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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.si CCC 00295981/98/02032936$17.50Received 12 August 1996 ( 1998 John Wiley optimization; topology optimization; discrete variable optimization; Mixed-Integer Non-linear Programming; MINLP; the Modifi ed OA/ER algorithm; MINLP strategy; hydraulic gate; sliding gate; roller gate; Aswan 1. INTRODUCTION This paper describes the Mixed-Integer Non-linear Programming (MINLP) approach to the synthesis of roller and sliding gate structures, i.e. the simplest types among vertical-lift hydraulic steel gates, see Figure 1. Roller and sliding gates are also regarded as the most frequently Figure 1. Vertical-lift hydraulic steel gate structure manufactured types of hydraulic steel gates for headwater control. They are used to regulate the water stream on hydro-electric plants, dams or spillways. As hydraulic steel gates are very special structures, only a few authors have discussed their optimization, e.g. Kravanja et al.,13 Jongeling and Kolkman4 as well as Almquist et al.5 Particular interest was shown in the optimization not of these (roller and sliding gates) but of similar structures. In such investigations, Vanderplaats and Weisshaar6 as well as Gurdal et al.7 optimizedstiff enedlaminated compositepanels, Butler8 and Ringertz9 optimized stiff enedpanels, Farkas and Jarmai10 optimizedwelded rectangular cellular plates, Finckenoret al.11 treated skin stringer cylinders and Gendy et al.12 stiff ened plates. Almost all authors used Non-linear Programming (NLP) techniques. Gurdal et al.7 proposed the genetic algorithm, while Kravanja et 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 the simultaneous topology and parameter optimization of structures is presented. Part II describes the extension to the simultaneous standard dimension optimization. Based on the superstructure approach defi ned in Parts I and II, the main objective of this paper (Part III) is the MINLP synthesis of roller and sliding hydraulic steel gate structures, obtained at minimal gate costs and subjected to defi ned design, material, stress, defl ection and stability constraints. As the MINLP approach enables simultaneous topology, parameter and standard dimension optimization, a number of gate structural elements (girders and plates), the gate global geometry, intermediate distances between structural elements and all continuous and standard dimensions are obtained simultaneously.This last part of the three-part series of papers is divided into three main sections: 1. Section 2 describes how diff erent topology and standard dimension alternatives are postu- lated and how their interconnection relations are formulated by means of explicit logical constraints in order to perform topology and standard dimension alterations within the optimization procedure. 330S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIK Int. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley (b) minimal structure with only fi xed structural elements girders. As the modelling of vertical girders is simplifi ed and needs no special interconnection logical constraints, the modelling of discrete decisions regarding horizontal girders proved to be more sophisticated. 2.2.1. Modelling of topology alterations Let us consider the vertical cross-section of the gate element superstructure with fi xed and alternative horizontal girders, see Figure 3(a). The number of fi xed and alternative girders and their locations in the superstructure can be described by the following logical constraints: M.*/) + m|M ym)M.!9(9) ym)ym1,m2, 3, . . . , M!1(10) yM.!91(11) Logical constraint (9) defi nes the minimal (M.*/) and maximal (M.!9) number of structural elements (girders). While number M.*/ represents the number of fi xed structural elements, the diff erence between the maximal and minimal number of elements (M.!9!M.*/) gives the number of alternative structural elements. Constraint (10) defi nes the direction of the removal of alternative elements: from the top down the superstructure. From Figure 3 is evident that the mostupperelementis the fi xed one, whichis setby the constraint(11). It then furtherfollows from 334S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIK Int. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley here they are proposed to be calculated by constraints (21). The next constraints (24)(28) express normal and shear stresses, acting in the section of horizontal girders. The combinations between these stresses are 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 and constraints (30) are actuated when water acts on the inner side of the skin plate. The plate buckling reduction coeffi cient s, dependent on the plate slenderness ratio jP , is defi ned by EUROCODE-3:17 s0)6/Jj2 P !0)13. In this paper, the buckling coeffi cient s is determined by the equation s1/(!0)22703#1)75365j1 ). Lateral buckling of horizontal girders is verifi ed by constraints (32) in the case of compressed girder fl anges when water acts on the inner side of the skin plate. All stability constraints (29)(32) are defi ned according to EUROCODE-3. The horizontalgirder defl ectionis expressed by means of the inequality constraint (33). All constraints of the lower and upper horizontal girders are defi ned in the same manner as intermediate girders, with the exception that diff erent coeffi cients of the eff ective width ln,mare used here, see constraints (34) and (35). THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III339 ( 1998 John Wiley m2, 3, . . . , (M!1) for the girder, m1, 2, . . . , (M!2) for the lower and m2, 3, . . . , (M!1) for the upper skinplate element p2 1-3n,m1#(!p1n,m#k)p1-3n,m1)2!p1-3n,m1)(!p1n,m#k)p1-3n,m1)p23% mM for the girder and mM!1 for the skin plate element: p2 1-3n,M1#(!p1n,M#k)p1-3n,M1)2!p1-3n,M1)(!p1n,M#k)p1-3n,M1)p23% 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|N h0n(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 constraints The set of constraints (80)(82) represents integer logical linear constraints. Constraints (80) determine the number of fi xed structural elements (fi xed horizontal girders and skin plate elements) for each gate element. Constraints (81) make sure that the number of horizontal girders and skin plate elements of the upper gate elements is equal to or higher than the one of the lower gateelements.Constraints(82) explainthat the upper alternative(intermediate)horizontalgirders and skin plate elements disappear in the direction from the top to the bottom of the gate before the lower ones do. This, in eff ect, reduces the combinatorics of the MINLP problem since it THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III345 ( 1998 John Wiley m2, 3, . . . , M!1(82) 3.5. Interconnection logical constraints Interconnection logical mixed-integer linear constraints start with constraint (83), which defi nesthe gate partitionin the horizontal direction.Constraints (84)(87) defi ne vertical distance hsnbetween the uppermost selected intermediate horizontal girder and the sill of the gate element. Consequently, the uppermost fi xed horizontal girder is always connected to the uppermost selected intermediate alternative horizontal girder and disconnected from the disappearing ones. Constraints (88)(91) determine the minimal openings between fl anges of horizontal girders. dg2#2+ v|V yn,v,n1(83) hsn*hn,m!h61 4/ )(1!yn,m1)!h61 4/ )(1!yn,m)!h61 4/ )yn,m1, n1, 2, . . . , N;m2, . . . , M!2(84) hsn)hn,m#h61 4/ (1!yn,m1)#h61 4/ )(1!yn,m)#h61 4/ )yn,m1, n1, 2, . . . , N;m2, . . . , M!2(85) hsn*hn,M1!h61 4/ )(1!yn,M1),n1, 2, . . . , N(86) hsn)hn,M1#h61 4/ )(1!yn,M1),n1, 2, . . . , N(87) hn,2!bfn,2/2!h-08/!bf*-08n*hm3, 4, . . . , M!1(89) hn,M!bf*upn!(hn,m#bfn,m/2)*hm1, 2, . . . , M!1(90) hn,M!bf*upn!(hn,M1#bfn,M1/2)*h if not, the bounds are set to zero. bfn,m)b61 fn,m )yn,m,n1, 2 , . . . , N; m1, 2, . . . , M(92) bfn,m*b-08,%9 fn,m )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(93) twn,m)t61 wn,m )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(94) twn,m*t-08,%9 wn,m )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(95) dhn,m)d61 hn,m )yn,m,n1, 2, . . . , N; m1, 2, . . . , M!1(96) dhn,m*d-08,%9 hn,m )yn,m,n1, 2, . . . , N; m1, 2, . . . , M!1(97) bwin,m)b61 wn )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(98) bwin,m*b-08 wn )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(99) tfin,m)t61 fn )yn,m,n1, 2, . . . , N; m1, 2, . . . , M(100) tfin,m*t-08 fn )yn,m, n1, 2, . . . , N; m1, 2, . . . , M(101) 3.6.2. ogical relations for common variables Mixed-integer linear constraints (102)(105) defi ne the logical relations of the common design variables. These constraints enforce upon the common design variables of the horizontal girders bwin,mand tfin,mthe values of the corresponding common variables of the entire gate bwnand tfn. bwin,m)bwn#b61 wn )(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(102) bwin,m*bwn!b61 wn )(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(103) tfin,m)tfn#t61 fn )(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(104) tfn,m*tfn!t61 fn )(1!yn,m),n1, 2, . . . , N; m1, 2, . . . , M(105) 3.7. ogical constraints for standard variables 3.7.1. Standard dimension logical constraints Standard dimension logical constraints defi ne four standard dimensions: thickness of the skin plate tsn , fl ange thickness of the horizontal girder tfn, the web thickness of the outer horizontal girder t065 wn,m and the web thickness of the inner horizontal girder t*/ wn,m. Each standard dimension is determined by a scalar product between its corresponding vector of binary variables y and its assigned vector of discrete constants q (constraints (106), (108), (110), (111), (113) and (114), where only one discrete value can be selected (+y1), see constraints (107), (109), (112) and (115). Note thatd#03denotescorrosionadditionto the sheet-ironplates. In the synthesis,these dimensionsare fi rst 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 t065 wn,m and t*/ wn,m. As standard dimensions tsn and tfncorrespond to the common standard design variables for THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III347 ( 1998 John Wiley m1 or mM(110) t065 wn,m*+ p|P (q/,1!d#03)yn,p!t065,UP wn,m (1!yn),n1, 2, . . . , N; m1 or mM(111) + p|P yn,p1,n1, 2, . . . , N(112) t*/ wn,m)+ r|R (q/,3!d#03)yn,r#t*/,UP wn,m (1!yn),n1, 2, . . . , N; m1, 2, . . . , M(113) t*/ wn,m*+ r|R (q/,3!d#03)yn,r!t*/,UP wn,m (1!yn),n1, 2, . . . , N; m1, 2, . . . , M(114) + r|R yn,r1n1, 2, . . . , N(115) 3.8. Constraints for the MILP phase only Additional MILP constraints (116)(121), considered only in the MILP phase of the MINLP optimization, are included in the model in order to perform the optimization faster and more effi ciently. To enforce faster MINLP convergence, equal vertical distances between the selected horizontal 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. BEDENIK Int. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley 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,M2 n1, 2, . . . , N; m1, 2, . . . , M!3(121) 3.9. Heuristic constraints Heuristic constraints additionally increase the effi ciency of the MINLP optimization and are added only for the MILP phase. Constraints (122)(125) show that the thicknesses of each upper gate element must not exceed the ones of the lower gate elements. Each upper main gate element is 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 variables The initialization of continuous and binary variables is also included, see list of symbols in AppendixI. Only design variableshave to be initialised, while non-design variables are calculated on the basis of the initialized design variables. 4. SYNTHESIS OF THE INTAKE GATE IN ASWAN Acomprehensivecomparativedesign 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 existing roller gate, named Intake Gate, see Figure 6. The Intake Gate was constructed and erected in Aswan II, Egypt, by the Slovenian company Metalna19 from Maribor. Eight identical turbine intakes 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 simplifi ed static system for the structural analysis of the gate is used. The material actually used, i.e. St 44-2, has been considered in the optimization. The technical data 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 been calculated for the entire gate at the actual topology of 4-4-4/11 (4 horizontal girders for each of the three main gate elements and 11 vertical girders for the entire gate with the corresponding number of skin plate elements). The task of the synthesis of the Intake Gate is to fi nd the optimal topology, geometry as well as continuous and standard sizes with respect to the minimum of self-material and labour costs, subjected to design, stress, defl ection, and stability constraints. The synthesis is performed THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESISPART III349 ( 1998 John Wiley (b) horizontal cross-section through three steps: it begins with the generation of the gates superstructure, continues with the development of the gates MINLP optimization model and ends by solving the defi ned MINLP problem. 4.1. Simultaneous topology and parameter optimization of the Intake Gate Let us fi rst consider the synthesis of the Intake Gate followed by a simultaneous topology and continuous parameter optimization. Such synthesis has been performed by means of the MINLP approach introduced in Part I. This approach enables the obtaining of the optimal number of horizontal and vertical girders and skin plate elements as well as all continuous optimal dimensions simultaneously. Thegenerated superstructureof the IntakeGate is the same as the one in Figure 2. It comprises n 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 corresponding number (m!1)(2#2v) of skin-plate elements per each main gate element. 350S. KRAVANJA, Z. KRAVANJA AND B. S. BEDENIK Int. J. Numer. Meth. Engng. 43, 329364 (1998)( 1998 John Wiley (b) minimal gate topology 4-4-4/5: only fi xed structural elements () are included, alternative elements (- - -) are removed determine the number of (3#2+l|Vy1,v) vertical girders for the entire gate. The number of skin plate elements is given automatically when the number of horizontal and vertical girders is known. Each gate topology is therefore expressed by the correspondingvector of binaryvariables yMyn,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, y1,3 N. Note that the smallest fi rst index represents the lowest main gate element and the smallest second index the lowest horizontal girders. In order to obtain the whole linear approximation for the MILP master problem of the modifi ed OA/ER algorithm, the fi rst NLP has to be performed for the entire superstructure and the 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 the MILP steps, clearly show which structural elements are selected and which are rejected through the optimization. The gate has been modelled by means of the proposed general optimization model for roller and sliding gate structures GATOP (GATe OPtimization). The self-manufacturing costs, such as material, 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 given design, material, stress, defl ection and stability constraints. For real comparison between the obtainedoptimal gate and the actually erected IntakeGate, identical economic parameters, static system, material, stress and defl ection allowances as well as all functions from the structural analysis of the Intake Gate have been considered as constraints in the model. As standard dimensions are in this chapter not explicitly