the constraints of normal stresses which act in the plateelements of the skin plate. Constraints (42) and (50) interpret the requested ratio between plateelement height and its thickness to provide the calculation of the plates according to thefirst-order theory. Constraints (43) and (51) verify the local buckling in the case of compressedskin plate elements, when water acts on the outer side of the skin plate. It shouldbe noted that thevertical distance between the selected upper intermediate horizontal girder and the sill of the gateelement hsnhas been considered in the constraints of upper horizontal girder and upper skin plateelements.Combined stresses between horizontal girders and skin plate elements are defined by con-straints (52)(67), see also Figure 5. All possible combinations in the middle of the gate areconsidered.Each combination is proposed to be realised for four different points: at the outer andinner side of the skin plate, in the middle of the longer skin plate element border near thehorizontal girder web, and in the middle of the shorter skin plate element border between thewebs of the treated adjoining horizontal girders.Constraints (68), (69) explain that flanges of the upper and lower horizontal girders must beconstructed on the inside of the vertical gate profile. Constraints (70), (71) define the verticaldistances between two adjoining horizontal girders. Constraints (72), (73) explain that skin plate340S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.elements must be made of a shorter vertical border and a longer horizontal border. The verticalrelations between main gate elements are defined by constraints (74)(78), and finally, constraints(79) enforce the same width upon all the gate elements.All the substituted expressions (S1)(S29) are used several times in the mentioned constraints.Constraints of the intermediate horizontal girder, n1, 2, . . . , N and m2, 3, . . . , (M!1):ln,m:!0)02812exp(3)75898!0)36113)4L4/(hn,m1!hn,m1)#0)97848(21)egn,mS4&An,m/A4&n,m(22)In,m(bfn,m)t3fn#twn,m)b3wn#b4&sn,m)t3sn)/12#bfn,m)tfn(tfn/2)!egn,m)2#bwn)twn,m)(tfn#(bwn/2)!egn,m)2#b4&sn,m)tsn)(tfn#(bwn#(tsn/2)!egn,m)2(23)p1n,mM4&.!9n,m/4&1n,m(24)p2n,mM4&.!9n,m/4&2n,m(25)p1n,m)p!-(26)Msf.!9n,m/4&4n,m)p!-(27)Q4&.!9n,m)S4&.!9n,m/(In,m)twn,m)q!-(28)p2n,m)cQ)(c4&p)fy)/(!0)22703#1)75365)j4&p,p2n,m)(29)cQ)M4&.!9n,m/sf3n,m)(c4&pfy)/(!0)22703#1)75365)j4&p,p3n,m)(30)cQQ4&.!9n,m/(twn,m)bwn)(c4&q)fy/J3)/(!0)22703#1)75365)j4&p,qn,m)(31)bfn,m*(L4/dg)J12/40)Jfy/23)5(32)2)Q4&.!9n,m(8)L34!4L4)L25#L35)/(384)E)In,m)d!-(33)where the substituted expressions areA4&n,m bfn,m)tfn#bwn)twn,m#b4&sn,m)tsn(S5)where for b4&sn,msee equation (S6)b4&sn,m:ln,m(hn,m1!hn,m1)/2forL4(hn,m1!hn,m)(S6)c4&p1)25(S7)c4&q1)25(S8)M4&.!9n,m(Q4&.!9n,m/4)(2L4!L5)(S9)where for Q4&.!9n,msee equation (S10)Q4&.!9n,mw4&n,m)L5/2(S10)wherew4&n,m(p4&n,m1/6)(hn,m!hn,m1)#(p4&n,m/3)(hn,m1!hn,m1)#(p4&n,m1/6)(hn,m1!hn,m)(S10.1)THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III341( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)in whichp4&n,m1(hwn!hn,m1)c8(S10.1.1)p4&n,m(hwn!hn,m)c8(S10.1.2)p4&n,m1(hwn!hn,m1)c8(S10.1.3)S4&An,m(bfn,m)t2fn/2)#bwn)twn,m)(tfn#bwn/2)#b4&sn,m)tsn)(tfn#bwn#tsn/2)(S11)where for b4&sn,msee equation (S6)S4&.!9n,mbfn,m)tfn)(egn,m!tfn/2)#twn,m)(egn,m!tfn)2/2(S12)where for b4&sn,msee equation (S6)4&1n,mIn,m/(tfn#bwn#tsn!egn,m)(S13)4&2n,mIn,m/(tfn#bwn!egn,m)(S14)4&3n,mIn,m/(egn,m!tfn,m)(S15)4&4n,mIn,m/egn,m(S16)j4&p,p2n,m(bwn/(1)9twn,m)J4 fy/(E)k4&p2)(S17)wherek4&p223)9(S17.1)j4&p,p3n,m(bwn/(1)9twn,m)J4 fy/(E)k4&p3n,m)(S18)wherek4&p3n,m7)64!6)26)t4&n,m#10)t4&2n,m(S18.1)in whicht4&n,m !(tfn#bwn!egn,m)/(egn,m!tfn)(S18.1.1)j4&p,qn,m(bwn/(1)9twn,m)J4 fy/(J3)E)k4&qn,m)(S19)k4&qn,m5)34#(4/a4&2n,m)(S19.1)anda4&n,m(L4/dg)/bwn(S19.1.1)Constraints of the lower horizontal girder, n1, 2, . . . , N and m1. All the equality/inequal-ity constraints are defined exactly as those for the intermediate horizontal girder, exceptln,1 !0)02812exp(3)75898!0)36113)2L4/(hn,2!hn,1)#0)97848(34)and all the substituted expressions are written down exactly as above, except:b4&sn,1hn,1#ln,1)(hn,2!hn,1)/2(S20)w4&n,1(p4&-08n#p4&n,1)/2)hn,1#(p4&n,1/3)(hn,2!hn,1)#(p4&n,2/6)(hn,2!hn,1)(S21)342S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.wherep4&-08nhwn)c8(S21.1)p4&n,1(hwn!hn,1)c8(S21.2)p4&n,2(hwn!hn,2)c8(S21.3)Constraints of the upper horizontal girder, n1, 2, . . . , N and mM. All the constraints aredefined exactly as those for intermediate horizontal girder above, exceptln,M !0)02812exp(3)75898!0)36113)2L4/(hn,M!hsn)#0)97848(35)and all the substituted expressions are written down exactly as above, exceptb4&sn,Mh*4/#ln,M(hn,M!h4n)/2(S22)w4&/,M(p4&upn#p4&n,M)/2)(h*5/!h*/)#(p4&n,M/3)(hn,M!hsn)#(p4&n,M1/6)(hn,M!hsn)(S23)wherep4&upn(hwn!hogn!h*5/)c8(S23.1)p4&n,M(hwn!hn,M)c8(S23.2)p4&n,M1(hwn!hsn)c8(S23.3)Constraints of the plate elements of the skinplate, n1, 2, . . . , N and m1,2, . . . , (M!2):k3n,m !6)88282expM!1)22436 L4/(dg)(hn,m1!hn,m)2#0)13604)L4/(dg)(hn,m1!hn,m)#2)11677N#50)17457(36)k4n,m !76)70431expM!8)05965)L4/(dg(hn,m1!hn,m)#4)96141N#34)36154(37)ppl3n,m(k3n,m/100)p4&1-n,m)(hn,m1!hn,m)2/t2sn(38)ppl4n,m(k4n,m/100)p4&1-n,m)(hn,m1!hn,m)2/t2sn(39)ppl3n,m)p!-(40)ppl4n,m)p!-(41)(hn,m1!hn,m)/tsn)100(42)p1n,m)cQ)(c4&p1-)fy)/(!0)22703#1)75365)j4&p,p1-n,m)(43)where the substituted expressions arec4&p1-1)0(S24)p4&1-n,mhwn!(hn,m#hn,m1)/2)cw(S25)j4&p,p1-n,m(hn,m1!hn,m)/(1)9tsn)J4fy/(E)k4&p1-)(S26)in whichk4&p1-4)0(S26.1)THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III343( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Constraints of the upper plate elements of the skin plate, n1, 2, . . . , N and mM!1:k3n,M1 !6)88282expM!1)22436)L4/(dg)(hn,M!hsn)2#0)13604)L4/(dg(hn,M!hsn)#2)11677N#50)17457(44)k4n,M1 !76)70431expM!8)05965L4/(dg(hn,M!hsn)#4)96141N#34)36154(45)p1-3n,M1(k3n,M1/100)p4&1-n,M1(hn,M!hsn)2/t2sn(46)ppl4n,M1(k4n,M1/100)p4&1-n,M1(hn,M!hsn)2/t2sn(47)ppl3n,M1)p!-(48)ppl4n,M1)p!-(49)(hn,M!hsn)/tsn)100(50)p1n,M)cQ)(c4&p1-) fy)/(!0)22703#1)75365)j4&p,p1-n,M)(51)where the substituted expressions arec4&ppl1)0(S27)p4&1-n,M1hwn!(hn,M#hsn)/2)cw(S28)j4&p,p1-/,M(hn,M!hsn)/(1)9tsn)J4fy/(E)k4&p1-)(S29)in whichk4&p1-4)0(S29.1)Constraintsof combinedvon Mises stresses betweenthe intermediatemth horizontal girder and itsadjoining lower (m!1)th and upper mth skin plate elements, n1, 2, . . . , N; m2, 3, . . . ,(M!1) for the girder, m1, 2, . . . , (M!2) for the lower and m2, 3, . . . , (M!1) for theupper skinplate elementp21-3n,m1#(!p1n,m#k)p1-3n,m1)2!p1-3n,m1)(!p1n,m#k)p1-3n,m1)p23%&,!-(52)(!p1-3n,m1)2#(!p2n,m!k)p1-3n,m1)2!(!p1-3n,m1)(!p2n,m!k)p1-3n,m1)p23%&,!-(53)(k)p1-4n,m1)2#(!p1n,m#p1-4n,m1)2!(k)p1-4n,m1)(!p1n,m#p1-4n,m1)p23%&,!-(54)(!k)p1-4n,m1)2#(!p2n,m!p1-4n,m1)2!(k)p1-4n,m1)(!p2n,m!p1-4n,m1)p23%&,!-(55)p21-3n,m#(!p1n,m#k)p1-3n,m)2!p1-3n,m)(!p1n,m#k)p1-3n,m)p23%&,!-(56)(!p1-3n,m)2#(!p2n,m!k)p1-3n,m)2!(!p1-3n,m)(!p2n,m!k)p1-3n,m)p23%&,!-(57)(k)p1-4n,m)2#(!p1n,m#p1-4n,m)2!(k)p1-4n,m)(!p1n,m#p1-4n,m)p23%&,!-(58)(!k)p1-4n,m)2#(!p2n,m!p1-4n,m)2!(!k)p1-4n,m)(!p2n,m!p1-4n,m)p23%&,!-(59)Constraints of combined von Mises stresses between the lower horizontal girder and its adjoiningupper skin plate element of the nth main gate element, n1, 2, . . . , N and m1:p21-3n,1#(!p1n,1#k)p1-3n,1)2!p1-3n,1)(!p1n,1#k)p1-3n,1)p23%&,!-(60)(!p1-3n,1)2#(!p2n,1!k)p1-3n,1)2!(!p1-3n,1)(!p2n,1!k)p1-3n,1)p23%&,!-(61)344S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.(k)p1-4n,1)2#(!p1n,1#p1-4n,1)2!(k)p1-4n,1)(!p1n,1#p1-4n,1)p23%&,!-(62)(!k)p1-4n,1)2#(!p2n,1!p1-4n,1)2!(!k)p1-4n,1)(!p2n,1!p1-4n,1)p23%&,!-(63)Constraints of combined von Mises stresses between the upper horizontal girder and its adjoininglower skin plate element of the nth gate element, n1, 2, . . . , N; mM for the girder andmM!1 for the skin plate element:p21-3n,M1#(!p1n,M#k)p1-3n,M1)2!p1-3n,M1)(!p1n,M#k)p1-3n,M1)p23%&,!-(64)(!p1-3n,M1)2#(!p2n,M!k)p1-3n,M1)2!(!p1-3n,M1)(!p2n,M!k)p1-3n,M1)p23%&,!-(65)(k)p1-4n,M1)2#(!p1n,M#p1-4n,M1)2!(k)p1-4n,M1)(!p1n,M#p1-4n,M1)p23%&,!-(66)(!k)p1-4n,M1)2#(!p2n,M!p1-4n,M1)2!(!k)p1-4n,M1)(!p2n,M!p1-4n,M1)p23%&,!-(67)Constraints of the entire gate:bfn,1)bf*-08n#h-08n,n1, 2, . . . , N(68)bfn,M)bf*61n#h*sn,n1, 2, . . . , N(69)dhn,mhn,m1!hn,m,n1, 2, . . . , N; m1, 2, . . . , M!1(70)dhsnhn,M!hsn,n1, 2, . . . , N(71)hn,M!hsn)L4/dg,n1, 2, . . . , N(72)hn,m!hn,m1)L4/dg,n1, 2, . . . , N;m2, 3, . . . , M!1(73)hwn/1H8,(74)hwnhwn1!h0n1,n2, 3, . . . , N(75)H505+n|Nh0n(76)hgnh0n#h*gn#h*sn,n1, 2, . . . , N(77)hn,Mh0n#h*gn,n1, 2, . . . , N(78)tfn#bwn#tsntfn1#bwn1#tsn1,n2, 3, . . . , N(79)3.4. Pure integer logical constraintsThe set of constraints (80)(82) represents integer logical linear constraints. Constraints (80)determine the number of fixed structural elements (fixed horizontal girders and skin plateelements) for each gate element. Constraints (81) make sure that the number of horizontal girdersand skin plate elements of the upper gate elements is equal to or higher than the one of the lowergateelements.Constraints(82) explainthat the upper alternative(intermediate)horizontalgirdersand skin plate elements disappear in the direction from the top to the bottom of the gate beforethe lower ones do. This, in effect, reduces the combinatorics of the MINLP problem since itTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III345( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)prevents the obtaining of an equal number of alternative elements determined at quite differentvalues of the assigned binary variables.+m|Myn,m*M.*/#2,n1, 2, . . . , N(80)+m|Myn,m* +m|Myn1,m,n2, 3, . . . , N(81)yn,m)yn,m1,n1, 2, . . . , N;m2, 3, . . . , M!1(82)3.5. Interconnection logical constraintsInterconnection logical mixed-integer linear constraints start with constraint (83), whichdefinesthe gate partitionin the horizontal direction.Constraints (84)(87) define vertical distancehsnbetween the uppermost selected intermediate horizontal girder and the sill of the gate element.Consequently, the uppermost fixed horizontal girder is always connected to the uppermostselected intermediate alternative horizontal girder and disconnected from the disappearing ones.Constraints (88)(91) determine the minimal openings between flanges of horizontal girders.dg2#2+v|Vyn,v,n1(83)hsn*hn,m!h614/)(1!yn,m1)!h614/)(1!yn,m)!h614/)yn,m1,n1, 2, . . . , N;m2, . . . , M!2(84)hsn)hn,m#h614/(1!yn,m1)#h614/)(1!yn,m)#h614/)yn,m1,n1, 2, . . . , N;m2, . . . , M!2(85)hsn*hn,M1!h614/)(1!yn,M1),n1, 2, . . . , N(86)hsn)hn,M1#h614/)(1!yn,M1),n1, 2, . . . , N(87)hn,2!bfn,2/2!h-08/!bf*-08n*h&,.*/)yn,2,n1, 2, . . . , N(88)hn,m!bfn,m/2!hn,m1!bfn,m1/2*h&,.*/)yn,m,n1, 2, . . . , N;m3, 4, . . . , M!1(89)hn,M!bf*upn!(hn,m#bfn,m/2)*h&,.*/!d61)/,.)yn,M1!d61)/,.)(1!yn,m)!d61)/,.)yn,m1n1, 2, . . . , N;m1, 2, . . . , M!1(90)hn,M!bf*upn!(hn,M1#bfn,M1/2)*h&,.*/!d61)/,M1)(1!yn,M1),n1, 2, . . . , N (91)3.6. ogical constraints for continuous variables3.6.1. Bound logical constraintsMixed-integer linear constraints (92)(101) define upper and lower bounds for the individualdesignvariablesbfn,m, twn,m, and dhn,mas well as for the commondesign variablesbwin,mandtfin,mif their346S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.associated structural element exists, (yn,m1); if not, the bounds are set to zero.bfn,m)b61fn,m)yn,m,n1, 2 , . . . , N; m1, 2, . . . , M(92)bfn,m*b-08,%9fn,m)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(93)twn,m)t61wn,m)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(94)twn,m*t-08,%9wn,m)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(95)dhn,m)d61hn,m)yn,m,n1, 2, . . . , N; m1, 2, . . . , M!1(96)dhn,m*d-08,%9hn,m)yn,m,n1, 2, . . . , N; m1, 2, . . . , M!1(97)bwin,m)b61wn)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(98)bwin,m*b-08wn)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(99)tfin,m)t61fn)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(100)tfin,m*t-08fn)yn,m,n1, 2, . . . , N; m1, 2, . . . , M(101)3.6.2. ogical relations for common variablesMixed-integer linear constraints (102)(105) define the logical relations of the common designvariables. These constraints enforce upon the common design variables of the horizontal girdersbwin,mand tfin,mthe values of the corresponding common variables of the entire gate bwnand tfn.bwin,m)bwn#b61wn)(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(102)bwin,m*bwn!b61wn)(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(103)tfin,m)tfn#t61fn)(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(104)tfn,m*tfn!t61fn)(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(105)3.7. ogical constraints for standard variables3.7.1. Standard dimension logical constraintsStandard dimension logical constraints define four standard dimensions: thickness of the skinplate tsn, flange thickness of the horizontal girder tfn, the web thickness of the outer horizontalgirder t065wn,mand the web thickness of the inner horizontal girder t*/wn,m. Each standard dimension isdetermined by a scalar product between its corresponding vector of binary variables y and itsassigned vector of discrete constants q (constraints (106), (108), (110), (111), (113) and (114), whereonly one discrete value can be selected (+y1), see constraints (107), (109), (112) and (115). Notethatd#03denotescorrosionadditionto the sheet-ironplates. In the synthesis,these dimensionsarefirst treated as continuous and later as standard dimensions. The bound logical constraints (94)and (95) for the design variable twn,mare therefore suitable for the standard dimensions t065wn,mandt*/wn,m. As standard dimensions tsnand tfncorrespond to the common standard design variables forTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III347( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)the superstructure or its part d45,#0.t, they do not need bound logical constraints to be defined forthem (these dimensions always exist):tsn+i|I(q/,*!d#03)yn,i,n1, 2, . . . , N(106)+i|Iyn,i1n1, 2, . . . , N(107)tfn+k|K(q/,!d#03)yn,k,n1, 2, . . . , N(108)+k|Kyn,k1,n1, 2, . . . , N(109)t065wn,m)+p|P(q/,1!d#03)yn,p#t065,UPwn,m(1!yn),n1, 2, . . . , N; m1 or mM(110)t065wn,m*+p|P(q/,1!d#03)yn,p!t065,UPwn,m(1!yn),n1, 2, . . . , N; m1 or mM(111)+p|Pyn,p1,n1, 2, . . . , N(112)t*/wn,m)+r|R(q/,3!d#03)yn,r#t*/,UPwn,m(1!yn),n1, 2, . . . , N; m1, 2, . . . , M(113)t*/wn,m*+r|R(q/,3!d#03)yn,r!t*/,UPwn,m(1!yn),n1, 2, . . . , N; m1, 2, . . . , M(114)+r|Ryn,r1n1, 2, . . . , N(115)3.8. Constraints for the MILP phase onlyAdditional MILP constraints (116)(121), considered only in the MILP phase of the MINLPoptimization, are included in the model in order to perform the optimization faster and moreefficiently. To enforce faster MINLP convergence, equal vertical distances between the selectedhorizontal girders are proposed:dhn,m)(hn,M!h-08/)/(M.!9!1)#d61)/,.)(1!yn,M1),n1, 2, . . . , N; m1, 2, . . . , M!1(116)dhn,m*(hn,M!h-08/)/(M.!9!1)!d61)/,.)(1!yn,M1),n1, 2, . . . , N; m1, 2, . . . , M!1(117)dhn,m)(hn,M!h-08/)/(M.!9!2)#d61)/,.)yn,M1#d61)/,.)(1!yn,M2),n1, 2, . . . , N;m1, 2, . . . , M!2(118)348S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.dhn,m*(hn,M!h-08/)/(M.!9!2)!d61)/,.)yn,M1!d61)/,.)(1!yn,M2),n1, 2, . . . , N;m1, 2, . . . , M!2(119)dhn,m)(hn,M!h-08/)/(M.!9!3)#d61)/,.)yn,M2n1, 2, . . . , N; m1, 2, . . . , M!3(120)dhn,m*(hn,M!h-08/)/(M.!9!3)!d61)/,.)yn,M2n1, 2, . . . , N; m1, 2, . . . , M!3(121)3.9. Heuristic constraintsHeuristic constraints additionally increase the efficiency of the MINLP optimization and areadded only for the MILP phase. Constraints (122)(125) show that the thicknesses of each uppergate element must not exceed the ones of the lower gate elements. Each upper main gate elementis higher than its adjoining lower main gate elements, see constraints (126):tsn)tsn1,n2, 3, . . . , N(122)tfn)tfn1,n2, 3, . . . , N(123)twn,m)twn1,m,n2, 3, . . . , N; m1 or mM(124)twn,m)twn1,m/2,n2, 3, . . . , N; m2, 3, . . . , M!1(125)hon*hon1n2, 3, . . . , N(126)3.10. Initialization of variablesThe initialization of continuous and binary variables is also included, see list of symbols inAppendixI. Only design variableshave to be initialised, while non-design variables are calculatedon the basis of the initialized design variables.4. SYNTHESIS OF THE INTAKE GATE IN ASWANAcomprehensivecomparativedesign research work has been performedon some of the construc-ted roller and sliding gates. This paper shows the example of the synthesis of the already existingroller gate, named Intake Gate, see Figure 6. The Intake Gate was constructed and erected inAswan II, Egypt, by the Slovenian company Metalna19 from Maribor. Eight identical turbineintakes were erected, each regulated by an Intake Gate consisting of three vertical main elements.The IntakeGate was designed by means of a classicalmethod, where a simplified static system forthe structural analysis of the gate is used.The material actually used, i.e. St 44-2, has been considered in the optimization. The technicaldata of the Intake Gate and the economic data for the optimization are presented in Tables IIV.The actual self manufacturing and transportation costs in the amount of 70518 $ have beencalculated for the entire gate at the actual topology of 4-4-4/11 (4 horizontal girders for each ofthe three main gate elements and 11 vertical girders for the entire gate with the correspondingnumber of skin plate elements).The task of the synthesis of the Intake Gate is to find the optimal topology, geometry as well ascontinuous and standard sizes with respect to the minimum of self-material and labour costs,subjected to design, stress, deflection, and stability constraints. The synthesis is performedTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III349( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Figure 6. Actual structure of the Intake Gate: (a) vertical cross-section; (b) horizontal cross-sectionthrough three steps: it begins with the generation of the gates superstructure, continues with thedevelopment of the gates MINLP optimization model and ends by solving the defined MINLPproblem.4.1. Simultaneous topology and parameter optimization of the Intake GateLet us first consider the synthesis of the Intake Gate followed by a simultaneous topology andcontinuous parameter optimization. Such synthesis has been performed by means of the MINLPapproach introduced in Part I. This approach enables the obtaining of the optimal number ofhorizontal and vertical girders and skin plate elements as well as all continuous optimaldimensions simultaneously.Thegenerated superstructureof the IntakeGate is the same as the one in Figure 2. It comprisesn main gate elements, n1, 2, 3; m horizontal girders per each nth gate element, m1, 2, . . . , 6;an odd number (3#2v) of vertical girders for the entire gate, v1, 2, 3; and the correspondingnumber (m!1)(2#2v) of skin-plate elements per each main gate element.350S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Table I. Technical data of the opening of theIntakeWater head on the sillHw30)10 mClear height of openingH5058)00 mClear width of opening7)90 mTable II. Design data of the Intake GateSealing height of the gateH58)00 mSealing width of the gate58)00 mSpacing of gate wheels48)50 mDistance between outer side girders48)20 mWeb thickness of the vertical girdert781)0 cmFlange width of the vertical girderb7&10)0 cmWeb thickness of the side vertical girdert43)5 cmFlange width of the side vertical girderb4&20)0 cmCorrosion addition thicknessd#032)0 mmAllowable deflectionsd1)0 cmSafety coefficientcQ1)5Distances: h-08110)0 cmh-0823)5 cmh-0833)5 cmDistances:h*t10h*t20h*t30Distances:h*s13)5 cmh*423)5 cmh*4312)5 cmDistances: h*1 !3)5 cmh*2 !3)5 cmh*30Table III. Allowable stresses for material St 44-2Ultimate stress of materialf642)00 kN/cm2Yield stress of materialf:25)00 kN/cm2Allowable normal stressp!-14)70 kN/cm2Allowable shear stressq!-7)80 kN/cm2Allowable reference stressp3%&16)20 kN/cm2Table IV. Economic data for optimizationMaterial costs for St 44-2c.!5%30)60 $/kgTransportation costsc53!/410)35 $/kgSheet-iron cutting costsc#651)60 $/m1Welding costsc8%-$5)00 $/m1Anti-corrosion resistant paintingc!3#117)50 $/m2Within the given sets of structural elements, the permitted topology variations are between4 and 6 horizontal girders for the lower, intermediate and upper main gate elements separately,while 5, 7 or 9 vertical girders can be selected for the entire gate, see Table V. Thus, the optimaltopology was proposed to be found inside the defined superstructure (4-6)-(4-6)-(4-6)/(5-9), whichcontains 3481 different topology alternatives, see Figure 7.To each mth horizontal girder of each nth main gate element, an extra binary variable yn,m,n1, 2, 3 and m1, 2, . . . , 6, is assigned, while binary variables y1,v, v1, 2, 3, are used toTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III351( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Table V. Topology boundaries of the Intake Gate superstructureNumber of main gate elementsN3Max of horizontal girders per main gate elementM.!96Min of intermediate horiz. girders per main gate elementM.*/2Max of vertical girders per entire gate.!99Min of vertical girders per entire gate.*/5Figure 7. Gate topology bounds: (a) maximal gate topology 6-6-6/9: all structural elements are included; (b) minimal gatetopology 4-4-4/5: only fixed structural elements () are included, alternative elements (- - -) are removeddetermine the number of (3#2+l|Vy1,v) vertical girders for the entire gate. The number of skinplate elements is given automatically when the number of horizontal and vertical girders isknown. Each gate topology is therefore expressed by the correspondingvector of binaryvariablesyMyn,m; y1,lN, i.e. yMy1,m;y2,m;y3,m;y1,lN, i.e. yMy1,1, y1,2, y1,3, y1,4, y1,5, y1,6; y2,1, y2,2, y2,3,y2,4, y2,5, y2,6; y3,1, y3,2, y3,3, y3,4, y3,5, y3,6; y1,1, y1,2, y1,3N. Note that the smallest first indexrepresents the lowest main gate element and the smallest second index the lowest horizontalgirders.In order to obtain the whole linear approximation for the MILP master problem of themodified OA/ER algorithm, the first NLP has to be performed for the entire superstructure andthe initial vector of binary variables has to include the full set of structural elements, i.e. yM1, 1,1,1, 1, 1; 1, 1, 1, 1, 1, 1; 1, 1, 1, 1, 1, 1; 1, 1, 1N. All further vectors of binaryvariables,obtainedby theMILP steps, clearly show which structural elements are selected and which are rejected throughthe optimization.The gate has been modelled by means of the proposed general optimization model for rollerand sliding gate structures GATOP (GATe OPtimization). The self-manufacturing costs, such asmaterial, welding, sheet-iron cutting and anti-corrosion resistant painting as well as transporta-tion costs have been accounted for in the economic type of objective function, subjected to givendesign, material, stress, deflection and stability constraints. For real comparison between theobtainedoptimal gate and the actually erected IntakeGate, identical economic parameters, staticsystem, material, stress and deflection allowances as well as all functions from the structuralanalysis of the Intake Gate have been considered as constraints in the model. As standarddimensions are in this chapter not explicitly considered in the optimization, standard dimensionlogical constraints are removed from the model. The MINLP model of the Intake Gate for352S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Table VI. Convergence to the optimal topology of the Intake GateIterationvector of binary variables yMy1,m; y2,m; y3,m; y1,vNTopology1y1M1, 1, 1, 1, 1, 1; 1, 1, 1, 1, 1, 1; 1, 1, 1, 1, 1, 1; 1, 1, 1N6-6-6/9Initial topology2y2M1, 1, 1, 0, 0, 1; 1, 1, 1, 0, 0, 1; 1, 1, 1, 1, 0, 1; 1, 0, 0N4-4-5/53y3M1, 1, 1, 1, 0, 1; 1, 1, 1, 1, 0, 1; 1, 1, 1, 1, 0, 1; 1, 0, 0N5-5-5/54y4M1, 1, 1, 0, 0, 1; 1, 1, 1, 1, 0, 1; 1, 1, 1, 1, 0, 1; 1, 0, 0N4-5-5/5Optimal topology5y5M1, 1, 1, 0, 0, 1; 1, 1, 1, 0, 0, 1; 1, 1, 1, 0, 0, 1; 1, 0, 0N4-4-4/5Table VII. Convergence to the optimal result of the Intake GateMINLPMINLPResultho1ho2ho3tsn#bwn#tfin,miterationsubphase$topologymmmm1Initialization6-6-6/91. NLP559566-6-6/92)6672)6672)6670)96621. MILP476044-4-5/52)3852)6153)0000)9222. NLP493944-4-5/52)4672)5333)0000)96832. MILP469845-5-5/52)6672)6672)6670)9663. NLP495205-5-5/52)6062)6062)7880)94243. MILP495004-5-5/52)3852)7982)8170)9404. NLP493874-5-5/52)3852)7782)8370)94754. MILP501204-4-4/52)5092)6172)8741)0115. NLP496304-4-4/52)5382)6992)7630)997continuous sizes contains 1045 (in)equality constraints, 335 continuous and 27 binary 01variables.The synthesis of the Intake Gate was carried out by a user friendly version of the MINLPcomputer package TOP (Topology Optimization Program).1 The computer package TOP isa more general version of the computer package PROSYN, an MINLP process synthesizer, seeKravanja and Grossmann.20,21 TOP is the implementation of many advanced optimizationtechniques, a most important of which is the Modified OA/ER algorithm. MINOS by Murtaghand Saunders22 is used to solve NLP subproblems and OSL23 is used to solve MILP masterproblems. In terms of complexity, the TOPs synthesis problems can range from a simple NLPoptimization problem of a single structure up to the MINLP optimization of a complexsuperstructureproblem. TOP runs automatically or in an interactive mode and thus provides theuser with a good control and supervision of the calculations.The optimal solution yields the costs of 49387 $ or 30)0 per cent of net savings. The optimumwas reached at the topology of 4-5-5/5 (4 horizontal girders for the lower, 5 for the intermediateand 5 for the upper main gate element as well as 5 vertical girders for the entire gate with thecorresponding number of skinplate elements). The Modified OA/ER algorithm converged con-siderably fast: only 4 major MINLP iterations were needed (3 MILP and 4 NLP subproblems),see Tables VI and VII. It took only 161 s of CPU on the computer VAX 4600 to obtain theoptimal result. Beside the optimal topology of 4-5-5/5 of the gate structure, the optimization alsogave an optimal gate geometry and all the sizes. Figure 8 shows the initial and three otherTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III353( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Figure 8. Convergence to the optimal gate topology: (a) initial topology 6-6-6/9, 1. iteration; (b) obtained topology4-4-5/5, 2. iteration; (c) obtained topology 5-5-5/5, 3. iteration; (d) obtained optimal topology 4-5-5/5, 4. iterationtopologies (the last one is the optimal), automatically obtained and sequentially calculated ateach MINLP iteration.It should be noted that the obtained result of 49387 $ has a drawback in that it is reached onthe assumption that all dimensions, including the thicknesses of sheet-iron plates, are continuous.Standard dimensions are thus not simultaneously dealt with in the optimization. We call suchresults theoretical optima.However, to upgrade the practical relevance of the obtained results, standard dimensions mustalso be considered. We call such practical results real optima. The simplest way of accomplishingthis task is to round up the thicknesses to their close standard dimension values. To obtaina feasible result, the rounding up has to be directed to the closest higher standard values. ThefollowingNLP optimization, performed at the fixed optimal topology of 4-5-5/5, thus involves allthe standard dimensions set to the appropriate values, while other dimensions are varied. Itshouldbe noted that this sequential optimization of standard dimensions may lead to suboptimalresults and that special care should be taken to avoid infeasible results when the dimensions arerounded up to their standard values. It will be seen in the following subchapter that, in order toavoid the drawbacks, all standard dimensions should be optimized simultaneously, together withthe topology and continuous parameters of the gate.Figure 9 shows the Intake gate with the rounded up thicknesses of sheet-iron plates and otheroptimal dimensions. This real gate optimum yields the costs of 50578 $ or 28)3 per cent of netsavings.354S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Figure 9. Optimal structure of Intake gate with rounded dimensionsWhen only the NLP parameter optimization (sizing and geometrical optimization) has beenperformed at the actual fixed topology 4-4-4/11 and at varied parameters, the optimal solution inthe amount of 56715 $ has been reached, which corresponds to 19)6 per cent of net savings incomparison with the already constructed gate.4.2. Simultaneous topology, parameter and standard dimension optimization of the Intake GateThe synthesis of the Intake Gate is now performed by a simultaneous MINLP optimizationapproach, extended to standard dimensions. In the previous section, where only topology andcontinuous dimensions were obtained, some continuous dimensions were simply rounded up totheir closest higher standard values to gain the real optimum. In this section, topology, parameterand standard dimensions are simultaneously optimized on the basis of the MINLP approachfrom Part II.This overall MINLP optimization approach requires the generation of a gate superstructureof various topology and standard dimension alternatives. The generated gate superstructure(4-6)-(4-6)-(4-6)/(5-9),alreadypresentedinprevious section,hasthus beenextendedto include alsoTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III355( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)standard dimensions. In addition, 11 different standard dimension alternatives have been pro-posed for different structural elements including the standard thicknesses of sheet-iron platesfrom 10 to 40 mm. In this example, each standard dimension tsnis proposed to be expressed by3 different standard dimension alternatives i3I, IM1, 2, 3N, each standard dimension tfnby4 different alternatives k3K, KM1, 2, 3, 4N, eachstandard dimensiont065wn,mby 2 alternativesp3P,PM1, 2N, and each standard dimension t*/wn,malso by 2 alternatives r3R, RM1, 2N. Hence, thevector of 3 binary variables yn,iMyn,1, yn,2, yn,3N and the vector of 3 discrete values qn,iMqn,1,qn,2, qn,3NM2)0, 2)5, 3)0N (cm) are assigned to the variable tsn, the vectors of 4 binary variablesyn,kMyn,1, yn,2, yn,3, yn,4N and 4 discrete values qn,kMqn,1, qn,2, qn,3, qn,4NM2)5, 3)0, 3)5, 4.0N(cm) to the variable tfn, the vectors of 2 binary variables yn,pMyn,1, yn,2N and 2 discrete valuesqn,pMqn,1, qn,2NM1)0, 1)2N (cm) to the variablet065wn,mand, finally,the vectors of 2 binaryvariablesyn,rMyn,1, yn,2N and 2 discrete values qn,rMqn,1, qn,2NM1)2, 1)5N (cm) to the variable t*/wn,m,where the common vector of binary variables for standard dimensions is yMyn,i, yn,k, yn,p, yn,rN.Each possible combined solution of gate topology and standard dimensions is expressed by thecorresponding overall vector of 54 binary variables: yMyn,m, y1,v, yn,i, yn,k, yn,p, yn,rN. The definedsuperstructure comprises 8)9579106 different topology and standard dimension alternatives.The synthesiswas carried out by a user friendly version of the MINLP computer packageTOPusing the Modified OA/ER algorithm and the Linked Two-Phase MINLP Strategy. The optimalsolution yields the costs of 49783 $ at the obtained optimal topology of 4-5-5/5, which represents29)4 per cent of net savings when compared to the costs of the erected gate. The optimal structureof the Intake Gate is shown in Figure 10.The Modified OA/ER algorithm accompanied with the Linked Two-Phase MINLP Strategyconverged very fast: only 5 major MINLP iterations were needed (4 MILP and 5 NLPsubproblems), see the convergence and results in Table VIII. The theoretical optimum of 49387$ was found in the 4th major MINLP iteration (3 MILP and 4 NLP) and the real optimum of49783 $ in the next, i.e. the 5th iteration. The optimal solution, i.e. the real optimum, was reachedin the first MINLP iteration of the second stage, i.e. of the real optimization phase. The firstphase, that is the theoretical optimization, corresponds to the optimization of the gate performedin the previous section.In the first phase, all standard dimensionstsn, tfn, t065wn,mand t*/wn,mwere relaxedinto continuous parameters. When the theoretical optimum with an optimal topology wasfound, the standard dimensions were re-established and the simultaneous topology, parameterand standard dimension optimization then continued on the basis of the obtained linear globalapproximation until the optimal solution, i.e. the real optimum, was found. In the 6th and all thenext MINLPiterations,the upper boundof the variable cost, costUP49783 $ (thebest real NLPsolution), was used. Table VIII shows only the first two major MINLP iterations of the realoptimization stage (all next solutions are not as good). While in the first phase, only the vector ofbinary variables was assigned for topology determination yMyn,m, y1,vN, the overall binaryvector yMyn,m, y1,v, yn,i, yn,k, yn,p, yn,rN was in the second stage used to determine the topologyand standard dimension alternatives. Note that the obtained standard dimensions tsn, tfn, t065wn,mandt*/wn,mfrom the second phase included a corrosion addition for sheet-iron plates d#032 mm,resulting in that the mentioned dimensions are actually reduced by 2 mm.The MINLP model of the Intake Gate contains 1093 (in)equality constraints, 335 continuousand 60 binary 01 variables. 869 equations with 3169 non-zero elements, 327 continuous and 15currently fixed discrete variables are included in the first NLP subproblem and 2032 equationswith 1764 continuous and 54 discrete variables are included in the MILP master problem wherethe optimal solution was found. Only 293 s of CPU time were spent on the VAX 4600 computer.356S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Figure 10. Optimal structure of the Intake GateAlongside the optimal costs and optimal topology of the gate, all the necessary standarddimensions and other continuous dimensions (the gates global geometry, the intermediate distan-ces between structural elements and their continuous dimensions) were also obtained. The outlineof the actual gate structure and of the one with the optimal topology is shown in Figure 11.A comparison between the obtained result and the one of rounded up continuous dimensionsfrom the previous section shows that additional savings have been obtained due to the explicitand thorough consideration of standard dimensions in the simultaneous optimization (49783$or 29)4 per centnet savings vs. 50578 $ or 28)3 per cent net savingsin Section 4.1).The differencebetween rounded and optimized standard dimensions is clearly shown in Figures 9 and 10. Thelower gate element remains of the same values of standard dimensions as in the design of roundeddimensions, while in the element in the middle almost all standard dimension are reduced. In theupper gate element only the thickness of flanges has been changed. All continuous dimensionshave been modified. Note that in this example, the optimal topology obtained at the theoreticaloptimum is identical to the one gained at the real optimum.THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III357( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)Table VIII. Convergence to the optimal result of the Intake GateMINLPMINLPResultGatetsntfnt065wn,mt*/wn,miterationsubphase$TopologyelementcmcmcmcmPhase 1: Theoretical optimization1Initialization6-6-6/9Upper1)9013)3040)600)761. NLP559566-6-6/9Middle2)0783)6750)620)81Lower2)2773)8000)670)710)9121. MILP476044-4-5/5Upper2)3522)3960)710.761)012. NLP493944-4-5/5Middle2)7122)5430)831)26Lower2)7742)6740)840.911)3032. MILP469845-5-5/5Upper2)2652)4690)690.730)963. NLP495205-5-5/5Middle2)3013)0780)730)97Lower2)4503)4310)770.841)0743. MILP495004-5-5/5Upper2)2802)4280)690.730)974. NLP493874-5-5/5Middle2)4182)6180)751)05Lower2)7402)7750)830.901)284*MILP501294-4-4/5Upper2)7632)3480)780.831)20NLP496304-4-4/5Middle2)6992)5150)851)30Lower2)5382)6070)920.851)30Phase 2: Real optimization (including standard dimension)54. MILP501464-5-5/5Upper2)52)51)01)2Middle2)53)01)01)25. NLP497834-5-5/5Lower3)03)01)21)565. MILP497834-5-5/5Upper2)53)01)01)2Middle2)53)01)01)26. NLP497934-5-5/5Lower3)03)01)21)55. CONCLUSIONSThe paper presents the Mixed-Integer Non-linear Programming (MINLP) approach to thesynthesis of roller and sliding hydraulic steel gate structures. The overall synthesis is performed ina single uniform calculating process, where topology, parameters and standard dimensions areconsidered simultaneously in order to reach the minimum of structure costs.For this purpose, the superstructure of different topology and standard dimension alternativesfor roller and sliding gates was generated and the MINLP mathematical optimization modelGATOP (GATe OPtimization) was developed according to the proposed MINLP-MS modelformulation for mechanical superstructures, introduced in Parts I and II. Special logical con-straints for topology alterations and interconnection relations between the alternative and fixedstructural elements are formulated and embedded into the model. The synthesis of the gate wascarried out by a user friendly version of the MINLP computer package TOP, the latter being the358S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.Figure 11. The Intake Gate structure, vertical and horizontal cross-sections: (a) actual topology 4-4-4/11; (b) optimaltopology 4-5-5/5, 29.4% of net savingsimplementation of the Modified OA/ER algorithm. To solve highly combinatorial, non-linearand non-convex problems, the Linked Two-Phase MINLP Strategy was introduced in order toaccelerate the convergence of the Modified OA/ER algorithm.This simultaneous MINLP optimization approach has been applied to the synthesis of a rollergate, the so-called Intake Gate, erected in Aswan. The obtained results show that this methodenables 29)4 per cent of savings in investment costs when compared to the actual design obtainedby the classicalmethod. Beside the optimal self manufacturingcosts (material, welding, sheet-ironcutting, anti-corrosion resistant painting and transportation costs), an optimal gate topologywith the optimal number of girders and plate elements, optimal continuous dimensions (globalgeometry,all intermediatedistances betweenstructural elementsand their crossectionalsizes) hasbeen obtained as well as optimal standard dimensions (all the necessary standard thicknesses ofsheet-iron plates).The best solution of 29)4 per cent net savings is calculated when topology, parameters andstandard dimensions are optimized simultaneously. In the case when only topology and continu-ous parameters (dimensions) are considered and some continuous dimensions are sequentiallyrounded up to their closest higher standard values, a slightly less desirable result of 28)3 per centof net savings is reached.Both MINLP results become significantly better when compared to the 19.6 per cent netsavings obtained by NLP parameter optimization at an actual fixed topology.APPENDIXist of symbols for the optimization model GAOP (GAe OPtimization)Sets:nset for the main gate elements,n3Nmset for the horizontal girders of the nth main gate element,m3MTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III359( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)vset for the generation of vertical girders, v3iset for the standard dimension alternatives of the dimension tsn, i3Ikset for the standard dimension alternatives of the dimension tfn, k3Kpset for the standard dimension alternatives of the dimension t065wn,m, p3Prset for the standard dimension alternatives of the dimension t*/wn,m, r3RConstants (input data):b4&side girders flange widthb7&intermediate vertical girders flange webb*4horizontal distance between the outer vertical web of the side girders and the edgeof the skin platec!#31price of anti-corrosion resistant painting, determines painting costsc#65price of sheet-iron cutting, determines cutting costsc.!5%3price of used material, determines material costsc53!/41price of transporting, determines transportation costsc8%-$price of welding, determines welding costsd#03corrosion addition to the thicknesses of sheet-iron platesfyyield strength for the used steelh&,.*/minimalverticaldistance(opening)between two adjoiningflangesof the horizontalgirdersh-08/vertical distance between the first horizontal girder and the sill of the nth gateelementh*/vertical distance between the upper point of the opening and the upper horizontalgirder of the nth gate elementh*s/vertical distance between the upper horizontal girder and the upper point of theskin-plate of the nth gate elementh*5/vertical distance between the upper point of the opening and a upper point of thewater acting on the nth gate elementq/,*discrete value of standard dimension alternative for the dimension tsnq/,discrete value of standard dimension alternative for the dimension tfnq/,1discrete value of standard dimension alternative for the dimension t065wn,mq/,3discrete value of standard dimension alternative for the dimension t*/wn,mt4thickness of the side vertical girders webt78thickness of the intermediate vertical girders webEYoungs modulesH8water column on the gate structures sillH505height of opening (height of water action on the gate structure)L4static length of the gate structure (and of horizontal girders)L4horizontal distance between the outer vertical webs of the side girdersL5length of water action on the gate structureM.!9M.!9M, maximal number of horizontal girders of the nth gate element,(M.!9!1) skinplate elementsM.*/minimal number of intermediate horizontal girders of the nth gate elementcQsafety coefficient, usually 1)5c8specific weight of the waterd!-allowable deflection of the horizontal girder360S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIKInt. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley & Sons, Ltd.kPoissons ratio for steel, 0)3odensity of steel, 7850 kg/m3p!-allowable normal stressp3%&,!-allowable reference stressq!-allowable shear stressariables:bfn,mflange width of the mth horizontal girder and the nth gate elementbf*-08/vertical distance between the centre of the web and the upper point of the flange,related to the lower horizontal girderbf*upnvertical distance between the centre of the web and the upper point of the flange,related to the upper horizontal girderbwncommon web height of the horizontal girders of the nth gate elementbwin,mweb height of the mth horizontal girder and the nth gate element, i.e. the elementscommon variable; it is equal to the common web height if the related horizontalgirder existscostself-manufacturing and transportation costs of the entire gate structuredgnumber of horizontal partitionings of the gate, (dg#1) vertical girdersdhn,mvertical distance between the webs of two adjoining horizontal girdersdnsnverticaldistance betweenthe webs of the upper andthe upper-selectedintermediateadjoining horizontal girdersegn,meccentricity distance from the outer edge of the flange to the gravity centre ofa section of the mth horizontal girder and the nth main gate elementhn,mvertical distance (height) between the mth horizontal girder and sill of the nth maingate elementhgnheight of the skin plate of the nth main gate elementhonpart of the opening height H505; belongs to the nth main gate elementhsnvertical distance between the selected upper intermediate horizontal girder and the sillhwnwater column on the nth main gate elements sillk3n,mcoefficient of a normal stress of the mth skin plate element and the nth main gateelement (k3, DIN 19704)k4n,mcoefficient of a normal stress of the mth skin plate element and the nth main gateelement (k4, DIN 19704)tfncommon flange thickness of the horizontal girders of the nth main gate elementtfin,mflange thickness of the mth horizontal girder and the nth main gate element, i.e.elementscommon variable;it is equal to the commonflange thicknessif the relatedhorizontal girder existstsncommon skin plate thickness of the nth main gate elementtwn,mweb thickness of the mth horizontal girder and the nth main gate elementt065wn,mweb thickness twn,mof the outer (upper and lower) horizontal girders of the nth maingate elementt*/wn,mweb thickness twn,mof the inner horizontal girders of the nth main gate elementyn,mbinary variable assigned to the mth horizontal girder or skin plate element and thenth main gate elementyn,vbinary variable for the determination of the number of vertical girders of the entiregateTHE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III361( 1998 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Engng. 43, 329364 (1998)yn,ibinary variable assigned to the standard dimension alternative of the dimension tsnyn,kbinary variable assigned to the standard dimension alternative of the dimension tfnyn,pbinary variable assigned to the standard dimension alternative of the dimensiont065wn,myn,rbinary variable assigned to the standard dimension alternative of the dimensiont*/wn,mIn,mmoment of inertia of the mth horizontal girder and the nth main gate elementln,mcoefficientof the effectivewidth of the skin plate for the flange of the mth horizontalgirder and the nth main gate element (lI, DIN 19704)p1n,mnormal stress in the mth horizontal girder and the nth main gate element at theouter edge of the skin plate flangep2n,mnormal stress in the mth horizontal girder and the nth main gate element at theinner edge of the skin plate flangep1-3n,mnormal stress of the mth skin plate element and the nth main gate element (DIN19704)p1-4n,mnormal stress of the mth skin plate element and the nth main gate element (DIN19704)S
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