A TRIAL DESIGN OF STEEL FRAMED OFFICE BUILDING BASED ON AN OPTIMUM DESIGN METHOD

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1 ABTRACT : A TRIAL DEIGN OF TEEL FRAED OFFICE BUILDING BAED ON AN OPTIU DEIGN ETHOD. Ke 1, K. Iago 2, Y. Lee 3 and K. Uetan 4 1 General anager, Dept. o tructural Engneerng, NIKKEN EKKEI, Toyo Japan 2 Assocate Proessor, Graduate chool o Engneerng, Tohou Unversty, enda Japan 3 Researcher, Graduate chool o Engneerng, Kyoto Unversty, Kyoto Japan 4 Proessor, Graduate chool o Engneerng, Kyoto Unversty, Kyoto Japan Emal: ago@arch.tohou.ac.jp Untl now, many researches on optmum desgn o buldng structures have been made, and mprovng perormance o computers enabled us to solve large and complcated optmum desgn problems numercally. However, the optmum desgns gven by conventonal optmum desgn methods don t necessarly provde drectly acceptable optmum solutons or practcal structural desgn at the present tme. In ths paper, we clary the pont to buld an optmum desgn system whch s to gve optmum desgns sutable or practcal structural desgn by showng an example usng an optmum desgn method consderng constrants requred by actual desgn n practce wthn the ramewor o the Buldng tandard Law o Japan. KEYWORD: Optmum desgn, Dscrete-varable optmzaton, Gradent projecton method, Desgn support tool 1. INTRODUCTION Untl now, many researches on optmum desgn o buldng structures have been made, and mprovng perormance o computers enabled us to solve large and complcated optmum desgn problems numercally. However, the optmum desgns gven by conventonal optmum desgn methods don t necessarly provde drectly acceptable optmum solutons or practcal structural desgn at the present tme. any case studes are requred to obtan an acceptable and desrable desgn whch satses varous constrants requred by archtectural plannng, buldng code, standards or gudelnes provded by archtectural socetes. Purposes o optmum desgn programs vary dependng on the desgn stages; a prelmnary desgn stage n whch structural desgners study ecent layouts o structural members, a basc desgn stage n whch they choose materals and structural orm, and nal desgn stage n whch structural desgners decde the szes o each members. And each desgn stage requres many case studes or decson-mang. Despte o requrements or many case studes, many structural desgners would le to cope wth such problems dependng on ther experence wthout mang any case study, because they can t aord the tme and cost to mae such many case studes. Whle conventonal optmum desgn methods can t meet the aorementoned requrements, we show n ths paper that an acceptable and desrable desgn can be obtaned perormng many case studes usng a practcal optmum desgn program whch taes varous practcal constrants nto account. A tral desgn o a steel ramed oce buldng shown n ths paper shows that we can easly perorm comparatve case studes on requrements quanttatvely or decson-mang at a detal desgn stage. Although many problems must be solved to put ths optmum desgn system a structural desgn support tool to practcal use, we expect that we can establsh a new desgn method by whch structural engneers don t need to depend on ther ntuton derved rom ther experences when they desgn buldng structure any more.

2 2. FORULATION OF AN OPTIU DEIGN PROBLE In ths paper we ormulate an optmum desgn problem or a realstc steel ramed structure whose; desgn varables are szes o structural members; constrants consst o nequaltes on perormance whch desgn solutons should satsy; objectve uncton, whch s closely related to cost o the structure, s total weght o the steel gven by the desgn varables. The ormulaton o the problem ODPF (Optmum Desgn Problem or a teel Frame) s shown below. ODPF nd x x to mnmze W ( x ) subject to g x g j ( ) 0 Where x (unt: mm) denotes a sze o a steel member,.e., depth or wdth o the steel member, or thcness o steel plate. W ( x ) s total mass (unt: ton) o the structure and s uncton o x. The nequalty g( x) 0 stands or constrants that desgn varable x have to satsy. Detaled explanaton or those varables and unctons are descrbed below. 2.1.Objectve uncton, desgn varables and constrants Desgn varables A desgn varable x s a vector whose -th component x (unt: mm) stands or depth or wdth o a steel member, or thcness o a steel plate. Each component o the desgn varable x has to be dscrete, because szes o steel members used or buldngs have dscrete value. For example, depth or wdth o the member may have such dscrete values as; 100, 125, 150, 175,, and thcness o steel plates may have such dscrete values as; 6, 9, 12, 16,. At the rst step the desgn varable s handled as a contnuous varable to relax the dscrete optmzaton problem nto a contnuous one, whch s easer to solve, to obtan an ntal desgn to be nputted to the dscrete optmzaton problem next step Objectve uncton In the practcal structural desgn, engneers usually grasp and control the total mass o the steel they use to desgn a buldng rame. o, the total mass o the steel s very mportant ndex to evaluate the desgn. Let area and length o a -th steel member column, grder or brace be A ( x ) and l respectvely, the objectve uncton W ( x ) may be expressed as ollows: Where s densty o steel (=7.8ton/m 3 ) Inequalty constrants W ( x) A ( x ) l (1) Requrements n practcal structural desgn can be expressed as nequaltes as ollows. g j ( x ) 0 (2) For example, let j-th desgn requrement be a constrant o a member stress ( x ) whch s to be smaller than allowable stress A, ollowng nequalty holds. g ( x) ( x ) 0 (3) j A

3 Thus, we can also express other desgn requrements as nequaltes, and can dene a constrant vector g( x ). 3. OPTIIZATION ETHOD Yoshtom et.al.[2] proposed a soluton method whch dvde a dscrete optmum desgn problem nto two problems, contnuous (relaxed) optmum desgn problem whose contnuous desgn varable approxmate the dstrbuton o the standard sectons, and dscrete optmum desgn problem usng the soluton obtaned rom the ormer problem as an ntal desgn. We heren employ ths method to solve the optmum desgn problem. The proposed method eatures n usng a contnuous optmzaton n seeng dscrete sectons. At rst, as shown n Fgure 1, we solve a contnuous optmzaton problem and obtan canddate sectons, whose szes are dscrete, near the contnuous soluton. equentally, we perorm another optmzaton to see another set o canddates untl the desgn doesn t mprove the objectve uncton any more. x 2 tandard secton x 2 Canddate secton Relaxed space or the desgn varable (1) (2) (3) (4) (6) (5) (8) (7) (2) (1) (3) x 1 Dstrbuton o standard sectons Relaxed space or the desgn varable x 1 Fgure 1 Relaxaton o the dscrete optmum desgn problem I the constrant unctons have strong non-lnearty, the senstvtes o the objectve uncton are eectve only n the close area around the present desgn. Gradent projecton method [3] we employ heren s very eectve to solve such problems. As llustrated n Fgure 2, an optmum desgn s sought repeatng a set o modcatons o present desgn; (1) a modcaton based on projecton move vector X proj, whch s projecton o the steepest descent vector o objectve uncton W to the equalty constrant plane and, (2) another modcaton based on restoraton move vector X whch s to compensate the error caused by the non-lnearty o the constrant uncton. resto x 2 Contour o objectve uncton Contour o objectve uncton g 0 X proj X resto x 1 Fgure 2 Gradent Projecton ethod

4 4. Example 4.1. Analyss model or a tral desgn A desgn problem or 14-storey steel ramed oce buldng model s employed to dscuss ponts on practcal usage o the optmzaton method. As Fgure 4 shows, ths model has typcal plan as an oce buldng: elevator shats and star-cases are located on one sde o the plan, and the other sde s located 16-m spanned oce space. torey-heght o the rst storey s 6 m and those o typcal loors (2nd to 14 th loor) are 4m Desgn requrements and assumptons Desgn requrements and assumptons we made to create the desgn example model are lsted below. (1)teel o 490 N/mm 2 tensle class s employed or columns and grders. hapes used or column sectons are bult-up wde-lange sectons (or external columns) and bult-up steel box sectons (or nternal columns). Only bult-up wde-lange sectons are selected or grders. (2)Desgn o each secton member s based on the allowable stress concept, that s, each secton member s desgned n such a manner that member stresses caused by permanent loads and combnatons o permanent and temporary loads are to be smaller than allowable stresses. When we calculate long-term stress at each end o the members, we use bendng moments at jonts. tresses derved rom member ace moment by temporary loads (X and Y-drectonal, orward and bacward horzontal loads) are added to long-term stresses to obtan short-term stresses. (3)Composte eect o steel grder unted wth renorced concrete slab s taen nto account and grder stness s augmented. (4)Desgn o oundaton grder s gven (renorced concrete secton whose wdth s 900mm and depth s 1500mm), so szes o oundaton grders are not desgn varables. We assume that the gven secton o the oundaton grders have enough strength so that they won t yeld even the structure s subjected to horzontal orces. (5)Desgn o ples s also gven. Ples are modeled as oundaton sprngs whch have characterstcs as llustrated n Fgure 3. Axal orces o ples caused by moderate earthquaes are constraned to be smaller than 2/3 o the ultmate strength whch s dened short-term allowable strength or ples or compresson and tenson respectvely, whle no constrant s ntroduced or severe earthquaes. ple axal orce (N) vertcal dsplacement at ple top (mm) Fgure 3. Restorng orce model o ples (6)Permanent load and sesmc load are set as shown n Table 1. External grders bear 10N/m o exteror wall weght dstrbuted along them. Desgn sesmc load s determned accordng to Buldng tandard Law o Japan.

5 4.3. ember szes and desgn varables In concrete, desgn varables we have already explaned n secton 2.1. may be set as ollows. (1)Desgn varables or columns: zes o a wde-lange secton, depth, wdth, web thcness and lange thcness are set as varables ndependent o each other. For a box secton, depth and plate thcness are set as ndependent varables. (2)Desgn varables or grders: zes o a wde-lange secton, depth, wdth, web thcness and lange thcness are set as ndependent varables. (3)Desgn varables or braces: Brace area s set as a desgn varable ,000 9,600 6,400 6,400 9,600 3,200 " " " " " " " " 3,200 A B C D 25,600 9,600 16,000 3,200 " " " " " " 3,200 G04 G04 G04 G04 G04 G04 G04 G04 C02 G01 G01 G01 G01 G01 G01 G01 G01 G01 G01 B3 B3 B3 B3 C01 C01 C02 C01 C02 C01 C02 C01 C01 B1 B1 B1 B1 B2 B2 G02 B3 C05 G07 G02 B01 B3 B2 B2 B2 B02 C05 G02 G01 G01 G01 G01 G01 G01 G01 G01 G01 G01 C01 C01 C01 C01 C01 C01 C01 C01 C01 B3 B2 oce space B3 B3 B2 B3 B1 G02 elevator shats star case Fgure 4. Plan o the example model G08 B3 B3 B2 G07 B3 B1 B1 B1 B3 B3 B3 G04 G04 G04 G04 G04 G04 G04 G04 Oce concrete slab t=145mm 3,480N/m 2 dec plate 200N/m 2 nsh 1,000 N/m 2 t=100 (ree access loor) total 4,680 N/m 2 Roo concrete slab t=145mm 3,480N/m 2 dec plate 200N/m 2 nsh (asphalt water proo ) t=100 2,600 N/m 2 total 6,280 N/m 2 tar case steel star case 2,500 N/m Constrants Table.1 Desgn Load [Unt:N/m 2 ] A. Dead Load B. Lve Load Untl now, ew researches have amed at broad practcal usage o optmum desgn method, because most o conventonal researches on optmum desgn have conned ther nterest to ndvdual structural problems. ome o them are hard to apply to practcal desgn problems drectly because ther nterest s ar rom practcal nterest. We heren am at consderng most o the constrants whch s requred wthn the ramewor o room or loor desgn or rame desgn or sesmc desgn oce 2,900 1, star case, corrdor 1,800 1, roo 1,800 1,

6 Buldng tandard Law o Japan and can be expressed by nequaltes. Constrants we employ n the example are lsted n Table 2. bounds o szes (sde constrants) constrants on wdth-thcness rato stress constrants or moderate earthquaes constrants on nter-storey drt constrants on ple axal orces constrants on column-grder strength rato constrants on horzontal resstant orces grders columns Table 2. Constrants Constrants on grder depth H. 300mm H 750mm (G01-) 300mm H 900mm (G07,G08) Constrants on grder wdth B. 150mm B 400mm (G01-) 200mm B 500mm (G07,G08) Constrants on plate thcness. t w :web thcness, t :lange thcness 6mm tw 16mm, 9mm tw 36mm (G01-); 6mm t 19mm, 9mm t 32mm (G07-G09) 300mm H 600mm, 300mm B 600mm 9mm t 30mm, 9mm t 70mm (Wde-lange) w 200mm D 800mm,16mm t 40mm (Box secton) 2 2 braces Constrant on brace area A. 5cm A 100cm grders columns grders columns or each storey at ple top or each storey or each storey 2 t / B 7.7, t /( H 2 t ) 51.0 w 2 t / B 8.1, t /( H 2 t ) 36.6 (Wde-lange) w t / D 28.1 (Box secton) y z N 1.0 Z Z A b y t z c y : bendng moment around strong axs z : bendng moment around wea axs N : axal orce : allowable bendng stress b t : allowable tensle stress : allowable compressve stress c / h 1/ 200 : nter-storey o -th loor, h : -th storey heght 30000N R 15000N R : ple axal orce 1.5 B CL CU Q UN 1.3 B CL CU PP CL CU : plastc bendng moment o grder :plastc bendng moment at bottom o column :plastc bendng moment at top o column Q UN Q :requred horzontal resstant strength Q U :ultmate horzontal resstant orce o the structure U 5.Result o the optmzaton Table 3. shows the optmum desgn obtaned rom the proposed method. In case A. D ends o the long spanned grders (G07,G08) are pn connected whle n case B they are rgdly connected. Obvously, the condton o the connectons at grder end aects the result o column szes connected to the long spanned grders.

7 Table 3. Result o optmzaton Case A. D end o long spanned grders (G07,G08) are pn connected. TORY 4 F 7 F 8 F 10 F Roo Lst o Grder H(mm) B(mm) tw(mm) t(mm) TORY Lst o Column H(mm) B(mm) tw(mm) t(mm) TORY Lst o Brace Area(cm 2 ) G01 H C01 H F G02 H C02 H H F G04 H H H C05 H F H G07 H F G08 H G01 H C01 H G02 H C02 H H F F G04 H H H C05 H H F G07 H F G08 H G01 H C01 H F G02 H C02 H H F G04 H H F H C05 H H F G07 H G08 H G01 H C01 H F G02 H C02 H H F G04 H H H C05 H G07 H-550 H G08 H G01 H C01 H G02 H C02 H H H C05 H G04 H-650 H H G07 H G08 H Case B. D end o long spanned grders (G07,G08) are rgdly connected. TORY 4 F 7 F 8 F 10 F Roo Lst o Grder H(mm) B(mm) tw(mm) t(mm) TORY Lst o Column H(mm) B(mm) tw(mm) t(mm) TORY Lst o Brace Area(cm 2 ) G01 H C01 H B F G02 H C02 H B02 45 H F H B01 40 G04 H B02 55 H C05 H B01 45 H F B02 35 G07 H F B01 45 G08 H B02 35 G01 H C01 H B01 40 G02 H C02 H B02 40 H F H F B01 40 G04 H B02 45 H C05 H B01 35 H F B02 45 G07 H B F G08 H B02 35 G01 H C01 H F B01 30 G02 H C02 H B02 30 H F H B01 45 G04 H F B02 45 H C05 H B01 45 H F B02 55 G07 H B01 35 G08 H B02 35 G01 H C01 H B F G02 H C02 H B02 30 H F H B B02 45 C05 H G04 H-400 H H G07 H G08 H G01 H C01 H G02 H C02 H H H C05 H G04 H-500 H H G07 H G08 H

8 The dscrete optmum desgn obtaned rom present method drectly satses the requrements o practcal desgn. And structural engneers n practce can reer to and use the optmum desgns obtaned rom present method when they conduct a nal desgn o buldngs. What we realzed through the example s noted as ollows. (1)ome trals are needed or us to obtan the nal soluton shown n Table 3. For example, t too a lot o tme to solve and we obtaned a checerboard pattern soluton the secton gven by the soluton vary extremely between adjonng members when we assgned ndependent varables to each member. In practcal desgn, checerboard pattern soluton s unacceptable because t s common that members whch have smlar lengths or areas are grouped together and gven same secton. (2)I a desgn varable has small eect to reduce the objectve uncton, the desgn varable won t also be reduced. Thereore unacceptably large secton may be obtaned as an optmum desgn we assgn an ndependent desgn varable to a member sze. Ths means groupng o desgn varables s mportant. To avod such problems noted above and to obtan a reasonable and practcal desgn soluton, we need to nd out eectve constrants and groupngs o desgn varables. Control o constrants and groupngs whch determne qualty o the soluton s the pont structural desgners have to be nvolved. In the present example, we obtaned the good practcal desgn soluton through the member groupng shown n Fgure 3. whch s derved rom some tral optmzatons. Further studes whch we thn have to be done are lsted below. (1)Whle derent sectons are used or end part and central part o a grder member n common practcal desgn, the grder member n the present example has uned secton. o, ndependent desgn varable should be ntroduced to end and central part o a grder member. (2)Chec o panel zone strength at beam-column jont should be taen nto account. (3)Eectve member groupng and eectve constrants to obtan reasonable desgn soluton should be proposed by conductng many optmzatons usng present method. 6.Conclusons ost o conventonal researches on optmum desgn have conned ther nterest to ndvdual structural problems. ome o them are hard to apply to practcal desgn problems drectly because ther nterest s ar rom practcal nterest. We proposed an optmzaton method consderng most o the constrants whch s requred wthn the ramewor o Buldng tandard Law o Japan and can be expressed by nequaltes. It s shown by an example usng a realstc buldng model that we can drectly obtan a practcal desgn consstng o dscrete member szes. We ponted out that approprate member groupng and control o constrants are ey actor to obtan a desrable soluton. Especally, constrants have close relatonshp to perormance o a buldng structure such as saety actor o members and deormaton o structures. Thereore, control o constrants means control o perormance We are gong to tae more detaled constrants nto account and wden the coverage o present method. It s expected that ths optmzaton program s wdely used n practcal structural desgn. Reerences [1]Yujn Lee,hnta Yoshtom and Koj Uetan, A research on structural plannng usng characterstcs o optmum desgns o steel ramed plane rames. AIJ Journal o structural engneerng, Vol. 50B,pp ,2004(n Japanese) [2]hnta Yoshtom, aoto Yamaawa, Koj Uetan, A ethod For electng Optmum Dscrete ectons O teel Frames Usng Two-tep Relaxaton, Journal o structural and constructon engneerng, Transactons o AIJ No.586, pp , 2004(n Japanese) [3]Rosen,J.B.:The Gradenet Projecton ethod or Nonlnear Programmng - Part Ⅱ:Nonlnear Constrants,IA J., 9,pp ,1961