Paper received: 07.02.2007 Paper accepted: 07.07.2008 MINLP Optimization of a Single-Storey Industrial Steel Building Tomaž Žula1* - Zdravko Kravanja2 - Stojan Kravanja1 1 University of Maribor, Faculty of civil engineering, Slovenia 2 University of Maribor, Faculty of chemistry and chemical engineering, Slovenia The paper presents the topology and standard sizes optimization of a single-storey industrial steel building, made from standard hot rolled I sections. The structure consists of main portal frames, connected with purlins. The structural optimization is performed by the Mixed-Integer Non-linear programming approach (MINLP). The MINLP performs a discrete topology and standard dimension optimization simultaneously with continuous parameters. Since the discrete/continuous optimization problem of the industrial building is non-convex and highly non-linear, the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm has been used for the optimization. Alongside the optimum structure mass, the optimum topology with the optimum number of portal frames and purlins as well as all standard cross-section sizes have been obtained. The paper includes the theoretical basis and a practical example with the results of the optimization. © 2008 Journal of Mechanical Engineering. All rights reserved. Keywords: industrial buildings, topology optimization, sizing optimization, non-linear programming, MINLP 0 INTRODUCTION Single-storey frame structures are extensively used for industrial, leisure and commercial buildings. In order to obtain efficient frame designs, researchers have introduced various optimization techniques, suitable either for continuous or discrete optimization. O'Brien and Dixon [1] have proposed a linear programming approach for the optimum design of pitched roof frames. Guerlement et al. [2] have introduced a practical method for single-storey steel structures, based on a discrete minimum weight design and Eurocode 3 [3] design constraints. Recently, Saka [4] has considered an optimum design of pitched roof steel frames with haunched rafters by using a genetic algorithm. One of the latest researches reported in this field is the work of Hernández et al. [5], where the authors have considered a minimum weight design of the steel portal frames with software developed for the structural optimization. It should be noted that all the mentioned authors deal with the discrete sizes optimization only at fixed structural topologies. This paper discusses the simultaneous topology, standard sizes and continuous parameter optimization of an unbraced single- storey industrial steel building. The optimization of the portal frames and purlins was performed by the Mixed-Integer Non-linear Programming approach (MINLP). The MINLP is a combined discrete and continuous optimization technique. In this way, the MINLP performs the discrete topology (i.e. the number of frames and purlins) and the standard dimension (i.e. the standard cross-section sizes of the columns, beams and purlins) optimization simultaneously with the continuous optimization of the parameters (e.g. the structure mass, internal forces, deflections, etc.). The MINLP discrete/continuous optimization problems of frame structures are in most cases comprehensive, non-convex and highly non-linear. The optimization is proposed to be performed through three steps. The first one includes the generation of a mechanical superstructure of different topology and standard dimension alternatives, the second one involves the development of an MINLP model formulation and the last one consists of a solution for the defined MINLP optimization problem. The objective of the optimization is to minimize the mass of the single-storey industrial building. The mass objective function is subjected to the set of equality and inequality constraints *Corr. Author's Address: University of Maribor, Faculty of Civil Engineering, Smetanova 17, 2000 Maribor, Slovenia, tomaz.zula@uni-mb.si known from the structural analysis and dimensioning. The dimensioning of steel members is performed in accordance with Eurocode 3. The Modified Outer-Approximation /Equality-Relaxation algorithm is used to perform the optimization, see Kravanja and Grossmann [6], Kravanja et al. [7] and [8]. The two-phase MINLP optimization is proposed. It starts with the topology optimization, while the standard dimensions are temporarily relaxed into continuous parameters. When the optimum topology is found, the standard dimensions of the cross-sections are reestablished and the simultaneous discrete topology and standard dimension optimization of the beams, columns and purlins is then continued until the optimum solution is found. 1 SINGLE-STOREY INDUSTRIAL BUILDING The paper presents the topology and standard sizes optimization of unbraced rigid single-storey industrial building steel structures, Fig. 1. The columns, beams and purlins are proposed to be built up of standard hot rolled steel I sections. The considered portal frame structures are optimized under the combined effects of the self- weight of the frame members, a uniformly distributed surface variable load (snow and wind), a concentrated horizontal variable load (wind) and an initial frame imperfection. The purlins are designed to transfer the permanent load (the self-weight of the purlins and the weight of the roof) and the variable load (snow and wind). The internal forces are calculated by the elastic firstorder method. The dimensioning of the steel members is performed in accordance with Eurocode 3 for the conditions of both the ultimate limit state (ULS) and the serviceability limit state (SLS). When the ULS is considered, the elements are checked for the axial, shear and bending moment resistance, for the interaction between the bending moment and the axial force, the interaction between the axial compression/buckling and the buckling resistance moment. The total deflection 8max subject to the overall load and the deflections d2 subjected to the variable imposed load are calculated to be smaller than the limited maximum values: span/200 and span/250, respectively. The frame horizontal deflections are also checked for the recommended limits: the relative horizontal deflection of the portal frame should be smaller then the frame height/150. Fig. 1. Single-storey industrial steel building 2 MINLP MODEL FORMULATION FOR MECHANICAL SUPERSTRUCTURES It is assumed that a non-convex and nonlinear discrete/continuous optimization problem can be formulated as a general MINLP problem (MINLP-G) in the form: min z = cT y + f (x) s.t. h(x ) = 0 g(x) < 0 (MINLP-G) By + Cx < b n x e X = {x e R : xLO < x < xUP} y e Y ={0,1}m where x is a vector of continuous variables specified in the compact set X and y is a vector of discrete, mostly binary 0-1 variables. Functions fx), h(x) and g(x) are non-linear functions involved in the objective function z, the equality and inequality constraints, respectively. All functions fx), h(x) and g(x) must be continuous and differentiable. All functions fx), h(x) and g(x) must be continuous and differentiable. Finally, By+Cx< b represents a subset of mixed linear equality/inequality constraints. The above general MINLP-G model formulation has been adapted for the optimization of mechanical superstructures. The resulting MINLP formulation for mechanical superstructures (MINLP-MS) that is more specific, particularly in variables and constraints, can be used also for the modelling the steel industrial buildings. It is given in the following form: min z = cT y + f ( x ) s.t. h(x ) = 0 g(x)< 0 A(x)< a Ey < e (MINLP-MS) Dye + R (x)< r Kye + L (dcn )< k Py + S(dst)< s n x e X = {x e R : xLO < x < xUP} y e Y ={0,1}m The MINLP model formulation for mechanical superstructures is proposed to be described as follows: - Included are continuous variables x={d, p} and discrete binary variables y={ye, yst}. Continuous variables are partitioned into design variables d={d°n, d3'} and into performance (non-design) variables p, where subvectors dcn and dst stand for continuous and standard dimensions, respectively. Subvectors of the binary variables ye and yst denote the potential existence of structural elements inside the superstructure (the topology determination) and the potential selection of standard dimension alternatives, respectively. - The mass or economical objective function z involves fixed mass or cost charges in the linear term cTy and dimension dependant mass or costs in the termfx). - Parameter non-linear and linear constraints h(x)=0, g(x) < 0 and A(x) < a represent a rigorous system of the design, loading, resistance, stress, deflection, etc. constraints known from the structural analysis. - Integer linear constraints Ey < e are proposed to describe the relations between binary variables. - Mixed linear constraints .Dye+^(x) < r restore interconnection relations between currently selected or existing structural elements (corresponding ye=1) and cancel relations for currently disappearing or nonexisting elements (corresponding ye=0). - Mixed linear constraints Kye+L(dcn) < k are proposed to define the continuous design variables for each existing structural element. The space is defined only when the corresponding structure element exists (ye=1), otherwise it is empty. - Mixed linear constraints Py+S(dst) < s define standard design variables d3'. Each standard dimension d1 is determined as a scalar product between its vector of i, ze/, discrete standard dimension constants q={q1, qi} and its vector of subjected binary variables /-{A /2, A..., /1}, see Eq. (1). Only one discrete value can be selected for each standard dimension since the sum of the binary variables must be equal to 1 Eq. (2): jst V^ st d =E q y i E I E ¿ =1 (i) (2) 3 OPTIMIZATION MODEL FRAMEOPT The MINLP optimization model FRAMEOPT (FRAME OPTimization) for the optimization of the single storey industrial steel buildings has been developed with relating to the above MINLP model formulation for mechanical structures. The following assumptions and simplifications have been defined in the model FRAMEOPT and considered in the optimization: - Considered was a single load case only, where the partial safety factors and combination of actions were defined according to Eurocodes. The optimization of the structure was performed under the combined effects of: - the self-weight of the structure (the line uniform load of columns, beams and purlins) and the weight of the roof (the vertical surface load) plus - snow and vertical wind (the uniformly distributed vertical surface variable load) plus - horizontal wind (the horizontal force at the top of the columns). - Equal steel portal frames and equal purlins were proposed to compose the structure. - Steel members were proposed to be made from standard hot rolled European wide flange I sections (HEA sections). - The global portal frame geometry including the span, height and the beam inclination was proposed to be fix through the optimization. - Vertical and horizontal bracing systems as well as wall sheeting rails were not included in this calculation/optimization. - The internal forces and deflections were calculated by the elastic first-order method. - The portal frames were classified as non-sway steel portal frames. The ratio between the design value of the total vertical load NSd and the elastic critical value for failure in a sway mode Ncr was constrained: NSd/Ncr< 0.1. - The portal frame was calculated as a laterally supported frame. Hereby, the steel members were checked only for the in-plane instability. Columns were designed for the compression/buckling resistance plus the lateral torsional buckling. Beams were checked for the in-plane bending moment resistance. - Buckling lengths of columns were calculated as the in-plane buckling lengths for the non-sway mode. As an interface for mathematical modelling and data inputs/outputs GAMS (General Algebraic Modeling System), i.e. a high level language, was used [9]. The proposed optimization model includes the structure's mass objective function, parameter structural nonlinear and linear constraints, integer and mixed integer logical constraints, sets, input data (constants) and variables. 3.1. Mass objective function The mass objective function of the industrial building structure is defined by Eq. (3). The mass of the structure MASS comprises the masses of columns, beams and purlins. AC, AB and AP represent the cross-section areas of the column, beam and purlins, respectively. h denotes the height of the column, LB is the length of the frame beam and Ll is the length of the industrial building (and purlins). NOFRAME represents the number of portal frames and NOPURLIN denotes the number of purlins. Each portal frame is constructed from two columns and two beams, see Fig. 2. MASS = 2 • (A ■ h • p) ■ NOFRAME + 2 • (Ab • Lb • p) ■ NOFRAME + (A • Ll • p) NOPURLIN (3) 3.2. Parameter structural non-linear and linear constraints The first constraints of the model represent the constraints (4) to (7) which determine the relations between the continuous cross-sectional dimensions and the cross-sectional height of the column hC. These equations accelerate the convergence of the optimization when standard dimensions are re-established. They define the section breadth bC, the flange thickness f,C, the 1 E I Ab Ib Ab Ib Ac A c tw,c be A b : z ■£"1 — y _ tw,B bB Fig. 2. Portal frame and cross-sections of elements web thickness twC and the cross-section area AC (see Fig. 2) for the column. The second moments of the area about the y-y and z-z axis, Iy, C and Iz,C, the torsional constant It, C and the warping constant Ira, C for the frame column are given by Eqs. (8) to (11). Similar cross-sectional constraints are defined for the frame beam, Eqs. (12) to (16), and for the purlins, see Eqs. (17) to (24). be = - 8.7681 ■ 10-12 ■ hC + 3.5913 ■ 10-9 ■ h6c - 5.9883 ■ 10-7 ■ he5 + 5.1897■ 10-6 ■ h4c - 2.4578-10-3 ■ hC + +6.007■ 10-2 ■ hC -5.8757■ he + 29.294 tf, e = 1.5801 ■ 10-8■ hC + 3.4958^ 10-6 ■ hC + 2.3488^10-4 ■ hC-1.9322■10-3 ■ he + 0.76681 tw, e = -1.0598^ 10-5 ■ hC+2.4652■ 10-3 ■ he + 0.23804 Ae = 2 ■ bc ■ tf, c + (hc - 2 ■ tf, c ) tw, c (4) (5) (6) (7) (hc - 2 ■ tf 12 2 ■ tf, C ■ bc 12 -2 ■ t — ' I. 2 ■ bc ■ tf, c he - h 2- 12 12 11, e = 1 ■( bc ■ t3 e) + 3■ (he " 1c, e = ~4r■( -2■ tf, e)2 2 ■ tf, e ) 3 w, e bB = -8.7681 ■10-12 ■ hB + 3.5913■ 10-9 ■ hj - 5.9883 ■10-7 ■ hB5 + 5.1897■10-6 ■ hj - -2.4578^ 10-3 ■ hB + 6.007■ 10-2 ■ hB - 5.8757■ hB + 29.294 tf,B = 1.5801 ■ 10-8■ hB + 3.4958■10-6 ■ hB + 2.3488^10-4 ■ hB -1.9322■10-3 ■ hj + 0.76681 tw, B =-1.0598■ 10-5 ■ hB +2.4652■ 10-3 ■ hB + 0.23804 AB = 2 ■ bB ■ tf, B +(hB - 2 ■ tf, B )■ tw, B (8) (9) (10) (11) (12) (13) (14) (15) z A y L I z, C 2 ' bB ' f B . tw, B '(hB 2 ' f I 12 12 2 ' b. ' tf K = -8.7681 10-12 ' hp7 + 3.5913'10-9 - hp-5.9883 10-7 ' hp5 + 5.1897 '10-6 - hp4 --2.4578' 10-3 ' hp + 6.007' 10-2 ' hp2 - 5.8757' h + 29.294 f p = 1.5801'10-8' hP + 3.4958'10-6 ' hp + 2.3488'10-4 ' hp-1.9322'10-3 ' hp + 0.76681 = -1.0598 ' 10-5 ' h p +2.4652 ' 10-3 ' hp + 0.23804 2 ' bp ' /f, p + (bp - 2 ' f p )tw, p C „ = Ap = 2 ' bc ' /f3 p *w, p '(^«p - 2 ' if, p j 12 12 -2 ' bp ' tf, p ' hp f p 2 ' f p ' bp , (bp - 2 ' /f, p )) 12 12 =3'(2-bp 'tf3 p) + 3'(hp -2'/f, p)) '(hp -2'tf, The length of the frame beam LB is calculated according to Eq. (25) and the angle of the inclination of the beam a is defined by Eq. (26). L represents the frame span and f denotes the overheight of the frame beam: Lb =7(L/ 2)2 + f 2 a=arctan (//(L/ 2)) (25) (26) The uniformly distributed vertical surface variable load qz, the uniformly distributed horizontal surface variable load qy, the self-weight per unit length of the portal frame g, the concentrated design horizontal variable wind load P (for the ULS) and wind load Pw (for the SLS) are defined by Eqs. (27) to (31): qz =( cos2(a) + wv ) -ef qy = s ' cos(a) ' sin(a) ' ef g=Ab 'p + (Ap 'p-et)/ep P = Yq ' Wh Pw ' h/ 2 ' h/ 2 (27) (28) (29) (30) (31) (16) (17) (18) (19) (20) (21) (22) (23) (24) Where 5, wv and wh represent snow, the vertical and horizontal wind per m2 (the variable imposed load); ef stands for the intermediate distance between the portal frames, p is the density of steel, yq is the partial safety factor for the variable load and h represents the height of the columns. The number of the portal frames NOFRAME, the number of purlins NOPURLIN and the maximal intermediate distance between the purlins ep are determined by Eqs. (32) to (38), where LL represents the length of the industrial building, MINNOfmme and MAXNObsme denote the minimal and maximal number of defined portal frames, and MINNOpurlm and MAXNOpmlm stand for the minimal and maximal number of purlins. NOFRAME=L,/ ef +1 NOFRAME > MINNOi"me NOFRAME < MAXNOf"me NOPURLIN = 2 ' (LB/eD +1 ) NOPURLIN > MINNOpurlin NOPURLIN < MAXNOpurlin ep < 250 [cm] (32) (33) (34) (35) (36) (37) (38) I z, P Eqs. (39) to (45) represent the constraints which determine the portal frames to be non-sway frames. The column stiffness coefficient KC, the effective beam stiffness coefficient KB, the distribution factor of the column for the sway frame nf, and the plane buckling length of the column for a sway frame mode ¿#sway are calculated by Eqs. (39) to (42). The value of the distribution factor n2 is taken to be 1 because of the pinned connection of the columns. Kc = ^ C h K = Ib nS = Kc Kc +1.5-KB Psway i-0.2-( + n )-0.12-tf-n 1-0.8-( + n2 ) + 0.6-nS-n (39) (40) (41) (42) Eq. (43) represents the elastic critical load ratio (Nsd/Ncr) which defines the steel portal frame to be a non-sway frame. The distribution factor of the column for the non-sway frame nNS and the plane buckling length of the column for the non-sway frame mode ft non-sway are given by Eqs. (44) to (45): P-h (y+ Y g - g)-L n -E-Iy (Psway -h) C < 0.1 (43) Pno K C + 0.5- Kb , = 0.5 + 0.14 - ( nNS + n2 ) + 0.055 - ( nN n, (44) (45) The ULS constraints for the frame columns are defined by Eqs. (46)-(52). Eq. (46) represents the condition for the design bending moment resistance of the column (Msdk=i (96) 3.4. Sets, input data (constants) and variables The following sets, input data (constants: scalars and parameters) as well as continuous and binary variables are involved in the optimization model FRAMEOPT: Sets: i set for the standard dimension alternatives for columns, ieI j set for the standard dimension alternatives for beams, jeJ k set for the standard dimension alternatives for purlins, keK m set for the number of purlins, m eM n set for the number of portal frames (columns and beams), neN Scalars (constants, input data): f denotes the overheight of the frame beam [cm] f yield the strength of the structural steel [kN/cm2] h height of the column [cm] k effective length factor [-] kw effective length factor [-] mr mass of the roof plates [kg/cm2] 5 snow (variable imposed load) [kN/cm2] wv vertical wind (variable imposed load) [kN/cm2] wh horizontal wind (variable imposed load) [kN/cm2] CbC2 equivalent uniform moment factors [-] E elastic modulus of steel [kN/cm2] G shear modulus of steel [kN/cm2] L frame span [m] Ll length of the industrial building [m] minimum number of defined portal frames [-] maximum number of defined portal frames [-] minimum number of purlins [-] MINNO MAXNO11 MINNOpurlin -frame k k MAANOpurlm maximum number of purlins [-] ab imperfection factor [-] aLT imperfection factor [-] yq partial safety factor for the variable load [-] Yg partial safety factor for the permanent load [-] yM0 resistance partial safety factor [-] n2 distribution factor [ 1 ] A1 slenderness [-] n Ludolf s number [-] p density of steel [kg/m3] Parameters (constants, input data): qAC vector of i, ieI, discrete standard constants for cross-section area of the column q Ab vector of j, jeJ, discrete standard constants for cross-section area of the beam q AP vector of k, keK, discrete standard constants for cross-section area of the purlin qibc vector of i, ieI, discrete standard constants for overall breadth of the column q bB vector of j, jeJ, discrete standard constants for overall breadth of the beam qbP vector of k, keK, discrete standard constants for overall breadth of the purlin qf C vector of i, ieI, discrete standard constants for flange thickness of the column q j- B vector of j, jeJ, discrete standard constants for flange thickness of the beam q kP vector of k, keK, discrete standard constants for flange thickness of the purlin qSC vector of i, ieI, discrete standard constants for web thickness of the column q tw- B vector of j, jeJ, discrete standard constants for web thickness of the beam q kw- P vector of k, keK, discrete standard constants for web thickness of the purlin qJ1, C vector of i, ieI, discrete standard constants for torsional constant of the column q kP vector of k, keK, discrete standard constants for torsional constant of the purlin q Jy' C vector of i, i eI, discrete standard constants for second moment of area about the y - y axis of the column qJy' B vector ofj,jeJ, discrete standard constants for second moment of area about the y - y axis of the beam q ky-P vector of k, keK, discrete standard constants for second moment of area about the y - y axis of the purlin q Jz- C vector of i, ieI, discrete standard constants for moment of area about the z - z axis of the column q kz-P vector of k, keK, discrete standard constants for moment of area about the z - z axis of the purlin q J"- C vector of i, ieI, discrete standard constants for warping constant of the column q k"-P vector of k, keK, discrete standard constants for warping constant of the purlin Continuous variables: bB overall breadth of the beam [cm] bC overall breadth of the column [cm] bP overall breadth of the purlin [cm] ef intermediate distance between the portal frames [cm] ep intermediate distance between the purlins [cm] g self-weight of the portal frame [kN/cm] hB cross-sectional height of the beam [cm] hC cross-sectional height of the column [cm] hP cross-sectional height of the purlin [cm] qz uniformly distributed horizontal surface variable load [kN/cm] qy uniformly distributed vertical surface variable load [kN/cm] tfB flange thickness of the beam [cm] tfC flange thickness of the column [cm] tfP flange thickness of the purlin [cm] twB web thickness of the beam [cm] twC web thickness of the column [cm] twP web thickness of the purlin [cm] Ab cross section of the beam [cm2] AC cross section of the column [cm2] AP cross section of the purlin [cm2] Ite torsional constant of the column [cm4] ItP torsional constant of the purlin [cm3] IyB second moment of area about the y - y axis of the beam [cm4] Iy,e second moment of area about the y - y axis of the column [cm4] IyP second moment of area about the y-y axis of the purlin [cm4] Iz,e second moment of area about the z - z axis of the column [cm4] Iz,p second moment of area about the z-z axis of the purlin [cm4] Im,e warping constant of the column [cm6] ImP warping constant of the purlin [cm6] Kc stiffness coefficient of the column [m3] Kb stiffness coefficient of the purlin [m3] Lb length of the beam [cm] MeR elastic critical moment for lateral torsional buckling [kNcm] Mel,Rd design elastic moment resistance [kNcm] NOFRAME number of the portal frames [-] NOPURLIN number of the purlins [-] Npl,Rd design plastic axial resistance[kN] Nsd design axial force [kN] Msd design bending moment [kNcm] P concentrated horizontal variable load multiplied by the partial safety factor [kN] Pw concentrated horizontal variable load [kN] Vpl,Rd design plastic shear resistance [kN] Vsd design shear force [kN] a angle of the inclination of the beam [rad] A»n-sway plane buckling length of the column for a non-sway frame [-] j#sway plane buckling length of the column for a sway frame [-] c>F vertical deflection of the portal frame [cm] A horizontal deflection of the portal frame [m] distribution factors of the column for the non-sway frame [-] nS distribution factors of the column for the sway frame [-] k reduction factor due to the flexural buckling [-] klt reduction factor for lateral-torsional buckling [-] Binary variables: y¡ binary variable assigned to the i-th, i el, standard dimension alternative of the columns yj binary variable assigned to the j-th, jeJ, standard dimension alternative of the beams yk binary variable assigned to the k-th, keK, standard dimension alternative of the purlins ym binary variable assigned to the m-th, m eM, topology alternative of the purlins yn binary variable assigned to the «-th, neN, topology alternative of the frames Substituted expressions: A,B,C,D functions which are substituded in Eqs. 46 a,b,c and 49 a U,V functions which are substituded in Eqs. 61 a,b 4 OPTIMIZATION The Modified Outer-Approximation /Equality-Relaxation (OA/ER) algorithm (Kravanja and Grossmann [6]) was used to perform the optimization. The OA/ER algorithm consists of solving an alternative sequence of Non-linear Programming optimization subproblems (NLP) and Mixed-Integer Linear Programming master problems (MILP), Fig. 3. The former corresponds to the optimization of parameters for a building structure with a fixed topology and standard dimensions and yields an upper bound to the objective to be minimized. The latter involve a global linear approximation to the superstructure of alternatives in which a new topology and standard sizes are identified. When the problem is convex the search is terminated when the predicted lower bound exceeds the upper bound, otherwise it is terminated when the NLP solution can be improved no more. The OA/ER algorithm guarantees the global optimality of solutions for convex and quasi-convex optimization problems. The OA/ER algorithm as well as all other algorithms do not generally guarantee that the solution found is the global optimum. This is due to the presence of non-convex functions in the models that may cut off the global optimum. In order to reduce undesirable effects of nonconvexities, the following nonstructured and structured convexifications are applied for the MINLP Superstructure Combined optimization Fixed binary variables NLP Subproblem Continuous optimization New binary variables MILP Master problem Discrete optimization Convergence ? NO YES STOP Fig. 3. Steps of the OA/ER algorithm decomposition and the deactivation of the objective function linearization, the use of the penalty function, the use of the upper bound on the objective function to be minimized as well as the global convexity test and the validation of the outer approximations. By the use of the mentioned modifications, the likelihood of obtaining better results by the OA/ER algorithm, is significantly increased. A more extended information about these modifications may be found elsewhere, see Kravanja and Grossmann [6], and Kravanja et al. [7]. The optimum solution of a complex non-convex and non-linear MINLP problem with a high number of discrete decisions is in general very difficult to obtain. The optimization is thus proposed to be performed sequentially in two different phases to accelerate the convergence of the OA/ER algorithm. The optimization is proposed to start with the discrete topology optimization of the building, while the standard MILP master problem of the OA/ER algorithm: the deactivation of linearizations, the dimensions are temporarily relaxed into continuous parameters. Topology and continuous parameter optimization is soluble (a smaller combinatorial problem) and accumulates a good global linear approximation of the superstructure (a good starting point for the next phase overall optimization). When the optimum topology is found, the standard sizes of the cross-sections are re-established and the simultaneous discrete optimization of the topology and standard dimensions of the beams, columns and purlins is then continued until the optimum solution is found. The two-phase strategy requires that the binary variables should be defined in one uniform set. In the first phase, only the binary variables which are subjected to topology alternatives become active. Binary variables of standard dimension alternatives are temporarily excluded (set on value zero) until the beginning of the second phase, in which they participate in the simultaneous overall optimization. The same holds for standard dimension logical constraints. In the first phase they are excluded, while the second phase includes them into the optimization. The data and variables are initializated only once in the beginning of the optimization. An advantage of this strategy is also in the fact that binary variables for topology and standard dimensions need not be initialized: after the first NLP, the first phase always starts in the subspace of the topological binary variables only, while the second phase starts with the MILP master subproblem which then predicts a full set of binary variables for the successive NLP. Under the convexity condition, the two-phase strategy guarantees a global optimality of the solution. The optimization model may contain up to thousand binary 0-1 variables of the alternatives. Most of them are subjected to standard dimensions. Since this number of 0-1 variables is too high for a normal solution of the MINLP, a reduction procedure was developed, which automatically reduces the binary variables of alternatives into a reasonable number. The optimization at the second phase includes only those 0-1 variables which determine the topology and standard dimension alternatives close to the values, obtained at the first MINLP optimization phase. 5 NUMERICAL EXAMPLE The paper presents an example of the topology and standard dimension optimization of a single-storey industrial building. The building is 25 meters wide (L), 75 meters long (Ll) and 6 meters height (H), see Fig. 4. The structure consists of equal non-sway steel portal frames which are mutually connected with purlins. The overheight of the frame beam f) is 0.50 m. The portal frame is subjected to self-weight of the structure and the roof g, and to the variable loads of snow and wind. The mass of the roof is mr = 0.20 kg/m2. The variable imposed loads: 5 = 2.00 kN/m2 (snow), wv = 0.125 kN/m2 (vertical wind) and wh = 0.50 kN/m2 (total horizontal wind) are defined in the model input data. Both, the horizontal concentrated load at the top of the columns and the vertical uniformly distributed line load on the beams and purlins are calculated automatically through the optimization considering the calculated intermediate distance between the portal frames and purlins. The material used is steel S 355. The yield strength of the steel (f) is 35.5 kN/cm2, the density of steel (p) is 7.850T0-3 kg/cm3, the elastic modulus of steel (E) is 210 GPa and the shear modulus (G) is 80.76 GPa. The partial safety factor for the permanent load (yg) and for the combined (snow plus wind) variable load (yq) are both 1.35. The resistance partial safety factor (YM0) is 11. The imperfection factor (ab) is 0.34, the imperfection factor (aLT) is 0.21, the distribution factor (n2) is 1, slenderness for the steel S 355 is 76.4, the effective length factors (k and kw) are 1.0, the equivalent uniform moment factors for beams (Q) and (C2) are 1.879 and 0, respectively. While the defined minimum and maximum numbers of portal frames (MINNOiiame and MAXNOiiame) are 1 and 30, the minimal and maximal numbers of purlins (MINNOpurlin and MAXNOpurlin) are 1 and 20. L= 25 m Fig. 4. Global geometry of the single-storey industrial building The lower and upper bounds as well as the activity levels (starting points) of the independent continuous variables are shown in Table 1. The bounds and starting points of other dependant continuous variables are defined by using equations from the optimization model regarding the independent variables. Table 1. Bounds and activity levels of the independent variables Variable X Lower bound xlo Activity level xl Upper bound xup he 50 cm 80 cm 99 cm kB 30 cm 60 cm 70 cm kP 10 cm 20 cm 30 cm NOFRAMES 1 20 30 NOPURLINS 1 20 20 An industrial building superstructure was generated in which all possible structures were embedded by 30 portal frame alternatives, 10 various purlin alternatives and a variation of different standard sizes. The superstructure comprised 24 different standard hot rolled European wide flange I sections, i.e. HEA sections (from HEA 100 to HEA 1000) for each column, beam and purlin seperately. Vectors q of 24 discrete values for different standard sections were defined. For example, the vectors for the section's heights qh.C, qj, qhkp and the cross- section areas qjC, qjB , qjp are for the columns, beams and purlins defined as follows: q kC = q j = q kp = {9.6, 114, 13.3, 15.2, 17.1, 19.0, 23.0, 25.0, 27.0, 29.0, 31.0, 33.0, 35.0, 39.0, 44.0, 49.0, 54.0, 59.0, 64.0, 69.0, 79.0, 89.0, 99.0} q AC = q jB = q t = {21.2, 25.3, 31.4, 38.8, 45.3, 53.8, 64.3, 76.8, 86.8, 97.3, 113.0, 124.0, 133.0, 143.0, 159.0, 178.0, 198.0, 212.0, 226.0, 242.0, 260.0, 286.0, 321.0, 347.0} Regarding construction alternatives, the superstructure consists of a n possible number of portal frames, neN, N={ 1,2,3,...,30}, and 10 various even (2m) numbers of purlins, meM, M={1,2,3,...,10}, which give 30-10=300 different topology alternatives. Since i, j and k different standard sections are also defined for columns, beams and purlins seperately, iel, jeJ, keK, I=J=K={ 1,2,3,4,5,6,7,.......24}, there exist n-m-ij-k = 30-10-24-24-24 = 4147200 different discrete construction alternatives alltogether. The optimization was performed by the proposed MINLP optimization approach. The task of the optimization was to find the minimal structure mass, the optimum topology (the optimum number of portal frames and purlins) and the optimum standard sizes. The optimization was carried out by a user-friendly version of the MINLP computer package MIPSYN, the successor of PROSYN [6] and TOP [7], [8] and [10]. The Modified OA/ER algorithm and the two-phase optimization were applied, where GAMS/CONOPT2 (Generalized reduced-gradient method) [11] was used to solve the NLP subproblems and GAMS/Cplex 7.0 (Branch and Bound) [12] was used to solve the MILP master problems. The two-phase MINLP optimization was applied. After the first performed continuous NLP (the initialization), the first phase started with the discrete topology optimization at the relaxed standard dimensions, see also the convergence of the Modified OA/ER algorithm in Table 2. At this level, only the binary variables yn and ym for topology optimization, parameter structural nonlinear and linear constraints, Eqs.(4) to (65), and the logical constraints for topology optimization, Eqs. (66) to (69), were included. When the optimum topology was reached (110.161 tons at the 2nd MINLP iteration, all the following solutions were poorer), the optimization proceeded with a simultaneous discrete topology and standard dimension optimization at the second level. At this phase, the binary variables yi, yj and yk of standard sizes for columns, beams and purlins, as well as the logical constraints for standard dimensions, Eqs. (70) to (96), were added into the optimization. The final optimum solution of 122.144 tons was obtained at the 6th main MINLP iteration (all the following solutions were not as good). Table 2. Convergence of the Modified OA/ER algorithm MINLP Iteration MINLP Subphaze Result Mass [tons] Topology Cross-sections [cm2] Frames Purlins Column Beam Purlin Phase 1: topology optimization 1. Initialization 1.NLP 107.254 11.955 12.008 283.501 212.036 39.892 2. 1.MILP 107.763 13 14 321.721 176.232 30.527 2.NLP 110.161 276.759 206.027 32.137 3. 2.MILP 114.351 14 14 326.447 174.106 30.509 3.NLP 111.339 270.667 200.658 28.442 Phase 2: topology and standard dimension optimization 4. 3.MILP 125.260 14 14 321.00 198.00 38.80 4.NLP* 125.231 HEA 900 HEA 550 HEA 160 5. 4.MILP 115.708 12 14 321.00 212.00 38.80 5.NLP* 115.209 HEA 900 HEA 550 HEA 160 6. 5.MILP 122.144 13 14 321.00 212.00 38.80 6.NLP 122.144 HEA 900 HEA 550 HEA 160 7. 6.MILP 126.713 13 16 321.00 212.00 38.80 7.NLP 126.713 HEA 900 HEA 550 HEA 160 * Locally infeasible The optimum result represents the mentioned minimal structure mass of 122.144 tons, the obtained optimum topology of 13 portal frames and 14 purlins, see Fig. 5, and the calculated optimum standard sizes of the columns (HEA 900), beams (HEA 550) and purlins (HEA 160), see Fig. 6. At the second phase, where all the calculated dimensions were standard ones, a feasible optimum result was very difficult to be obtained. The optimization model contained a high number of 4147200 different discrete construction alternatives. Fig. 5: Optimum design of the single-storey industrial building 12 x 2.08 m 25.0 m Fig. 6: Optimum design of the portal steel frame The prescreening procedure of alternatives was thus applied, which automatically reduced the binary variables of alternatives into a reasonable number. The optimization at the second phase included only those 0-1 variables which determined the topology and standard dimension alternatives close to the (continuous) values, obtained at the first phase. For topology, column, beam and purlin only 3 binary variables were used (1 variables under and 2 over the continuous value). In this way, only 15 binary variables were used in the second phase instead of all 112 binary variables. The number of 4147200 discrete construction alternatives was significantly reduced to n-m-ij-k = 3-3-3-3-3 = 243 alternatives, which considerably improved the efficiency of the search. 6 CONCLUSIONS The paper presents the simultaneous topology and standard sizes optimization of a single-storey industrial steel building. The optimization is proposed to be performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The Modified OA/ER algorithm and the two-phase MINLP optimization strategy were applied. The proposed two-phase optimization starts with the topology optimization of the frames and purlins, while the standard dimensions are temporarily relaxed into continuous parameters. When the optimum topology is found, the standard dimensions of the cross-sections are re-established and the simultaneous topology and discrete standard dimension optimization of beams, columns and purlins is then continued until the optimum solution is found. 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