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1 Cty Research Onlne Cty, Unversty of London Insttutonal Repostory Ctaton: Mergos, P.E. (2016). Optmum sesmc desgn of renforced concrete frames accordng to Eurocode 8 and fb Model Code Earthquake Engneerng and Structural Dynamcs, do: /eqe.2851 Ths s the accepted verson of the paper. Ths verson of the publcaton may dffer from the fnal publshed verson. Permanent repostory lnk: Lnk to publshed verson: Copyrght and reuse: Cty Research Onlne ams to make research outputs of Cty, Unversty of London avalable to a wder audence. Copyrght and Moral Rghts reman wth the author(s) and/or copyrght holders. URLs from Cty Research Onlne may be freely dstrbuted and lnked to. Cty Research Onlne: publcatons@cty.ac.uk

2 Optmum sesmc desgn of renforced concrete frames accordng to Eurocode 8 and fb Model Code 2010 Panagots E. Mergos * Research Centre for Cvl Engneerng Structures, Department of Cvl Engneerng, Cty Unversty London, London EC1V 0HB, UK Abstract. Tradtonal sesmc desgn, lke the one adopted n Eurocode 8 (EC8), s force-based and examnng a sngle level of sesmc acton. In order to provde mproved control of structural damage for dfferent levels of sesmc acton, the new fb Model Code 2010 (MC2010) ncludes a fully-fledged dsplacement- and performancebased sesmc desgn methodology. However, the level of complexty and computatonal effort of the MC2010 methodology s sgnfcantly ncreased. Hence, the use of automated optmzaton technques for obtanng costeffectve desgn solutons becomes appealng f not necessary. Ths study employs genetc algorthms to derve and compare optmum sesmc desgn solutons of renforced concrete frames accordng to EC8 and MC2010. Ths s mportant snce MC2010 s meant to serve as a bass for future sesmc desgn codes. It s found that MC2010 drves to more cost-effectve solutons than EC8 for regons of low sesmcty and better or smlar costs for regons of moderate sesmcty. For hgh sesmcty regons, MC2010 may yeld smlar or ncreased structural costs. Ths depends strongly on the provsons adopted for selectng the set of ground motons. In all cases, MC2010 provdes enhanced control of structural damage. Keywords: Renforced concrete; sesmc desgn; Eurocodes; fb Model Code 2010; optmzaton; genetcalgorthms 1 Introducton Sesmc desgn of renforced concrete frames accordng to current codes, lke Eurocode 8 (EC8) [1], s based on forces. Structural members (.e. beams and columns) are dmensoned to wthstand nternal forces at the Ultmate Lmt State (ULS). Internal forces are calculated by conductng an elastc analyss for sesmc forces reduced by an emprcal behavour (forcereducton) factor q representng the ablty of the structural system to develop nelastc response [2]. Then, prescrptve rules are used (.e. member detalng rules, capacty desgn prncples) to ensure that the system s able to develop ductlty capacty adequate to justfy the behavour factor employed n the calculaton of nternal forces. Ths procedure s ndrect and opaque [3]. It s establshed that structural and non-structural damage s drectly related to member deformatons and lateral drfts [4, 5]. Hence, dsplacement- or deformaton-based desgn represents a more ratonal and drect approach for controllng nduced sesmc damage. A number of dfferent deformaton-based sesmc desgn methodologes (e.g. [6-8]) have been presented n the lterature and an nterestng comparatve study of them can be found n [9]. In addton to the above, n tradtonal sesmc desgn, as mplemented n Eurocode 8, a sngle level of sesmc acton s examned (typcally wth 10% probablty of exceedance n 50 * Correspondng author. Panagots E. Mergos, Lecturer n Structural Engneerng, Department of Cvl Engneerng, Cty Unversty London, EC1V 0HB, London, UK. E-mal address: panagots.mergos.1@cty.ac.uk, Tel (0)

3 years or return perod of 475 years). Only non-structural elements are checked for a more frequent sesmc acton at the Servceablty Lmt State (SLS). The need for mproved control of structural damage for dfferent levels of sesmc acton has led to the development of performance-based sesmc desgn [10]. Performance-based sesmc desgn s a transparent and drect desgn framework that requres a set of performance levels to be met for dfferent levels of sesmc hazard. Performance levels are related to the level of structural damage of the structure, whch n turn s drectly related to structural member deformatons and/or nter-story drfts. The new fb Model Code 2010 (MC2010) ncludes a fully-fledged deformaton- and performance-based sesmc desgn and assessment methodology for varous levels of sesmc hazard [3, 11]. MC2010 wll serve as a bass for future codes for concrete structures. It s worth notng that EC8-Part 3 [12] has already adopted a performance and dsplacement-based methodology smlar to MC2010. However, EC8-Part 3 s solely drected to sesmc assessment of exstng structures. MC2010 performance-based methodology covers both sesmc desgn of new and assessment of exstng structures [3]. In MC2010, each performance lmt state corresponds to a specfc physcal condton of the structure and t s expressed n terms of deformaton lmts of the structural members provdng drect control of allowable structural damage. The levels of sesmc hazard are dentfed by ther annual probablty of beng exceeded. Sesmc actons are specfed n terms of acceleraton tme-hstores of the ground moton components. The reference method for determnng sesmc demands s the most rgorous nelastc response hstory analyss wth stepby-step ntegraton of the equaton of moton n the tme doman [3]. In structural engneerng, the need for cost-effectve desgn solutons of complex problems n lmted tme has led to the development of automated structural optmzaton methodologes. These can be dvded n two categores: gradent-based and heurstc. Heurstc algorthms (e.g. Genetc Algorthms GA, Smulated Annealng SA, Partcle Swarm Optmzaton PSO, Taboo Search TS) are becomng more and more popular n structural optmzaton, because they can handle more complcated structural problems and they don t requre calculaton of dervatves [13]. Extensve research has been conducted over the past decades on optmum sesmc desgn of structures (e.g. [14, 15]). However, only a small part of ths research has been dedcated to renforced concrete structures. Ths can be partally attrbuted to the complex nature and detalng of renforced concrete structures that ncreases sgnfcantly the number of desgn varables [16]. Early efforts to optmse sesmc desgn of concrete structures were based on tradtonal sesmc desgn code approaches (e.g. [17]). The number of research studes on optmzaton of performance- and deformaton-based sesmc desgn of renforced concrete structures s rather lmted. Ganzerl et al. [18] were the frst, to the best of the author s knowledge, to consder sesmc optmzaton wth performance-based constrants. The constrants were expressed n terms of plastc rotatons at column and beam members ends based on FEMA-273 gudelnes (FEMA 1997). Pushover analyss was used to calculate sesmc demands. Materal cost was defned as the sngle desgn objectve. Secton dmensons and longtudnal renforcng steel areas were set as the desgn varables of the optmzaton problem. A smple portal frame case study was examned. Chan and Zou [19] examned optmum sesmc desgn of renforced concrete frames by employng optmalty crtera approach. The proposed soluton s dvded n two steps. Frst, member secton dmensons are selected to fulfll the servceablty performance level for frequent earthquakes. Then, member steel renforcement s desgned to wthstand demands of rare earthquakes for the ultmate performance level. Pushover analyss s used to calculate sesmc demands.

4 Lagaros and Papadrakaks [20] compared the provsons of EC8 for sesmc analyss of 3D renforced concrete structures wth a performance-based sesmc desgn methodology n the framework of mult-objectve optmzaton. For the latter approach, pushover analyss was employed to determne demands for dfferent levels of earthquake ntenstes. Storey drfts were used as performance level ndcators. Constructon cost and storey drfts for the 10% probablty of exceedance n 50 years (10/50) hazard level were set as the two desgn objectves. It was found that EC8 optmum desgns are more vulnerable to future earthquakes compared to optmum desgns obtaned by the performance-based methodology. Fragadaks and Papadrakaks [21] presented a performance-based optmum sesmc desgn methodology for renforced concrete frames based on nonlnear tme hstory analyses. Interstory drfts were used as performance crtera. Three performance levels (Immedate Occupancy, Lfe Safety and Collapse Preventon) were consdered. The sum of concrete and steel materal costs was set as the sngle desgn objectve. Desgn varables were determned by usng tables of concrete sectons and applyng the concept of mult-database cascade optmzaton. Both, a determnstc and a relablty-based approach, were mplemented. It was found that both approaches lead to structures of mproved sesmc resstance and reduced cost. Furthermore, relablty-based optmzaton may provde further economy compared to the determnstc soluton. Gencturk [22] nvestgated performance-based sesmc desgn optmzaton of renforced concrete and renforced engneered cementtous compostes (ECC) frames, by usng Taboo Search optmzaton algorthm. Intal cost and sesmc performance, n terms of nter-storey drfts for the 10/50 hazard level, represent the desgn objectves. Intal cost accounts for both materal and labor costs. Performance levels are determned by nter-story drfts threshold values. These values are taken ether as constant, n accordance wth FEMA-273 provsons, or by mappng local stran lmts to nter-story drfts after conductng sample pushover analyses. The lfe-cycle cost s also calculated for the optmal solutons. It s concluded that ECC can consderably mprove lfe-cycle performance of buldngs. It can be concluded from the above, that no study has been conducted so far on optmzaton of renforced concrete frames n accordance wth MC2010 sesmc desgn provsons. To fll ths gap, ths study presents optmum sesmc desgn solutons of renforced concrete frames obtaned by MC2010 and compares them wth optmum desgns followng EC8 gudelnes. To serve ths goal, a general computatonal optmzaton framework of renforced concrete frames s developed that makes use of a genetc algorthm able to track global optma of complcated problems wth dscrete desgn varables. The am here s to examne f and to what extent MC2010 provdes more cost effectve and safe desgn solutons wth respect to EC8. Ths s mportant snce MC2010 s meant to serve as a bass for future Eurocodes. In addton, topcs related to the complexty and computatonal cost of performng sesmc desgns based on the two standards as well as some open ssues n the sesmc desgn provsons of MC2010 are dscussed. 2 Optmzaton of renforced concrete frames wth genetc algorthms 2.1 Introducton In optmzaton problem formulatons, the goal s to mnmze an objectve functon C(x) subject to m number of constrants gj(x) 0 (j=1 to m). A desgn soluton s represented by the desgn vector x, whch contans n number of ndependent desgn varables x (=1 to n). In structural optmzaton the objectve functon C(x) s typcally the ntal cost of the structure. Constrants gj are ether related to engneerng demand parameters (EDPs) (e.g. forces,

5 dsplacements, rotatons, drfts) or to detalng rules set by desgn codes and constructon practce. Furthermore, to realstcally represent constructon practce, desgn varables x typcally take values from dscrete sets of values D=(d1, d2,, dk), where dp (p=1 to k) s the p-th possble dscrete value of desgn varable x and k s the number of allowable dscrete values of x. For renforced concrete structures, desgn varables are generally related to concrete secton dmensons and steel renforcement. The prevous can be wrtten as: Mnmze: C(x) Subject to: g j (x) 0, j = 1 to m (1) Where: x = (x 1, x 2,, x n ) x D = (d 1, d 2,, d k ), = 1 to n 2.2 Genetc Algorthms (GA) Genetc Algorthms (GA) [23] belong to the class of stochastc, nature-nspred heurstc algorthms. They are based on Darwn s theory of natural selecton and evoluton. GA can be easly mplemented and appled to advanced optmzaton problems snce they don t requre use of gradents of cost or constrants functons. Furthermore, they are able to dentfy global optma as opposed to local optmum solutons [13]. GA teratvely modfy populatons (generatons) of ndvduals n order to evolve toward an optmum soluton. An ndvdual x (genome) represents a canddate soluton to the optmzaton problem. The values of the desgn varables x (=1 to n) formng each ndvdual are called genes. The best objectve functon of a generaton s the smallest objectve functon of all ndvduals of the generaton. In order to create the next populaton, GA select certan ndvduals n the current populaton (parents) and use them to create ndvduals n the next generaton (chldren). The followng are the basc steps of GA: 1. A random ntal populaton s created. 2. New populatons are generated successvely by: ) Calculatng the objectve functons of all ndvduals. ) Selectng parents based on ther objectve functon. ) Makng chldren from selected parents. v) Formng new populaton from chldren. 3. Algorthm s termnated when one stoppng crteron s met. Three types of chldren can be created by GA: ) Elte chldren: These are the ndvduals wth the best objectve functons of the current populaton. They progress unchanged to the next populaton. ) Cross-over chldren: They are derved by mxng the genes of a par of parents. ) Mutaton chldren: They are created by alterng the genes of a sngle parent. In ths study, the mxed nteger GA as mplemented n MATLAB-R2015a [24] s employed. Ths algorthm can handle both contnuous and dscrete desgn varables. To serve ths goal, specal crossover and mutaton functons are used to ensure that dscrete varables take values only from pre-determned dscrete sets of values [25]. Furthermore, the algorthm s able to account for nonlnear constrants by usng the penalty functon approach. Accordng to ths approach, GA mnmze a penalty functon that s equal

6 to the objectve functon plus a term accountng for constrants volaton. More partcularly, the penalty functon s equal to the objectve functon for feasble desgns. For an unfeasble desgn, however, the penalty functon becomes equal to the maxmum value of the objectve functons of all feasble ndvduals of the populaton plus a sum of the constrant volatons of the unfeasble desgn [26]. The genetc algorthm n ths study s termnated when one of the followng stoppng crtera s met: ) Number of generatons exceeds a pre-specfed maxmum number of generatons. ) The mean relatve varaton of the best objectve functon value does not exceed a pre-specfed tolerance over a pre-specfed number of generatons. 2.3 Desgn parameters and varables The nput data of the optmzaton problem can be dvded n desgn parameters and desgn varables. Desgn parameters keep ther values fxed n the optmzaton process. In ths study, as desgn parameters are assumed the geometry (number and lengths of bays, story heghts and member connectvty), loadng and materal propertes of the renforced concrete frames as well as the concrete cover of the member sectons. On the other hand, desgn varables determne dmensons and steel renforcement of secton propertes. As shown n Fg. 1, desgn varables can be grouped n column and beam secton propertes desgn varable sub-vectors. Assembly of these sub-vectors forms the desgn varables vector x. Column secton propertes desgn varable sub-vectors are the heghts hc and wdths bc of the column sectons, the dameters dbc and numbers nc of man bars per sde, assumed heren the same for all column secton sdes for smplcty, the dameters dbwc, spacngs sc and numbers of legs nwc of transverse renforcement assumed agan the same n both column secton drectons heren for smplfcaton purposes. Beam secton propertes desgn varable sub-vectors are the heghts hb and wdths bb of the beam sectons, the dameters dbt and numbers of man bars ntb at the top, the dameters dbb and numbers nbb of man bars at the bottom, the dameters dbwb, spacngs sb and numbers of legs nwb of transverse renforcement parallel to beam secton heghts. It s mportant to menton here that the allocaton of desgn varables to the secton propertes s ndependent among the sub-vectors. Ths means, for example, that two column secton propertes can have the same heght and wdth desgn varables, but dfferent number of man bars or spacng of transverse renforcement desgn varables. Ths approach s effcent because t mnmzes the use of desgn varables and avods the applcaton of equalty constrants that complcates further the optmzaton problem. After defnng secton propertes, member propertes need to be determned. Member propertes nclude desgn parameters lke member lengths, concrete cover and materal propertes as well as the secton propertes of the members. In ths study, three secton propertes per member property are assgned. The frst two secton propertes determne the crtcal regons at the ends of members and the thrd secton property determnes the nternal part of the member between the two crtcal regons. However, the approach followed heren can be easly extended to consder an unlmted number of secton propertes per member property. Ths could be useful, for example, for beam members domnated by gravty loads, where the longtudnal and transversal renforcement may vary sgnfcantly nsde the member. Havng establshed member propertes, groups of members havng the same member propertes can also be defned. Ths can be very effectve for optmzaton problems snce t reduces mportantly the number of desgn varables. Furthermore, t s n agreement wth

7 typcal constructon practce, where several members are constructed n the same way for mprovng the effcency of constructon. On the other hand, ths approach leads to ncrease of the materal cost snce some members are over-desgned. Thus, a balance between cost and smplcty of constructon s necessary. ntb dbt bars nwc dbwc sc centers nc dbc bars hc nc dbc bars nwc dbwc sc centers hb nwb dbwb sb centers b c b b nbb dbb bars Fg. 1: Desgn varables: a) column sectons; b) beam sectons 2.4 Objectve functon In ths study, the objectve functon C(x) s the materal cost of the renforced concrete frames. The materal cost conssts of the cost of concrete, steel and the cost of the formworks of beam and column members. Hence, the total constructon cost s taken as n cols C(x) = =1 C c + C b n beams =1 (2) Where ncols and nbeams are the numbers of column and beam members and C c and C b are the costs of the th column and beam member respectvely. The cost of the th column and/or beam member can be determned as: C m = C cm + C sm + C fm (3) Where m stands for column or beam (m=c or m=b respectvely), C cm s the cost of concrete, C sm the cost of renforcng steel and C fm s the cost of formwork. The cost of concrete s calculated by C cm = h m b m L m C cu (4) Where L m s the member length and C cu s the cost of concrete per unt volume (Euros/m 3 ). Furthermore, the cost of renforcng steel s gven by: = j=1[(a sj L j ) + (V swj L wj /s wj )] C su ρ s (5) C sm 3 Where A sj and L j are the area and development lengths of man renforcng bars of secton j of member. It s noted that the lengths L j of secton man bars, n the case of beam members, can be dfferent for the top and bottom man renforcng bars. For smplcty, t s assumed n ths study that the lengths L j are equal to 25% of member length for the two end sectons and 50% of member length for the nternal secton.

8 In addton, V swj, L wj and s wj are the volume, development length and spacng of transverse renforcement of secton j of member. V swj s calculated by multplyng the total length of transverse renforcement at secton j by the area of one shear renforcement leg. C su s the cost of steel per unt mass (Euros/kg) and ρ s s renforcng steel densty n kg/m 3. L wj s taken equal to the lengths of the crtcal end regons for the two end sectons and equal to length of the member outsde the crtcal end regons for the nternal secton. The cost of formwork of each member s determned by the followng relatonshp, where C fu s the cost of formwork per unt area (Euros/m 2 ). C fm = 2 (h m + b m ) L m C fu (6) In the rest of ths study, the followng unt costs are assumed: C cu =100Euros/m 3, C su =1Euro/kg and C fu =15Euros/m Desgn constrants In sesmc desgn of renforced concrete frames, constrants gj(x) are ether related to engneerng demand parameters (EDPs) (e.g. forces, dsplacements, rotatons, drfts, etc.) or to detalng rules set by desgn codes and constructon practce. In the frst case, an EDP must reman below a lmt value EDPcap. Ths type of constrants can be wrtten n the followng normalzed form EDP EDP cap EDP EDP cap 1 0 (7) Regardng detalng requrements, the constrants can be expressed n terms of structural desgn parameters SDPs. It s noted that a SDP can be a desgn varable tself (e.g. column heght, man bar dameter) or a smple functon of the desgn varables lke the volumetrc ratos of steel renforcement. In some cases, t s requred that a SDP remans lower than or equal to a maxmum value SDPmax. Ths category of constrants s wrtten n the followng general form: SDP SDP max SDP SDP max 1 0 (8) In the other cases, t s requred that a SDP s greater than or equal to a mnmum value SDPmn. The latter famly of constrants s expressed n the normalzed form shown below: SDP SDP mn SDP mn SDP 1 0 (9) In the followng sectons, the constrants set by the dfferent desgn gudelnes wll be descrbed n more detal. In addton to them, constrants related to standard constructon practce should also be appled. Examples of these constrants are that the wdth of a beam cannot be greater than the wdth of the adjacent column; secton dmensons of the upper parts of a column cannot be greater than secton dmensons of the lower parts of the same column; number of legs of shear renforcement cannot be greater than the number of longtudnal bars and others.

9 3 Optmum desgn of RC frames accordng to EC2 Pror to desgnng for sesmc actons, RC frames must be desgned to resst dead and lve loads. Eurocode 2 [27] provsons are appled n ths study for desgnng aganst statc loads. EC2 provsons consst of a number of detalng rules and a number of requrements related to EDPs. Regardng detalng rules, desgn constrants of mnmum volumetrc rato of longtudnal renforcement, mnmum dameter of longtudnal and transverse renforcement, mnmum dstance between two longtudnal steel bars and mnmum volumetrc rato of transverse renforcement are expressed n the general form of Eq. (9). On the other hand, constrants of the maxmum volumetrc rato of longtudnal renforcement, maxmum spacng of shear renforcement and maxmum dstance of unrestraned next to restraned man bars of columns are wrtten n the form of Eq. (8). For the ULS, EDPs are member forces (moments and shear forces) derved by lnear elastc analyss for the followng load combnaton, where Gk represents the characterstc value of the permanent acton and Qk stands for the characterstc value of the varable acton. S d = 1.35G k Q k (10) EDPs constrants are wrtten n the general form of Eq. (7), where capactes are derved by usng characterstc materal strengths dvded by partal safety factors equal to γc=1.50 for concrete and γs=1.15 for renforcng steel. For bendng moments of column members, moment capactes are calculated for the axal load demand of the load combnaton under examnaton. For beam deflectons, the lmtng span to depth rato approach s used heren ensurng that deflectons are lmted to span/250. Moreover, crack control s acheved by lmtng maxmum bar sze or spacng. 4 Optmum sesmc desgn of RC frames accordng to EC8 In sesmc desgn of renforced concrete frames accordng to Eurocode 8 (EC8), structural members are desgned to meet the Lfe Safety (LS) performance level for a rare earthquake event wth 10% probablty of exceedance n 50 years (10/50) for ordnary structures. Collapse Preventon (CP) lmt state s later accomplshed by a number of capacty desgn prncples. Sesmc actons are defned through natonal zonaton maps n terms of peak ground acceleratons on rock agr. Sesmc desgn accordng to EC8 can be performed ether wthout provsons for energy dsspaton and ductlty (Ductlty Class Low DCL) or wth provsons for energy dsspaton and ductlty (Ductlty Classes Medum and Hgh DCM and DCH). DCM and DCH dffer n the levels of lateral strength and allowable nelastc response. DCH allows for further reductons n sesmc forces, but requres more demandng prescrptve rules for ncreasng ductlty capactes. For DCL, all sesmc EDPs are calculated from the sesmc load combnaton shown below, where desgn sesmc actons Ed are calculated by the desgn response spectrum that s derved from the elastc response spectrum reduced by the behavour factor q. ψ 2 s the quaspermanent load combnaton coeffcent of the varable acton. Reference analyss method of EC8 s the modal response spectrum analyss. However, for regular buldngs wth unmportant hgher modes the lnear elastc lateral force method can also be appled. S Ed = G k + ψ 2 Q k + E d (11)

10 For DCM and DCH, frst the dsspatve zones of structural members (typcally located at the ends) are desgned n bendng under the sesmc desgn load combnaton. Next, capacty desgn prncples are forced to ensure ductle structural response. In partcular, column sectons are desgned n bendng followng the strong column weak beam capacty rule to prevent soft storey falure mechansms. Moreover, capacty desgn n shear s appled to beam and column members to preclude brttle shear falures. In addton to the above, RC frames are checked for a frequent earthquake wth 10% probablty of exceedance n 10 years (10/10) to satsfy the Damage Lmtaton (DL) lmt state. Checks verfy that nterstorey drfts developed for the frequent earthquake are less than lmt values dependng on the type of non-structural elements (e.g. 1% for non-structural elements that don t nterfere wth structural response). P-delta (2 nd order) effects are consdered at the storey level wth calculatng rato θ from Eq. (12). In ths equaton, N tot and V tot are the total vertcal and shear load at the storey respectvely, δ s nterstorey drft and H s storey heght. It s requred that θ never exceeds 0.2. Furthermore, f θ exceeds 0.1 then 2 nd order effects are taken nto account by multplyng 1 st order effects by the magnfcaton factor 1/(1-θ ). θ = N tot δ V tot H (12) All prevous requrements are regarded as EDPs constrants and are ncluded n the optmzaton problem n the general form of Eq. (7). The EDPs are member bendng moments and shear forces, nterstorey drfts and θ ratos. Apart from EDPs constrants and EC2 detalng rules, DCM and DCH necesstate addtonal or strcter detalng rules n the crtcal regons to accommodate local ductlty demands. The addtonal column constrants of mnmum cross-secton sdes, mnmum volumetrc rato of longtudnal renforcement, mnmum dameter of transverse renforcement, mnmum number of bars per sde and mnmum confnement of transverse renforcement n crtcal regons are expressed n the general form of Eq. (9). The same holds for the addtonal beam constrants n crtcal regons such as mnmum volumetrc rato of longtudnal renforcement, mnmum longtudnal bar dameter for DCH, mnmum bottom renforcement at the supports and mnmum longtudnal bar dameters crossng nteror or exteror jonts. On the other hand, the more demandng column constrants n crtcal regons for maxmum spacng between restraned man bars and spacng of transverse renforcement are formulated n accordance wth Eq. (8). The same holds for the beam constrants of maxmum longtudnal renforcement volumetrc rato and spacng of transverse renforcement n the locatons of the crtcal regons. Fg. 2a presents the flowchart of the optmzaton soluton adopted n ths study for sesmc desgn of RC frames n accordance wth EC8 provsons. It can be seen that desgn canddate solutons are frst checked for constructon practce constrants and detalng constrants accordng to EC2 and EC8 provsons. Detalng constrants are examned frst because they requre less computatonal effort. If detalng constrants are not satsfed then EDPs constrants are not checked to avod the relatvely hgh computatonal cost related to fnte element analyses of RC frames. In addton, f the canddate solutons are not adequate aganst statc loads n accordance wth EC2 prncples they are not examned for sesmc loads to avod unnecessary analyses.

11 5 Optmum sesmc desgn of RC frames accordng to fb MC2010 fb MC2010 adopts a fully-fledged performance-based sesmc desgn methodology [3]. The code employs deformaton lmts, whch are drectly related to sesmc damage, n order to verfy 4 dstrct Lmt States. The Operatonal (OP) and Immedate Use (IU) Lmt States are related to servceablty of structures, whlst the Lfe Safety (LS) and Collapse Preventon (CP) are related to loss of lves and structural collapse (Ultmate Lmt States ULS). Lmt States are checked for dfferent levels of Sesmc Hazard. Deformaton lmts controllng Lmt States and correspondng levels of Sesmc Hazard recommended by fb MC2010 for ordnary structures are lsted n Table 1 [3]. a) Optmzer b) Optmzer Constructon practce, EC2 & EC8 detalng rules OK? NO Constructon practce & Statc loads detalng rules OK? NO YES YES EC2 EDPs constrants OK? NO Statc loads EDPs constrants OK? NO YES YES EC8 EDPs constrants OK? NO MC2010 CP Lmt State EDPs constrants OK? NO YES YES Desgn s feasble Desgn s unfeasble MC2010 LS Lmt State EDPs constrants OK? NO YES Optmzer MC2010 IU Lmt State EDPs constrants OK? NO YES MC2010 OP Lmt State EDPs constrants OK? NO YES Desgn s feasble Desgn s unfeasble Optmzer Fg. 2: Flowchart of optmum sesmc desgn accordng to: a) EC8; b) MC2010 The verfcaton of Lmt States entals comparsons of chord rotaton demands θed at member ends wth yeld chord rotatons θy at the same locatons for the OP Lmt State and twce θy for the IU Lmt State. Furthermore, the two ULS are checked by comparng the plastc part of chord rotaton demands at member ends θ pl Ed wth characterstc values (lower 5% percentle) of the cyclc ultmate plastc hnge rotaton capactes θ pl uk dvded by a factor of γ*r=1.35 for the LS Lmt State and wth θ pl uk wthout safety factor for the CP Lmt State. It s recommended [3] that for beams and rectangular columns wth rbbed bars yeld chord rotaton θy s taken from the followng equaton, where φ y s end secton yeld curvature, Ls the shear span of the member on the sde of the end secton, z s lever arm of end secton, ascr s a coeffcent equal to 1 f shear crackng precedes flexural yeldng or equal to 0 f not, h s end secton heght, dbl and fyl dameter and yeld strength of longtudnal renforcement (MPa) and fc member concrete strength n MPa. θ y = φ y (Ls+a scr z) ( h ) + φ yd bl f yl (13) L s 8 f c

12 Furthermore, characterstc ultmate plastc hnge rotaton capacty θ pl u,k s derved by the respectve mean value θ pl um dvded by safety factor γrd. When θ pl u,m s calculated by the followng emprcal relatonshp γrd can be taken equal to pl θ um = v f c 0.2 ( max(0.01;ω 2 ) max(0.01;ω 1 ) )0.3 (mn (9; L s h )) ( aρwfyw ) fc (14) In Eq. (14), ω 1 and ω 2 are mechancal ratos of renforcement n tenson and compresson zone respectvely, v s normalzed axal load rato, a s confnement effectveness factor and ρ w and f yw are volumetrc rato and yeld strength of transverse renforcement. It s noted that Eq. (14) s recommended for rectangular beams and columns wth ductle steel renforcement and wthout dagonal renforcement. In addton to chord rotaton checks, brttle shear falures are checked n terms of nternal shear force demands VEd and desgn shear force capactes VRd. VRd outsde plastc hnge regons s calculated as for statc loadngs. Insde plastc hnge regons, fb MC2010 specfes a strut nclnaton of 45 o when plastc rotaton θ pl exceeds 2 θy and 21.8 o for elastc response (θ pl =0). Interpolaton s allowed for ntermedate values of θ pl. The reference analyss method of fb MC2010 s nonlnear response hstory analyss wth step-by-step ntegraton of moton equatons n the tme doman. The fnte element model appled should use realstc estmates of the effectve elastc stffness of concrete members EIeff. It s recommended n MC2010 that EIeff of concrete members s taken by the followng relatonshp, where My represents member end secton yeld moment and the other parameters have been defned prevously. EI eff = M yl s 3θ y (15) Lumped plastcty fnte elements wth blnear moment-rotaton hysteretc models and realstc rules for stffness degradaton durng unloadng and reloadng may be employed to model nelastc response of renforced concrete members. It s worth notng that when conductng nonlnear analyss both types of sesmc demands (.e. deformatons and forces) are obtaned drectly by the analytcal soluton wthout addtonal consderatons for brttle modes of falure (.e. capacty desgn prncples). It s also mportant to clarfy that no addtonal prescrptve rules, lke detalng rules set by EC8 for DCM and DCH, need to be appled when desgnng n accordance wth MC2010 apart from the detalng rules requred for desgnng aganst statc loads. In MC2010, sesmc actons are represented by acceleraton tme-hstores of the ground motons. At least seven ground motons are requred to use average response values. All acceleraton tme hstores should be scaled such that ther elastc response spectrum s not lower than 90% of the target response spectrum for perods rangng between 0.2 T to 2 T, where T s the fundamental perod of the structure. As t wll be shown later n ths study, ths requrement set by MC2010 can be very onerous and may lead to mportant ncreases n the structural cost. It s remnded that EC8 specfes that the mean spectrum of the set of ground motons and not all spectra shouldn t be less than 90% of the target response spectrum n the same range of perods. It s also noted that pror to desgnng, T s not known and cannot be estmated wth accuracy because t depends on steel renforcement whch affects members yeld moments My and consequently effectve elastc stffness EIeff as defned n Eq. (15). Hence, a post-desgn check s requred to verfy that the set of ground motons satsfes the selecton crtera of MC2010 based on the actual T of the desgn soluton.

13 Table 1: Lmt States, Sesmc Hazard levels and Deformaton Lmts recommended by fb MC2010 for ordnary structures Lmt State Sesmc Hazard Deformaton Lmt Operatonal (OP) Frequent wth 70% probablty of exceedance n 50 years (70/50) Mean value of θ y Immedate Use (IU) Occasonal wth 40% probablty of Mean value of θ y may be exceeded by a exceedance n 50 years (40/50) factor of 2.0 Lfe Safety (LS) Rare wth 10% probablty of exceedance n 50 years (10/50) Safety factor γ* R of 1.35 aganst θ pl u,k Collapse Preventon (CP) Very rare wth 2% probablty of exceedance n 50 years (2/50) θ pl u,k capacty may be reached (γ* R =1) Fg. 2b presents the optmum desgn methodology adopted n ths study for sesmc desgn of RC frames n accordance wth MC2010. Intally, the desgn solutons are examned for constructon and statc loads detalng rules and EDPs. Ths s done n a manner smlar to optmum desgn accordng to EC2. These constrants are checked frst because they requre sgnfcantly less computatonal effort than the tme consumng nonlnear response hstory analyses. Later, the EDPs are examned successvely for each Lmt State of MC2010. If one Lmt State s not satsfed then the followng ones are not examned to avod unnecessary response hstory analyses. All EDPs constrants are wrtten n the general form of Eq. (7). Even wth ths approach, t s clear that MC2010 requres a large number of nelastc response hstory analyses to be conducted for each desgn soluton. Ths ncreases grossly the computatonal cost of the optmzaton task, where a sgnfcant number of tral desgns need to be examned n order to obtan the optmum soluton. Before closng ths secton, t s mentoned that no specfcatons of the MC2010 are provded regardng servceablty checks of non-structural components as well as some detalng rules concernng for example the length of the crtcal regons, where enhanced ductlty demands are expected. To fll ths gap n ths study, servceablty checks of nonstructural components are conducted accordng to EC8 recommendatons and crtcal end regon lengths are calculated n accordance wth EC8 DCM specfcatons. 6 Optmum sesmc desgn of RC frames applcatons In ths secton, applcatons of the optmum sesmc desgn methodologes descrbed prevously to RC plane frames are presented. In partcular, a smple portal frame and a concrete frame wth 4 storeys and 2 bays are examned. The buldngs are of ordnary mportance and rest on sol class B accordng to the classfcaton of EC8. The frames are desgned for 0.16g, 0.24g and 0.36g peak ground acceleraton values for the 10/50 sesmc hazard level n order to examne the nfluence of the level of sesmcty (low, moderate and hgh respectvely) on the optmum sesmc desgn solutons. The elastc (target) response spectrum wth 5% dampng of EC8 determned for the prevous specfcatons and 0.24g peak ground acceleraton s shown n Fg. 3. Peak ground acceleratons for the other sesmc hazard levels of MC2010 objectves are calculated by multplyng the 10/50 values by the mportance factor γi gven by the followng equaton proposed n EC8-Part 1, where PL s the target probablty of exceedance n 50 years and PLR s the reference probablty of exceedance n 50 years (=10%). γ I = ( P L P LR ) 1/3 (15)

14 The frames are desgned followng the provsons of EC8 for all three ductlty classes (.e. DCL, DCM and DCH) and n accordance wth MC2010. In the latter case and n order to evaluate the nfluence of ground motons selecton specfcatons, two dfferent cases are examned. In the frst case, desgnated as THA, the frames are desgned for a set of 7 scaled ground moton records satsfyng EC8-Part 1 recommendatons as descrbed n the prevous secton. In the second case, desgnated as THB, the frames are desgned for a set of 7 scaled ground moton records satsfyng MC2010 specfcatons. The goal here s to examne to whch extent the conservatve specfcatons of MC2010 on the selecton of ground moton records, descrbed n secton 5, can nfluence the cost of the optmal desgn solutons wth respect to EC8 ground moton selecton provsons. Fgure 3a presents the scaled and mean elastc spectra wth 5% dampng of the set of 7 ground motons selected and scaled followng EC8 provsons. In ths case, selecton and scalng was performed by employng computer program REXEL [28]. Because the fundamental perod of the structures s unknown pror to ther desgn t was decded to match the mean and target spectrum for perods between 0.1s and 4s n order to capture most possble solutons. The selected ground moton records can be seen n Table 2. They are all recorded on sol type B and have magntude Mw>5.5. It s evdent n Fg. 3a that the mean spectrum follows very closely the target spectrum. a) b) Fg. 3: Elastc spectra wth 5% dampng for ground moton sets selected and scaled n accordance wth a) EC8; b) MC2010 No computer tools exst for selectng record sets accordng to MC2010 gudelnes. To serve ths goal, n ths study, a smplfed procedure s appled. All records of the European Strong Moton Database [29] on sol type B wth Mw>5.5 are scaled so that ther scaled 5% dampng spectra are not less than 90% of the target spectrum n perods rangng between 0.1s and 4s. The scaled spectra are later ranked n accordance wth ther goodness-of-ft to the target spectrum as quantfed by the normalzed root-mean-square-error [30]. The frst 7 ground motons comprse the set of records used heren (Table 2). Fgure 3b presents the scaled and mean elastc spectra wth 5% dampng of the set of 7 ground motons selected and scaled followng MC2010. It can be seen that the mean spectrum mportantly exceeds the target spectrum leadng to serous overestmaton of sesmc demands. Ths reflects the level of conservatsm adopted n MC2010 specfcatons. For the optmum desgns, t s assumed that secton dmensons hc, bc, hb, bb take values from the followng dscrete set: (0.25m; 0.30m; 0.40m; ; 1.5m). Furthermore, longtudnal bars dbc, dbb, and dbt are defned n the followng dscrete values set: (12mm; 16mm; 20mm; 25mm). Transversal bars dbwc, and dbwb take values from: (8mm; 10mm; 12mm). Transverse renforcement spacng sc and/or sb may take the followng values: (0.1m; 0.15m; 0.20m; 0.25m; 0.30m). Fnally, numbers of man bars nc, ntb, nbb and legs of shear renforcement nwc and nwb may take any nteger value greater than one.

15 Table 2: Unscaled ground motons selected based on EC8 and MC2010 provsons Records selected based on EC8 Earthquake Name Staton Year Epcentral Dstance R (km) Magntude Kalamata ST X Montenegro (aftershock) ST Y Izmt ST Y South Iceland ST Y Umbra Marche ST X Frul (aftershock) ST Y Agon ST Y Records selected based on MC2010 Earthquake Name Staton Year Epcentral Dstance R (km) Mw Magntude Kalamata ST X Kalamata ST X South Iceland ST Y Campano Lucano ST X South Iceland ST Y Ano Losa ST Y Frul ST Y 6.1 Portal frame Mw PGA (g) PGA (g) Drecton Drecton In ths secton, a smple portal renforced concrete frame (Fg. 4a) s optmally desgned n accordance wth the methodologes descrbed prevously. The span of the frame s 4m and the heght 3m. Concrete C25/30 and renforcng steel B500C n accordance wth EC2 specfcatons are used. Concrete cover s assumed to be 30mm. Vertcal symmetrc concentrated loads are appled at the jonts equal to 120.0kN for permanent and 80.0kN for lve loadng. Storey mass for the sesmc combnaton s 29.4t. The frame conssts of two columns C1 and C2 and one beam B1. Due to symmetry, t s assumed that C1 and C2 have exactly the same sectons and renforcement, B1 has the same top and bottom longtudnal renforcement and member end sectons have the same transverse renforcement. Furthermore, due to constructon reasons, t s assumed that the longtudnal renforcement does not vary along beam and column members. However, end and ntermedate sectons may have dfferent transverse renforcement spacng to account for the addtonal desgn requrements n the crtcal end regons. Followng these observatons, two column and two beam sectons are used as shown n Fg. 4a. Sectons 1 are used for member end zones and sectons 2 for the rest of the element. Sectons 1 and 2 have exactly the same detalng apart from spacng of transverse renforcement. In total, 16 (8 for columns and 8 for the beam) ndependent desgn varables are used n ths problem. The results presented n the followng were obtaned by runnng GA wth populatons of 75 ndvduals. Iteratons were termnated when the mean relatve varaton of the best ftness value was neglgble for 100 generatons. MATLAB-R2015a default optons were used for GA operatons. Furthermore, a sgnfcant number of dfferent-ndependent GA runs for each desgn soluton were conducted and the mnmum cost obtaned s reported heren. Fgure 4b presents optmzaton hstores of the desgns obtaned by MC2010 methodology for the THA ground moton set and the three desgn peak ground acceleratons. It can be seen that optmum cost ncreases as desgn acceleratons ncrease. Fgure 5a compares optmum costs n Euros obtaned by all sesmc desgn methodologes for the three desgn peak ground acceleratons for the 10/50 sesmc hazard level. It can be seen that n all cases costs ncrease as desgn acceleratons ncrease. Desgns accordng to EC8 DCL and DCM yeld smlar costs for all desgn PGA values. On the other hand, DCH yelds sgnfcantly ncreased costs. Ths occurred because of the enhanced detalng rules of ths ductlty class and the dscrete desgn varable sets assumed n ths study. It s also worth notng that the optmum costs of DCH reman essentally the same for all desgn PGA values. Ths

16 shows the nfluence of detalng requrements on the fnal costs of renforced concrete structures. a) b) B1 1 3m 2 C1 C m 1 Fg. 4: a) Examned portal frame; b) Optmzaton hstores of desgns obtaned by MC2010 methodology for THA ground moton set and three dfferent desgn PGAs It s also evdent that desgns obtaned by the MC2010 for both ground moton sets (THA and THB) drve to sgnfcantly reduced desgn costs for the low 0.16g and moderate 0.24g desgn acceleratons. Furthermore, the MC2010 desgn wth THA moton set yelds slghtly smaller cost than the EC8 desgns for 0.36g. However, the same desgn methodology wth the THB moton set drves to sgnfcantly greater desgn costs than all EC8 desgns obtaned for 0.36g. The drect comparson of optmum costs obtaned by MC2010 methodology for the THA and THB ground moton sets shows the mportance of the appled accelerograms set. For 0.16g PGA both solutons yeld same optmum costs. Ths s because the desgn n ths case s controlled by mnmum detalng requrements. However, for hgher sesmcty levels the cost derved by selectng a ground moton data set n accordance wth MC2010 provsons s sgnfcantly hgher than the one derved by the EC8-Part 1 compatble set of accelerograms. Fgure 5b shows percentle contrbutons of constructon cost components to the total cost obtaned by the dfferent desgn methodologes for all desgn PGAs. It can be seen that for the 0.16g desgns accordng to MC2010 concrete and formwork domnate structural cost. However, ths changes as desgn PGA ncreases and for the 0.36g desgn for THB moton set the steel contrbutes more to the total cost. It s also worth notng the ncreased contrbuton of transverse steel for the DCH desgn wth respect to the other two EC8 ductlty classes. Table 3 presents secton dmensons, longtudnal renforcement rato ρl and rato of transverse steel parallel wth the shear force ρw of the optmum solutons. It can be seen n ths table that the THA desgn solutons have always smaller ρl values than the DCL solutons (for smlar secton szes) and smaller, smlar or even larger ρl values than the DCM solutons. Furthermore, they have equal or larger ρw values than the DCL solutons and smaller ρw values than the DCM solutons. It s also worth notng that the THA solutons have the same transverse renforcement ratos nsde and outsde the crtcal end zones. Ths s the case because the provded transverse renforcement s adequate to satsfy the rotaton and shear force constrants at the member ends and no addtonal detalng and confnement requrements are set by MC2010 nsde the crtcal end regons. Fgure 6 presents MC2010 checks of rotaton and shear force constrants (Eq. 7) for all Lmt States as obtaned by subjectng all 0.36g PGA optmum desgn solutons to the THA ground moton set. Column sectons are defned by the column member number (e.g. C1) and a letter desgnatng the locaton of the secton n the member (.e. B=bottom and T=top). Smlarly, beam sectons are defned by the beam member number (e.g. B1) and a letter desgnatng the locaton of the secton n the member (.e. L=left and R=rght). Lmt States are stated by the acronyms shown n Table 1.

17 a) b) Fg. 5: Optmum costs obtaned by dfferent desgn methodologes and desgn PGAs a) n Euros; b) percentle contrbutons Table 3: Secton propertes of optmum desgn solutons Members Columns Beams Sectons Secton 1 Secton 2 Secton 1 Secton 2 Property hc bc l w hc bc l w hb bb l w hb bb l w Unts m m % % m m % % m m % % m m % % DCL DCM g DCH THA THB DCL DCM g DCH THA THB DCL DCM g DCH THA THB Fg. 6: MC2010 rotaton and shear force constrants of beam and column secton optmum solutons obtaned by dfferent desgn methodologes for 0.36g desgn PGA. It can be concluded that all desgn solutons perform rather well. DCM and DCH desgns do not satsfy beam rotaton constrants for the OP Lmt State. It s recalled that EC8 does not

18 have any provsons for the OP Lmt State. Furthermore, t can be seen that the MC2010 desgn for THA moton set margnally satsfes beam and column rotaton constrants at the OP Lmt State and column rotaton constrants at the CP Lmt State. Ths shows that these where the controllng (actve) constrants of ths desgn. It s also evdent that MC2010 desgn for THB moton satsfes all constrants wth a hgh level of conservatsm. 6.2 Four-storey two-bay frame In ths secton, a four-storey two-bay renforced concrete frame (Fg. 7) s optmally desgned accordng to EC8 and MC2010 provsons. Span length s 3m and storey heght s 3m. Concrete C25/30 and renforcng steel B500C are used. Concrete cover s assumed to be 30mm. Vertcal concentrated loads of 144.0kN are appled at all exteror jonts and 288kN at the nteror jonts. All storey masses for the sesmc load combnaton are equal to 59.9t. Beam Secton 2 Beam Secton 2 4 x 3m B04 Beam Secton 1 Beam Secton 1 Column Secton 1 Column Secton 2 2 x 3m Column Secton 1 Fg. 7: Examned four-storey two-bay frame The frame conssts of 12 columns and 8 beams. Due to symmetry, t s assumed that the two exteror columns have exactly the same sectons and renforcement. Furthermore, for the smplcty of the calculatons, t s assumed that sectons and renforcement reman the same along columns heght. It s also assumed that one bar dameter s used for all longtudnal renforcement bars of the exterors and nteror column. The same holds for the dameter of transverse renforcement placed n all columns. Regardng beam members, t s assumed that the beams of the 1 st and 2 nd storey have the same secton and steel renforcement, whch s unform along ther length. The same assumpton s made for the beams of the 3 rd and 4 th storey. It s also assumed that one bar dameter s used for all beam longtudnal renforcng bars and one bar dameter for the transverse renforcement of all beam members. Due to symmetry, t s also assumed that beam sectons have the same top and bottom longtudnal renforcement. Followng the prevous observatons, two dfferent column secton propertes and two dfferent beam secton propertes are used n ths study. Column secton 1 s used for the exteror and column secton 2 for the nteror columns. Beam secton 1 s used for the beams of the frst 2 storeys and beam secton 2 for the beams of the last two storeys. In total, 23 ndependent desgn varables are employed for ths problem. The results presented n the followng were obtaned by runnng GA wth populatons of 100 ndvduals. Iteratons were termnated when the mean relatve varaton of the best ftness value was neglgble for 100 generatons. MATLAB-R2015a default optons were used for