DESIGN OPTIMIZATION OF 3D STEEL FRAME STRUCTURES

I DESIGN OPTIMIZATION OF 3D STEEL FRAME STRUCTURES 9.1 Objectives Two objectives are associated with this chapter. First is to ascertai the advatages, metioed i Chapter 8, of the developed algorithm cosiderig more complex problems. Secod is to ivestigate the effect of the approaches, employed for determiig the effective bucklig legth of a colum, o the optimum desig. This chapter therefore exteds the work to the discrete optimum desig of 3 dimesioal (3D) steel frame structures usig the modified geetic algorithm (GA) liked to desig rules to BS 5950 ad BS 6399. I the formulatio of the optimizatio problem, the objective fuctio is the total weight of the structural bers. The cross sectioal properties of the structural bers, which form the desig variables, are chose from three separate catalogues (uiversal beams ad colums covered by BS 4, ad circular hollow sectios from BS 4848). Chapter 2 ad 3 idicate that the theory ad methods for the evaluatio of the effective legth of colums are based o a secod order aalysis assumig that the

Desig Optimizatio of 3D Steel Frame Structures 274 bucklig of bers out of plae of the framework is preveted. This, however, could be correct for 3D structures as well if a rigid bracig system or shear wall, etc., is provided. Therefore, due to the eed for the use of a more accurately evaluated effective bucklig legth of colums i 3D steel frame structures, the fiite elemet method is employed. Followig the desig procedure of steel structures to BS 5950, the miimum weight desigs of two 3D steel frame structures subjected to multiple loadig cases are obtaied. These examples show that the modified GA i combiatio with the structural desig rules provides a efficiet tool for practicig desigers of steel frame structures. This chapter starts with describig the desig procedure for 3D steel frame structures accordig to BS 5950, the combies that procedure with the modified GA to perform desig optimizatio of bechmark examples. 9.2 Desig procedure to BS 5950 The local ad global coordiate systems show i Figure 9.1 are assumed i order to correlate betwee the idices give by BS 5950 ad that employed i the cotext. Figure 9.2a shows a isometric view of a 3D structure with coordiate system while Figures 9.2b ad 9,2c display the structural system as well as the deformed cofiguratio. Z Z Figure 9.1. Local ad global coordiate systems

Desig Optimizatio of 3D Steel Frame Structures 275 Trasverse beams Z Successive frameworks Bracig system (a) Isometric view Z U, c max N b δ s, 1 I x max b δ L, c s, Nb + 1 I x h s 1, 1 I x 1, Nb + 1 I x 2 h 1 B 1 B Nb (b) Z projectio of the 3D structure Z U, c L, δ max c b h s h 1 SP 1 SP N bb (c) Z projectio of the 3D structure Figure 9.2. 3D structure with the coordiate system ad deformed cofiguratio

Desig Optimizatio of 3D Steel Frame Structures 276 It is assumed that the successive frameworks are rigid joited. I additio, it is supposed that oe ed of each trasverse beam is free to rotate about its local axes, ad Z while the secod ed is free to rotate about ad axes. This assumptio has bee made because BS 5950 does ot cater for the desig of bers subjected to torsioal momet. Similarly, the structural system of the bracig bers is cosidered as show i Figure 9.2c. BS 5950 reuires the desiger to select appropriate stadard sectios for the bers of a steel structure i order to obtai a desig havig a sufficiet factor of safety. This is accomplished by cosiderig ultimate ad serviceability limit states. I elastic desig of rigid joited multi storey structures, BS 5950 recommeds that a liear aalysis of the whole structure is carried out. This is achieved by utilisig the fiite elemet package ANSS. The, the desig criteria are checked. This ca be summarised i the followig steps. Step 1. Preparatio of data files icludig structural geometry, loadig cases, etc. Step 2. Classificatio of the structure whether it is sway or o sway. This is achieved by applyig the otioal horizotal loadig case. A structure, aalysed without icludig the effect of claddig, is classified as o sway if each colum of the structure satisfies U, c L c 2000 L, c 1, (9.1) U, c L c 2000 L, c 1 ad c 1,Λ, N c =. (9.2)

Desig Optimizatio of 3D Steel Frame Structures 277 Step 3. Evaluatio of the effective legths eff, L ad eff, L of colums, beams ad bracig bers about the major () ad mior () local axes. I this work, the effective bucklig legth eff, L of colums has bee evaluated by the followig three c approaches: usig the charts from BS 5950; a more accurate method (SCI, 1988) based o fiite elemet aalysis (ANSS); selectio of the coservative (higher) value out of the two. I the secod approach, the effective legth L L x = (9.3) eff, FE c ( ), c F c P E, c where F ( x ) is the ormal force at the critical load of the structure, c P E, c 2 EI c 2 L c π =. (9.4) For a beam the effective bucklig legth eff, b L about the axis euals the urestraied legth of the compressio flage o the uderside of the beam (MacGiley, 1997). For colums ad beams, the effective legth eff, L about the axis euals the urestraied legth of the ber uder cosideratio. For bracig bers, BS 5950 specifies the effective legths eff, br L ad eff, br L depedig o the ed restraits of the bers. I this work, it is assumed that

Desig Optimizatio of 3D Steel Frame Structures 278 each bracig ber is ot restraied at either eds about the local axis. Therefore, eff, br L ad eff, br L ca be determied by eff 1. 0L, br br L =, (9.5) eff, br L = 0. 85L. (9.6) br Step 4. Calculatio of the slederess ratios λ x) ad λ ) of the ber usig (, eff,, ( x, i, j L λ ( x ) =, (9.7), r eff, L ( xi, j ) λ ( x ), i, j = (9.8) r, where r ad r are the radius of gyratios of the sectio ber about its,, ad axes respectively. Step 5. Check of the slederess costraits Sle s, G for each ber Sle s, G 1, s = 1, 2, (9.9) where G Sle 1, λ, = ad (9.10) 180 G Sle 2, ( x i, j λ ( x ), i, j ) =. (9.11) 180 Step 6. Aalysis of the structure uder each loadig case to obtai the ormal force, shearig forces ad bedig momets for each ber.

Desig Optimizatio of 3D Steel Frame Structures 279 Step 7. Check of the stregth criteria for each ber follows: uder the loadig case as a) Determiatio of the type of the sectio of the ber (e.g. sleder, semi compact, compact or plastic). b) Evaluatio of the desig stregth p of the ber. y, Str, c) Check of the stregth costraits G depedig o whether the ber is r, i tesio or compressio. This stage cotais five checks (r = 5) for each ber uder each loadig case. The stregth costraits are local capacity, overall capacity, shear capacities i ad directios ad the shear bucklig capacity. These ca be expressed as follows: where the local capacity G Str, 1, = A A e, g, F ( x ( x Str, G 1, r = 1, 2, 3, 4, 5 ad = 1,2, Λ, Q (9.12) r, i, j F ) p i, j ) p y, y, ( x ( x i, j i, j M + ) M M M + ) M M, C, M, C,, C, M, C, + ( x ) ( x + ( x ) ( x i, j i, j i, j i, j ) ) for tesio bers (9.13) for comprisso bers where F ( x ) is the axial force. The applied momet about the local axes ad (, (, are M x) ad M x ). The desig stregth is ). The momet p ( x y, i, j

Desig Optimizatio of 3D Steel Frame Structures 280 capacities of the ber about its ad axes are M x ) ad ( C, i, j M x ). It is assumed that A x ) ad A ( x ) ( C, i, j ( e, i, j g, i, j are eual. Accordig to clause 4.3.7 of BS 5950, the desiger is ot reuired to check the bracig bers for lateral torsioal bucklig whe they are i tesio, therefore the Str, overall capacity G is determied by 2, G Str, 2, ( x ) = m A g, M M b, ( x F, i, j ) p for tesio bers (beams ad colums) C, p m ( x y, i, j ( x m + ) M i, j ) Z,, M M b, ( x i, j, ) (9.14) + for compressio bers. Str, The shear capacities G ad G ( x ) are computed by 3, G G Str, 4, Str, x F ( ), ( x ) =, (9.15) P ( ) 3, x, i, j Str, x F ( ), ( x ) = (9.16) P ( ) 4, x, i, j where P x ) ad P x ) are the shear capacities of the ber i the (, i, j (, i, j (, ad directios. The critical shear forces are F x). Str, Each ber should also checked for shear bucklig G if d( x t( x i, j i, j 5, ) 63 ε ( xi, j ). (9.17) )

Desig Optimizatio of 3D Steel Frame Structures 281 Hece, G Str, x F ( ), ( x ) =. (9.18) V ( ) 4, x cr, i, j d) For a sway structure, the otioal horizotal loadig case is cosidered, this is termed the sway stability criterio. Step 8. Checks of the horizotal (i ad directios) ad vertical odal displacemets that are kow as serviceability criteria This is performed by: Ser t, G 1, t = 1, 2 ad 3. (9.19) a) Computig the horizotal odal displacemets due to the ufactored imposed loads ad wid loadig cases i order to satisfy the limits o the horizotal displacemets, G Ser 1, c = U, c L c 300 L, c, (9.20) G U Ser, c = 2, c L c 300 L, c ad c 1,Λ, N c =. (9.21) b) Imposig the limits o the vertical odal displacemets (maximum value withi a beam) due to the ufactored imposed loadig case usig G δ max Ser b = 2, b L b 360, b, 2 b = 1, Λ, N. (9.22) The flowchart give i Figure 9.3 illustrates the desig procedure of 3D steel frame structures to BS 5950.

Desig Optimizatio of 3D Steel Frame Structures 282 Start Apply otioal horizotal loadig case, compute horizotal odal displacemets ad determie whether the framework is sway or o sway usig step 2 Compute the effective bucklig legths accordig the reuired approach metioed i step 3 Apply loadig case = 1, 2, Λ, Q : if the framework is sway, the iclude the otioal horizotal loadig case Aalyse the framework, compute ormal forces, shearig forces ad bedig momets for each ber Desig of ber = 1, 2, Λ, N Determie the type of the sectio (sleder, semi compact, compact or plastic) utilisig Table 7 of BS 5950 Evaluate the desig stregth p ( x ) y, i, j of the ber Check the slederess criteria employig (9.30) (9.11) NO Tesio ber? ES A B C D Figure 9.3a. Flowchart of desig procedure of 3D steel frame structures

Desig Optimizatio of 3D Steel Frame Structures 283 A B C D Local capacity check Local capacity check Overall capacity check Lateral torsioal bucklig check Carry out the checks of shear applyig (9.15) (9.16) ad shear bucklig usig (9.18) if ecessary Is = N? NO ES NO Is = Q? ES Compute the horizotal ad vertical odal displacemets due to the specified loadig cases Check of the serviceability criteria usig (9.19) (9.22) Ed Figure 9.3b. (cot.) Flowchart of desig procedure of 3D steel frame structures

Desig Optimizatio of 3D Steel Frame Structures 284 9.3 Problem formulatio ad solutio techiue The geeral formulatio of the desig optimizatio problem ca be expressed by Miimize F( x ) = N = 1 W L Str, subject to: G 1, r = 1, 2, 3, 4, = 1, 2, Λ, Q r, Sle s, G 1, s = 1, 2 Ser t, G 1, t = 1, 2, 3 s, I b x s 1, b x I 1 = 1,,, N, =, 2, Λ, N 1 (9.23), s 2 Λ s b 1 b + T T T T 1, x2, x j, Λ, x J ) x = ( x, j = 1, 2, Λ, J x, D ad i j j where D j = ( d d d,, Λ, j, 1 j, 2 j, λ W is the mass per uit legth of the ber uder cosideratio ad is take from a catalogue. The stregth, slederess ad serviceability criteria are Sle x s, Ser x t, ) G Str, r,, G ( ) ad G ( ) respectively. The vector of desig variables x is divided ito J sub vectors x J. The compoets of these sub vectors take values from a correspodig catalogue D j. I the preset work, the cross sectioal properties of the structural bers, which form the desig variables, are chose from three separate catalogues (uiversal beams ad colums covered by BS 4, ad circular hollow sectios from BS 4848). Figure 9.4 demostrates the applied solutio techiue.

Desig Optimizatio of 3D Steel Frame Structures 285 Start Iput data files: GA parameters, FE model, loadig cases, etc. Radomly geerate the iitial populatio Desig set =1, 2, Λ o,n p Decode biary chromosomes to iteger values ad select the sectios from the appropriate catalogue accordig to their correspodig iteger values Apply the desig procedure illustrated i flowchart give i Figure 9.3 to check stregth, sway stability ad serviceability criteria to BS 5950 Save the feasibility checks of the desig set Desig ES set = N? o p NO New desig ES Evaluate the objective ad pealised fuctios o Select the best N p idividuals out of N p, ad impose them ito the first geeratio of GA algorithm A Figure 9.4a. Flowchart for the solutio techiue

Desig Optimizatio of 3D Steel Frame Structures 286 A Geeratio 1: Calculate the ew pealised objective fuctio, the carry out crossover ad mutatio Desig set = 2, 3 Λ, N p Decode biary chromosomes to iteger values ad select the sectios from the appropriate catalogue accordig to their correspodig iteger values Apply the desig procedure illustrated i flowchart give i Figure 9.3 to check stregth, sway stability ad serviceability criteria to BS 5950 Save the feasibility checks of the desig set New geeratio Desig set = N? p NO New desig ES Evaluate the objective ad pealised fuctios Covergece occurred? ES Stop NO Store the best idividuals, ad impose them ito the ext geeratio ad carry out crossover ad mutatio Figure 9.4b. (cot.) Flowchart for the solutio techiue

Desig Optimizatio of 3D Steel Frame Structures 287 9.4 Bechmark examples Havig itroduced the desig procedure accordig to BS 5950 liked to the GA procedure ad the formulatio of desig optimizatio problem, the process of optimizatio i ow carried out. Two steel frame structures are demostrated here to illustrate the effectiveess ad beefits of the developed GA techiue as well as ivestigatig the effect of the employed approach for the determiatio of the effective bucklig legth o the optimum desig attaied. The catalogue of available cross sectios i BS 4 iclude 64 uiversal beams (UB) ad 32 uiversal colums (UC). These sectios are give i Sectio 7.2.1. The catalogue of circular hollow sectios (CHS) is take from BS 4848 ad this icludes 64 sectios, listed i Table 9.1, varyig from 76.1 5.0 CHS to 457.0 40.0 CHS. I the preset work, the iitial populatio size o N p is assumed to be 1000 ad fixed populatio size N p of 70 durig successive geeratios, elite percetage 30 %, probability of crossover P c was 0.7, probability of mutatio E r was P m was 0.01 ad oe poit crossover is applied. I additio, the techiue described i Sectio 6.2 is utilised where the simple "exact" pealty fuctio employed is Miimize C - F( x), all costraits satisfied F ( x ) = (9.24) 0, ay of costraits violated. used where The covergece criteria ad termiatio coditios detailed i Sectio 5.6.3.7 are av C = 0.001, cu max = C = 0.001 ad ge 200.

Desig Optimizatio of 3D Steel Frame Structures 288 Table 9.1. The used circular hollow sectios Cross sectio Cross sectio Cross sectio 1 76.1 5.0 CHS 23 219.1 10.0 CHS 45 323.9 25.0 CHS 2 88.9 3.2 CHS 24 219.1 12.5 CHS 46 355.6 8.0 CHS 3 88.9 4.0 CHS 25 219.1 16.0 CHS 47 355.6 10.0 CHS 4 88.9 5.0 CHS 26 219.1 20.0 CHS 48 355.6 12.5 CHS 5 114.3 3.6 CHS 27 244.5 6.3 CHS 49 355.6 16.0 CHS 6 114.3 5.0 CHS 28 244.5 8.0 CHS 50 355.6 20.0 CHS 7 114.3 6.3 CHS 29 244.5 10.0 CHS 51 355.6 25.0 CHS 8 139.7 5.0 CHS 30 244.5 12.5 CHS 52 406.4 10.0 CHS 9 139.7 6.3 CHS 31 244.5 16.0 CHS 53 406.4 12.5 CHS 10 139.7 8.0 CHS 32 244.5 20.0 CHS 54 406.4 16.0 CHS 11 139.7 10.0 CHS 33 273.0 6.3 CHS 55 406.4 20.0 CHS 12 168.3 5.0 CHS 34 273.0 8.0 CHS 56 406.4 25.0 CHS 13 168.3 6.3 CHS 35 273.0 10.0 CHS 57 406.4 32.0 CHS 14 168.3 8.0 CHS 36 273.0 12.5 CHS 58 457.0 10.0 CHS 15 168.3 10.0 CHS 37 273.0 16.0 CHS 59 457.0 12.5 CHS 16 193.7 6.3 CHS 38 273.0 20.0 CHS 60 457.0 16.0 CHS 17 193.7 8.0 CHS 39 273.0 25.0 CHS 61 457.0 20.0 CHS 18 193.7 10.0 CHS 40 323.9 8.0 CHS 62 457.0 25.0 CHS 19 193.7 12.5 CHS 41 323.9 10.0 CHS 63 457.0 32.0 CHS 20 193.7 16.0 CHS 42 323.9 12.5 CHS 64 457.0 40.0 CHS 21 219.1 6.3 CHS 43 323.9 16.0 CHS 22 219.1 8.0 CHS 44 323.9 20.0 CHS

Desig Optimizatio of 3D Steel Frame Structures 289 9.4.1 Example 1: Two bay by two bay by two storey structure The first 3D steel frame structure, aalysed i this chapter, is the two bay by two bay by two storey structure show i Figure 9.5. It ca be observed from Figure 9.5a that the structure cosists of three successive frameworks, trasverse beams ad the bracig system. The spacig betwee the successive frameworks is 10.0 m while the distace betwee the successive trasverse beams is 5 m. Because BS 5950 does ot cater for the desig of bers subjected to torsioal momets, it is assumed that oe ed of each trasverse beam is free to rotate about its local axes, ad Z while the secod ed is free to rotate about ad axes. Similarly, the structural system of the bracig bers is cosidered. The structural system is show i Figures 9.5b 9.5d. The structure was desiged for use as a office block icludig projectio rooms. Nie loadig cases were take ito accout ad these represet the most ufavourable combiatios of the factored dead load (DL), imposed load (LL) ad wid load (WL) as reuired by BS 5950 ad BS 6399. Because the structure is doubly symmetric, two orthogoal wid cases have bee cosidered where the wid loads are factored by 1.2. The values of the ufactored DL ad LL are tabulated i Table 9.2. These values are calculated accordig to the recommedatios give by Owes et al (1992), MacGiley (1997) ad Nethercot (1995). Table 9.2. The values of dead load ad imposed load Value of the load o roof Value of the load o first floor DL 7.0 kn/m 2 9.0 kn/m 2 LL 2.0 kn/m 2 5.0 kn/m 2

Desig Optimizatio of 3D Steel Frame Structures 290 Framework B Framework A Trasverse beams 4 5.0 m = 20.0 m 2 5.0 m = 10.0 m 2 10.0 m = 20.0 m 2 10.0 m = 20.0 m Bracig bers Frot face (a) Isometric view View F 1 3 2 3 1 5 m 4 6 6 5 4 1 3 3 6 6 5 m 2 1 4 5 4 10 m 10 m 10 m 10 m (b) Framework A (c) Framework B 3 3 3 3 7 7 7 7 7 10 m 5 m 1 7 7 4 8 1 6 6 6 6 5 m 1 7 7 4 8 1 7 7 7 7 7 10 m 10 m (d) View F 10 m 3 3 3 3 5 m 5 m 5 m 5 m (c) Pla Figure 9.5. Two bay by two bay by two storey structure

Desig Optimizatio of 3D Steel Frame Structures 291 The reiforced cocrete slabs are assumed to trasmit the loads i oe directio because the ratio of the spacig betwee the successive frameworks ad the distaces betwee the successive trasverse beams eual 2 (see MacGiley, 1997). Accordig to BS 6399: Part 2, the stadard method was utilised to determie the values of the wid pressures o the vertical walls ad the flat roof assumig 1. the opeigs are domiat, 2. the buildig type factor eual 4.0, 3. the referece height of the structure (H r ) eual to the maximum height of the structure above the groud level (10.0 m), 4. the basic wid speed Vb is 23 m/s, 5. the terrai ad buildig factor S b is 1.58, 6. the altitude factor S a, the directioal factor probability factor Sp eual 1.0, S d, the seasoal factor S s ad the 7. the size effect factor C a is take as 0.90, 8. the exteral pressure coefficiet C pe for each surface of the buildig is determied accordig table 5 ad 8 of BS 6399 ad 9. the iteral pressure coefficiet C pi for each surface of the structure is C = 0 C (9.25) pi. 75 pe where C pe is the exteral pressure coefficiet of the surface uder cosideratio. At this stage havig itroduced the basic assumptios for evaluatig the desig loads, the values of these loads ca accordigly be computed depedig o the ie loadig cases summarised bellow: 1. the floors are subjected to the vertical uiform loads P = 1. 4DL + 1. 6LL, v

Desig Optimizatio of 3D Steel Frame Structures 292 2. the floors are subjected to the vertical uiform loads P = 1. 4DL + 1. 6LL, ad left had side (LHS) odes of each framework are subjected to the horizotal cocetrated loads due to the otioal horizotal loadig coditio, 3. the floors of the first bay (coutig from the left) are subjected to the vertical v uiform loads P v = 1. 4DL while the rest of the floors are subjected to v P = 1. 4DL + 1. 6LL, 4. the floors are subjected to a staggered arragemet of vertical uiform loads v P = 1. 4DL + 1. 6LL ad P v = 1. 4DL, 5. the floors are subjected to the vertical uiform loads P = 1. 2DL + 1. 2LL ad the structure is subjected to the first factored wid loadig case whe the LHS face of the structure is the widward face, 6. the floors are subjected to the vertical uiform loads P = 1. 2DL + 1. 2LL ad the structure is subjected to the secod factored wid loadig case whe the frot face of the structure is the widward face, 7. the floors are subjected to the vertical uiform load P v = 1.0LL, v v 8. the floors are subjected to the vertical uiform loads P v = 1.0LL ad the structure is subjected to the first ufactored wid loadig case whe the LHS face of the structure is the widward face ad 9. the floors are subjected to the vertical uiform loads P v = 1.0LL ad the structure is subjected to the secod ufactored wid loadig case whe the frot face of the structure is the widward face. A more accurately evaluated effective bucklig legth of colums was determied usig the fiite elemet method usig the loadig patter displayed i Figure 9.6. The

Desig Optimizatio of 3D Steel Frame Structures 293 fiite elemet model of the structure was assembled i ANSS usig 5 elemets for each beam ad colum while oe elemet for each ber of the bracig system was used. 0.01P P 0.01P P 0.01P P P (a) Loadig patter of the roof level 0.01P 0.01P 0.01P 8P (b) Loadig patter of the first floor level Figure 9.6. Loadig patter used for the stability aalysis The desig optimizatio processes were carried out cosiderig 8 desig variables. The likig of the desig variables is displayed i Figures 9.3.b 9.3d. Referrig to BS 4 ad BS 4848, the catalogue of the available cross sectios icludes 64 uiversal beams (UB), 32 uiversal colums (UC) ad 64 circular hollow sectios (CHS). This results i a total strig legth of 44. The problem was aalysed utilisig the solutio parameters as described i Sectio 9.4. Five rus of the desig optimizatio processes were carried out usig 5

Desig Optimizatio of 3D Steel Frame Structures 294 differet seed umbers to geerate the iitial populatio. Solutios are preseted i Table 9.3. The desig variables correspodig to the best desigs are give i Table 9.4. Table 9.3. Two bay by two bay by two storey structure: compariso of the best desig obtaied i five rus Total weight (kg) Ru First approach (code) Secod approach (FE) Third approach (coservative) 1 62023.89 69047.74 70856.30 2 62730.46 70563.89 72220.46 3 63079.28 69527.74 70547.74 4 63150.46 69867.74 70390.46 5 62730.46 70347.74 69383.89 Average weight 62742.91 69870.97 70679.77 Miimum weight 62023.89 69047.74 69383.89 Table 9.4. Two bay by two bay by two storey structure: compariso of the desig variables for the obtaied optimum desigs Desig variable First approach (code) Cross sectios Secod approach (FE) Third approach (coservative) 1 203 203 52 UC 356 368 129 UC 356 368 129 UC 2 305 305 97 UC 356 368 129 UC 356 368 202 UC 3 356 368 129 UC 356 368 177 UC 356 368 177 UC 4 356 368 153 UC 356 406 393 UC 356 406 235 UC 5 686 254 125 UB 686 254 125 UB 686 254 125 UB 6 838 292 176 UB 914 305 201 UB 914 305 201 UB 7 762 267 173 UB 686 254 170 UB 686 254 170 UB 8 168.3 6.3 CHS 168.3 5.0 CHS 168.3 6.3 CHS Weight (kg) 62023.89 69047.74 69383.89

Desig Optimizatio of 3D Steel Frame Structures 295 Durig the desig optimizatio process, the covergece characteristics of the solutios were examied. Figure 9.7 shows the covergece history of the best desigs. It ca be observed that the best solutios were obtaied withi 50 geeratios while the rest of the computatios were carried out to satisfy the covergece criteria. Best desig (kg) 90000 80000 70000 60000 First approach (code) Secod approach (FE) Third approach (coservative) 50000 0 10 20 30 40 50 60 70 80 Geeratio umber Figure 9.7. Two bay by two bay by two storey structure: best desig versus geeratio umber 9.4.2 Example 2: Three bay by four bay by four storey structure The ext 3D structure is the three bay by four bay by four storey structure show i Figure 9.8. The structure cosists of four successive frameworks, trasverse beams ad a bracig system as show i Figure 9.8a. The spacig betwee the successive frameworks is 8.0 m. The distace betwee the successive trasverse beams is 4.0 m. The structural system is show i Figures 9.8b 9.8f. The structure is desiged for use as a office block icludig projectio rooms. Te loadig cases represetig the most ufavourable combiatios of the factored DL, LL ad WL are take ito accout.

Desig Optimizatio of 3D Steel Frame Structures 296 Framework A Framework B 6 4.0 m = 24.0 m Trasverse beams 17.0 m Framework D Bracig system 4 8.0 m = 32.0 m Frot face Framework C View F (a) Isometric view 5 5 5 10 10 10 2 5 5 5 2 4 4 1 1 4 4 2 5 5 5 3 3 5 5 5 3 3 2 1 1 4 m 4 m 4 m 5 m 7 9 9 7 7 6 6 10 10 9 8 10 10 10 9 7 10 10 10 10 8 8 8 6 6 8 m 8 m 8 m 8 m 8 m 8 m (b) Framework A (c) Framework B 11 11 4 m 2 7 12 2 11 11 7 10 7 5 2 2 4 m 4 m 2 11 11 12 7 2 1 11 7 6 12 12 11 2 1 7 10 7 2 5 2 5 m 2 11 11 11 11 12 7 1 6 12 1 8 m 8 m 8 m 8 m 8 m 8 m (d) Framework C (e) Framework D (f) View F Figure 9.8. Three bay by four bay by four storey structure

Desig Optimizatio of 3D Steel Frame Structures 297 Because the structure is sigly symmetric, three orthogoal wid cases have bee cosidered where the wid loads are factored by 1.2. The values of the ufactored DL ad LL are tabulated i Table 9.2. Table 9.5. The values of deal load ad imposed load Value of the load o roof Value of the load o other floors DL 6.5 kn/m 2 8.0 kn/m 2 LL 2.0 kn/m 2 5.0 kn/m 2 Accordig to BS 6399: Part 2, the stadard method was utilised to determie the values of the wid pressures o the vertical walls ad the flat roof cosiderig: 1. the opeigs are domiat, 2. the buildig type factor euals 4.0, 3. the referece height of the structure (H r ) euals to the maximum height of the structure above the groud level (17.0 m), 4. the basic wid speed Vb is 23 m/s, 5. the terrai ad buildig factor S b is 1.71, 6. the altitude factor S a, the directioal factor probability factor Sp eual 1.0, S d, the seasoal factor S s ad the 7. the size effect factor C a is take as 0.87, 8. the exteral pressure coefficiet C pe for each surface of the buildig is determied (see Sectio 2.3.3) accordig Tables 5 ad 8 of BS 6399 ad 9. the iteral pressure coefficiet C pi is calculated by (9.25). The desig loads were computed accordig to the followig loadig cases:

Desig Optimizatio of 3D Steel Frame Structures 298 1. the floors are subjected to the vertical uiform loads P = 1. 4DL + 1. 6LL, v 2. the floors are subjected to the vertical uiform loads P = 1. 4DL + 1. 6LL, ad left had side (LHS) odes of each framework are subjected to the horizotal cocetrated loads due to the otioal horizotal loadig coditio (see Chapter 2), 3. the floors of the first bay (coutig from the left) are subjected to the vertical v uiform loads P v = 1. 4DL while the rest of the floors are subjected to v P = 1. 4DL + 1. 6LL, 4. the floors are subjected to a staggered arragemet of vertical uiform loads v P = 1. 4DL + 1. 6LL ad P v = 1. 4DL, 5. the floors are subjected to the vertical uiform loads P = 1. 2DL + 1. 2LL ad the structure is subjected to the first factored wid loadig case whe the LHS face of the structure is the widward face, 6. the floors are subjected to the vertical uiform loads P = 1. 2DL + 1. 2LL ad the structure is subjected to the secod factored wid loadig case whe the frot face of the structure is the widward face, 7. the floors are subjected to the vertical uiform loads P = 1. 2DL + 1. 2LL ad the structure is subjected to the third factored wid loadig case whe the rear face of the structure is the widward face, 8. the floors are subjected to the vertical uiform load P v = 1.0LL, v v v 9. the floors are subjected to the vertical uiform loads P v = 1.0LL ad the structure is subjected to the first ufactored wid loadig case whe the LHS face of the structure is the widward face ad

Desig Optimizatio of 3D Steel Frame Structures 299 10. the floors are subjected to the vertical uiform loads P v = 1.0LL ad the structure is subjected to the secod ufactored wid loadig case whe the frot face of the structure is the widward face. The fiite elemet method was employed (see Toropov et al., 1999) i order to evaluate the effective bucklig legth of colums. This was performed by utilisig the loadig patter displayed i Figure 9.9. I the fiite elemet model, the structure was assembled i ANSS usig 5 elemets for each beam ad colum while oe elemet for each ber of the bracig system. The optimizatio process was carried out cosiderig 12 desig variables. The likig of the desig variables is displayed i Figures 9.8.b 9.8f. Referrig to BS 4 ad BS 4848, the catalogue of the available cross sectios iclude 64 uiversal beams (UB), 32 uiversal colums (UC) ad 64 circular hollow sectios (CHS). This results i a total strig legth of 64. The problem was aalysed utilisig the solutio parameters described i Sectio 9.4. Five rus of the optimizatio process were carried out usig 5 differet seed umbers to geerate the iitial populatio. The optimizatio process was termiated whe ay of the covergece criteria is satisfied. Solutios are preseted i Table 9.6. The desig variables correspodig to the best desigs are give i Table 9.7.

Desig Optimizatio of 3D Steel Frame Structures 300 0.01P P 0.01P 0.01P P P P (a) Loadig patter of the roof level 0.01P 0.01P 0.01P 8P 8P 4 (b) Loadig patter of the third floor level 0.01P 0.01P 0.01P P 0.01P 5P 0.01P 8P 5P 8 4 P (c) Loadig patter of the secod floor level 0.01P 0.01P 0.01P 0.01P 6P 0.01P 8P 6P 8P (d) Loadig patter of the first floor level Figure 9.9. Loadig patter used for the stability aalysis

Desig Optimizatio of 3D Steel Frame Structures 301 Table 9.6. Three bay by four bay by four storey structure: compariso of the best desig obtaied i five rus Total weight (kg) Ru First approach (code) Secod approach (FE) Third approach (coservative) 1 168914.20 185520.60 188914.46 2 167214.50 185696.60 188522.56 3 169078.60 184881.00 187604.90 4 168492.20 185819.00 185566.40 5 169436.20 185852.60 188502.60 Average weight 168627.14 185553.96 187604.90 Miimum weight 167214.50 184881.00 185566.40 Table 9.7. Three bay by four bay by four storey structure: compariso of the desig variables for the obtaied optimum desigs. Desig variable First approach (code) Cross sectios Secod approach (FE) Third approach (coservative) 1 305 305 137 UC 356 368 177 UC 356 368 177 UC 2 305 305 137 UC 356 368 177 UC 356 368 153 UC 3 356 406 235 UC 356 406 287 UC 356 406 287 UC 4 356 406 235 UC 356 406 287 UC 356 406 287 UC 5 686 254 125 UB 914 305 201 UB 914 305 201 UB 6 356 368 153 UC 356 368 153 UC 305 305 198 UC 7 356 368 153 UC 305 305 137 UC 356 368 153 UC 8 356 406 235 UC 356 406 340 UC 356 406 393 UC 9 356 406 235 UC 356 406 340 UC 356 406 393 UC 10 914 305 201 UB 686 254 140 UB 762 267 147 UB 11 686 254 125 UB 686 254 125 UB 686 254 125 UB 12 139.7 10.0 CHS 139.7 8.0 CHS 139.7 5.0 CHS Weight (kg) 167214.5 184881.00 185566.4

Desig Optimizatio of 3D Steel Frame Structures 302 From Table 9.6, it ca deduced that the optimizer was able to obtai several solutios ad the differeces betwee them are small. Durig the optimizatio process, the covergece characteristics of the solutios were examied. Figure 9.10 shows the covergece history of the best desigs. Best desig (kg) 250000 230000 210000 190000 First approach (code) Secod approach (FE) Third approach (coservative) 170000 150000 0 10 20 30 40 50 60 70 80 90 100 Geeratio umber Figure 9.10. Three bay by four bay by four storey structure: best desig versus geeratio umber. From this figure, it ca be observed that the optimum solutios were achieved withi 50 geeratios while the rest of the computatioal effort was eeded to satisfy the termiatio coditios described i sectio 9.4. 9.5 Validatio of the optimum desig This sectio shows that the developed FORTRAN code for desig of 3D steel frame structures is successfully implemeted. As discussed i Sectio 8.5, to validate the applied FORTRAN code, the problem should be first ru whe 2). The, CSC software is used to check the calculated costraits. m is 1 (techiue

Desig Optimizatio of 3D Steel Frame Structures 303 The two bay by two bay by two storey structure (studied i Sectio 9.4.1) was aalysed. The loadig cases metioed i Sectio 9.4.1 are utilised. The optimizatio process was carried out usig the desig procedure discussed i Sectio 9.2 while the solutio parameters ad the covergece criteria were applied as cosidered i Sectio 9.4. Five rus were carried out whe applyig the first approach for determiig the effective bucklig legths. The desig variables correspodig to the best solutio are tabulated i Table 9.8. It is worth comparig the desig variables obtaied with those achieved i sectio 9.4.1 (techiue 1) whe a more accurate euatio for determiig m ( x ) was applied. This compariso is also preseted i Table 9.8. It ca be observed that whe applyig techiue 2, the optimizer succeeded i obtaiig a solutio (62570.47 kg) uite ear to that achieved whe usig techiue 1 (62023.89 kg). Table 9.8. Two bay by two bay by two storey structure: compariso of the desig variables for the optimum desigs. Desig variable Cross sectios Techiue 1 Techiue 2 1 203 203 52 UC 254 254 73 UC 2 305 305 97 UC 254 254 73 UC 3 356 368 129 UC 305 305 97 UC 4 356 368 153 UC 356 368 153 UC 5 686 254 125 UB 686 254 125 UB 6 838 292 176 UB 838 292 194 UB 7 762 267 173 UB 686 292 170 UB 8 168.3 6.3 CHS 219.1 6.3 CHS Weight (kg) 62023.89 62570.46

Desig Optimizatio of 3D Steel Frame Structures 304 The covergece characteristics were also examied. This was achieved by plottig the chages of the best desig with the umber of geeratios performed for each ru as show i Figure 9.11. Best desig (kg) 150000 130000 110000 90000 70000 First ru Secod ru Third ru Fourth ru Fifth ru 50000 0 10 20 30 40 50 60 70 80 90 Geeratio umber Figure 9.11. Two bay by two bay by two storey structure: best desig versus geeratio umber At this stage, the structural weight has bee optimized ad the sectio of each ber has bee determied. The ext step is to validate the code checks usig CSC software. This is achieved by usig the followig proceduce: 1) I S FRAME, the structural geometry, ber sectios ad loadig cases are defied. The, the bedig momets, shear forces, ad odal displacemets are calculated accordig to the aalysis type presupposed (liear aalysis). 2) Startig the S STEEL program. The desig checks are the carried out. 3) The desig results are the displayed o a separate widow as show i Figure 9.12.

Figure 9.12. The desig results of two bay by two bay by two storey structure (captured from S STEEL)

Desig Optimizatio of 3D Steel Frame structures 306 I this figure, the desig checks of each ber are idicated i colour i which the code utilisatio meu gives the rage for of each colour. It ca be observed that most of the bers reach their maximum capacities. This idicates that the developed algorithm is successfully icorporated i desig optimizatio. It is worth otig that the desig results vary betwee 0.7 ad 1.0. 9.6 Cocludig remarks Desig optimizatio techiue based o GA was preseted for 3D steel frame structures where the structures were subjected to multiple loadig coditios. The desig method obtaied a 3D steel frame structure with the least weight by selectig appropriate sectios for beams, colums ad bracig bers from the British stadard for uiversal beam sectios, uiversal colum sectios ad circular hollow sectios. The followig cocludig remarks ca be made. 1) It has bee prove that the developed GA approach ca be successfully icorporated i desig optimizatio i which the structural bers have to be selected from the available stadard sectios while the desig satisfies the desig criteria. This idicates that the developed approach ca be utilised by a practisig desigers. 2) I the preset chapter, the skills ad experiece of the desiger have bee reflected i the optimizatio problem by imposig costraits o the secod momet of area of two adjacet colums i two adjacet storey levels. This ca be implemeted usig other costraits such as sectioal dimesios, sectioal area, etc. This idicates that the optimizer is able to treat differet practical costraits depedig o the ature of the problem. 3) It has bee show that the developed GA provides the desiger with more tha oe solutio to choose from, ad the differece betwee them was small. This could be

Desig Optimizatio of 3D Steel Frame structures 307 a advatage whe a desiger eeds to choose a appropriate solutio depedig o the availability of the sectios. 4) From Tables 9.4 ad 9.7, it ca be observed that the same sectios have bee obtaied for differet bers of a structure eve though these bers are liked to differet desig variables. This idicates that it ca be beeficial to iclude the groupig of structural bers as a additioal criterio i the formulatio of the desig optimizatio problem. 5) I the preset study, computatio of the effective bucklig legth has bee automated ad icluded i the developed algorithm. This was achieved by employig three differet approaches. Applicatio of a modified GA to desig optimizatio of structural steelwork allows the best set from a appropriate catalogue of steel cross sectios to be chose. The optimizer has bee liked to a commercial fiite elemet code ad the British codes of practice i order to obtai optimum desigs accepted by practisig structural egieers.