Strength design optimization of structural steel members according to Eurocode3

Transcription

1 Srengh design opimizaion o srucural seel members according o Eurocode3 Juan Francisco Carbonell-Marquez 1, Luisa María Gil-Marín, and Enrique Hernández-Mones 3 Keords: Seel Srucures, Srucural Opimizaion, Cross-Secion Class, Local Buckling, Eurocode 3. (Published in Journal o Consrucional Seel Research, Volume 80, Januar 013, Pages 13 3) hp://dx.doi.org/ /j.jcsr Absrac In order o design a seel member subjeced o a bending momen and an axial load, here are an ininie number o possible soluions I- or H- seel cross-secions, he doubl-smmeric soluion being jus one o hem. This paper presens a procedure o obain he opimal seel cross-secion rom he ininie number o possible soluions. The process is based on he Reinorcemen Sizing Diagrams emploed in reinorced concree srengh design. The procedure looks or an pe o soluion regarding compac or non-compac seel secions. All aspecs relaed o local insabiliies ill be aken ino accoun, as ell as special consideraions in order o address he global insabiliies associaed ih he slenderness o he seel elemen. Noaion A Cross-secion area emploed o compue N b,rd A Cross-secion area A e Eecive cross-secion area or cross-secions in Class 4 A 1 A E M M b,rd M,Ed M,Rk N N b,rd N Ed Top lange area Boom lange area Seel elasic modulus Exernal in-plane bending momen Design buckling resisance momen o a laerall unresrained beam Exernal in-plane bending momen applied o he secion Criical cross-secion characerisic momen resisance abou - axis Exernal axial load Design buckling resisance o a compression member Exernal axial load applied o he secion 1 Ph.D. candidae. Universi o Granada, Campus de Fuenenueva Granada. Spain. jcarbonell@ugr.es. Associae Proessor, Universi o Granada, Campus de Fuenenueva Granada. Spain. mlgil@ugr.es. 3 Proessor, Universi o Granada, Campus de Fuenenueva Granada. Spain. emones@ugr.es

2 N Rk W e, W el, W pl, W b comp b b b d h k l b b χ Criical cross-secion characerisic resisance o normal orce Eecive secion modulus abou - axis, or Class 4 secions Elasic secion modulus abou - axis Plasic secion modulus abou - axis Appropriae secion modulus emploed in he compuaion o M b,rd Compressed lange idh Boom lange idh Top lange idh Web heigh Cenroid heigh Speciied seel ield srengh Facor emploed in he compuaion o he crierion o preven he compression lange buckling in he plane o he eb Unbraced lengh o he beam-column member Boom lange hickness Top lange hickness Web hickness Reducion acor or he relevan buckling mode in compression χ LT Reducion acor or laeral-orsional buckling γ M 1 Parial sae acor or he buildingζ Ineracion acor 1. Inroducion Tpical secions or beam-column members in seel ediicaions are usuall I- or H- rolled secions. Hoever, in oher ields o seel consrucions such as civil bridges, he seleced crosssecions ma be elded, since he higher demands o be suppored b he srucure calls or larger dimensions no possible or abulaed rolled secions. Wheher ediicaion or civil consrucion, designers end o proporion heir srucures using smmeric secions, hese being jus one o he muliple soluions. Neverheless, he opimal soluion ma no coincide ih he smmeric one and imporan savings in he amoun o seel used could be achieved. In his respec, environmenal concerns consiue an imporan role because savings in seel consumpion ma be ranslaed ino signiican reducions in greenhouse gas emissions. Figure 1. Condiions o he problem o be analzed The presen ork sudies he opimal design o beam-column members subjec o an exernal inplane bending momen, M, and o an axial load, N, iniiall considered o be applied a he cenroid o he eb o he secion (Figure 1). Figure shos he emploed nomenclaure or he

3 cross-secion o he elemen and he sign crieria or he applied exernal loads. Bending momen, M, acing on he srong axis o he cross-secion ill be considered posiive hen compressing he op lange o he secion. The applied axial load, N, ill be considered posiive in ension. For he sake o simplici, he illes in rolled secions and hroa hickness in elded secions have been ignored in he process. The dieren elemens o he secion are proporioned o provide suicien srengh and siness o resis he exernal acions and avoid premaure buckling o he member. For non-compac secions, he plasic capaci ill no be reached, so elasic capaci ill be emploed. b z A 1 d M N h d A b b b z Figure. Nomenclaure and sign crieria The problem sudied in his ork has alread been solved b Gil-Marín e al. [1] or Class 1 secions. Opimizaion as compleed b using he RSD design approaches [-3]. This mehodolog, originall conceived or reinorced concree, represens he required reinorcemen area or supporing a deermined exernal loading as a uncion o deph o neural axis in he concree secion (Figure 3). When appling RSD design approaches o opimizaion in seel secions, minor changes need o be made. Thereb, he graphics represen he crosssecion area, A, as a uncion o he eb heigh, d, and he opimal soluion corresponds o he one ih he loes value or A (Figure 4). Figure 3. Example o RSD in a reinorced concree secion, rom [4]

4 The presen paper explains he ell-developed process ha as olloed o obain he opimal soluion or an pair o ( M, N ). The process makes i possible or he designer o choose he Class o he adoped cross-secion; his is eiher compac or non-compac. Selecing he Class o he secion is ver imporan, or example, hen designing a building or earhquake resisance according o Eurocode 8 (EC8) [5]. EC8 saes ha, or an given building subjeced o an earhquake, he relaion beeen is resisance and capaci or dissipaing energ is relaed o he secion classiicaion (see Table 6.3 in EC8). Generall speaking, he more ducili needed he more compacness is required or he cross-secion. Figure 4. Example o RSD in a seel secion: opimizaion o IPE500 under M,Ed = 88 kn m and N Ed = kn ih =16 mm and =10, mm. Poin A represens he opimal soluion and poin B corresponds o IPE500. Taken rom [1] In conras o he previous case is a composie roada or higha bridge. These kinds o bridges, hich also called in-girder bridges, are composed o o longiudinal seel girders conneced o he concree slab o he deck b shear connecors. Tin-girder bridges are he mos economical soluion hen covering span lenghs in he range o 30 and 100 m [6], ih special suiabili beeen 60 and 80 m [7]. Considering hese span lenghs, sel-eigh becomes an imporan acion o be ihsood. Under his load, and beond he complexi involving a composie secion, cross-secions under posiive momen a mid-span regions o composie bridges are usuall in Class 1 or, since compression is carried mainl b he concree deck. Hoever, on inernal suppors, under negaive momen, secions end o be designed in Class 3 or 4 in order o avoid he excessive amoun o seel ha ould be needed i hose compressed secions ere o be in Class 1 or [8]. The pical secion or hese kinds o bridges is shon in Figure 5. The mos usual range or H/L, being H he heigh o he I-secion and L he covered span, is beeen 1/5 and 1/0 or higha or roada bridges and 1/15 or raila bridges [6-7][9]. For a higha bridge ih a span o 600 m, H ould be beeen.5 and 3.0 m. This is due o he ac ha he high dimensions o he secions do no allo he designer o choose hem rom he sandard rolled secions and a elded design is needed. For hese pes o girders, he algorihm developed ihin his ork les he designer impose an consrain relaed o he dimensions o a paricular elemen o he secion, in his case, eb heigh or even relaed o he Class o he cross-secion.

5 The algorihm used o opimize he secions has been implemened in a compuer program and some examples are presened here. The resuls obained ill be analzed in order o es he validi o he process. B H S = 0.50 o 0.55 B H/5 o H/3 Figure 5. Tpical secion or a in-girder composie bridge. The opimizaion procedure As above explained, he opimizaion procedure o be presened in he curren ork is based on RSD mehodolog. This approach consiss on he consideraion o all he possible soluions or a design problem hrough a graphical represenaion ha allos o choose he opimal one. In reinorced concree members, usuall he reinorcemen area is represened in uncion o he neural axis deph [-3]. In seel consrucion, as as observed ih reinorced concree, an ininie number o soluions exis or he design o a seel cross-secion subjeced o combined loads N and M. These soluions can be presened using graphics similar o hose used in he reinorced concree RSD represenaion. In his case, he area o srucural seel has been represened in uncion o he heigh o he eb [1]. The main advanage o his procedure is ha he engineering kno all he possible cross-secions ha resis a given combinaion o axial load and momen (N, M) making possible he choice, among all hem, o he opimal one considering minimum eigh, availabili o seel shapes, simplici on he job sie, Class o he cross-secion and so on. The process olloed during he opimizaion procedure is represened in he lo char in Figure 6. Secion iniial proporioning The irs sep in he process is o selec a ixed value or he eb hickness,, and a range o values or he heigh o he eb, d. The range o d is obained accouning boh shear srengh and shear buckling requiremens. The langes preliminar proporions are provided b equilibrium o orces acing on he cross-secion, appling he axial load a he cenroid o he eb. The equilibrium is esablished b ignoring he eb conribuion and assuming ha he orces carried b he op and boom langes ac a he ends o he eb and drive he langes o he ield sress. Thereore, he sum o momens on eiher ends o he eb resuls in Eq. 1:

6 d A1 d + M N = 0 d A d + M + N = 0 (1) Load combinaion M, N Range o values or d Preliminar proporion o lange areas (Eq. 1) Range o values or Preliminar proporion o lange idhs, b & b b (Eq. ) Secion Classiicaion Reducion o elemens in Class 4 Global insabiliies: - Laeral buckling N b,rd (Eq. 3) - Laeral-orsional buckling M b,rd (Eq. 4) - Flange idhs modiicaion General Mehod EC3 Ineracion acor ζ (Eq. 5) 0,95 ζ 1,00 NO YES Local buckling o lange in he plane o eb? (Eq. 10) NO YES Rejec he secion END OF PROCESS: sore he secion SELECT THE OPTIMUM SOLUTION FROM THE STORED CROSS-SECTIONS Figure 6. Flo char explaining he enire process Once A 1 and A are knon or each value o d, he nex sep is o choose anoher range o values or he lange hicknesses, and b. Thereore, or each value o and b he values o he lange idhs can be obained rom Eq. :

7 b = A A and b = () 1 b b In he olloing, ihou los o generali, he same hickness o boh langes has been considered, being, = b =. Secion classiicaion As described in Eurocode 3(EC3)[10], he role o cross-secion classiicaion is o ideni he exen o hich he resisance and roaion capaci o he cross-secion is limied b is local buckling resisance. The classiicaion o a deermined cross-secion ill depend on he slenderness, i.e. he idh o hickness raio, o he pars subjec o compression. According o EC3, here are our classes or seel secions: Class 1, hich can orm a plasic hinge ih he roaion capaci required rom plasic analsis ihou reducion o he resisance; Class, similar o class 1 bu ih limied roaion capaci due o local buckling; Class 3, hose secions in hich local buckling appears beore orming a plasic hinge and are assumed o ork ih an elasic disribuion o sresses reaching he ield srengh; and Class 4, in hich local buckling is reached beore elasic limi [11]. This classiicaion ma also be ound in oher codes as AISC Seel Consrucion Manual [1] ih oher erminolog and slenderness limi values. Thereb, according o AISC, Class 1 and secions are called compac secions; Class 3 secions are equivalen o non-compac secions; and Class 4 secions are similar o slender secions. The limi values or he slenderness o each componen o he secion are given b Tables 5.-1 and 5.-, presened in secion 5 o Par 1-1 in EC3. According o hese sandard codes, he cross-secion is classiied according o he highes (leas avorable) class o is compression pars. Widhs o he elemens o he cross-secion in Class 4 have o be reduced in order o heir eecive dimensions according o Par 1-5 o EC3. Global insabiliies a member level Once he class o he cross-secion is deermined, i is necessar o calculae he resisance o he beam-column member o laeral buckling and laeral-orsional buckling due o axial load and bending momen, respecivel. Folloing he ormulas given in EC3 [9], he design buckling resisance o a compression member should be aken as: N b, Rd χ A = (3) γ M 1 here A=A or cross-secions in Classes 1,, or 3, and A=A e or cross-secions in Class 4 hen subjeced o uniorm compression. The parameer χ is he reducion acor or he relevan buckling mode, compued as indicaed in secion in Par 1-1 o EC3. On he oher hand, Secion 6.3. o EC3 [9] provides he ormula o calculae he parameer χ, i.e. he reducion acor or laeral-orsional buckling. According o his, he design LT buckling resisance momen o a laerall unresrained beam should be aken as:

8 M b, Rd χ W LT = (4) γ M 1 Here, W is he appropriae secion modulus, aken as pl, W el, or Class 3 cross-secions, and e, he relevan axis is applied. W or Class 1 or cross-secions, W or Class 4 cross-secions hen onl momen abou When he buckling resisances o he member are calculaed, he General Mehod or laeral and laeral orsional buckling o srucural componens is applied. This mehod, explained in Secion o EC3 [9], allos he veriicaion o he resisance o he ormer global insabiliies o single members subjec o compression and mono-axial bending in he plane. The member mus ulill Eq. 5 in order o achieve sabili. N Ed M χ N γ χ M γ +, Ed Rk M 1 LT, Rk M 1 1 (5) here N Rk and, Rk M are he criical cross-secion characerisic resisance o normal orce and momen resisance abou - axis. In his ork, applied loads N Ed M,Ed are: ( ) M = M + e + e N (6) NEd, Ed 0 N = N (7) Being e N he shi o he relevan cenroidal axis o he cross-secion due o he idhs reducion in class 4 hen he member is subjeced o uniorm compression and e 0 he disance beeen he mid-heigh o he eb here he axial load is supposed iniiall applied a he gravi cener o he gross-secion (Figure 7), calculaed as: ( ) e = h d + (8) 0 In he above expression h is he heigh o he gravi cenre o he cross-secion. In his ork, he value or he sum presened in Eq. 5 has been called ineracion acor and is ζ χ NRk γ M 1 χlt M, Rk γ M 1 N M Ed, Ed represened b = +. Design adjusmens I is clear rom he lo char presened in Figure 6 ha he proposed procedure is ieraive. The dimensions o he cross-secion are preliminar proporioned and classiied. Aerards, he General Mehod is applied o evaluae he sabili o he member; because in mos o he cases he preliminar cross-secion ill no be able o sand he applied loads ihou buckling, dimensions need o be modiied. In his ork, or each pair o values d -, he idh o he langes, b and b b, are adjused unil he member does no buckle, i.e. ζ 1. Hoever, in order o gain opimal resuls, a loer limi has been imposed o ζ, so ha he adjusmens ill be compleed hen 0.95 ζ 1. The adoped process or providing a cross-secion o minimum

9 cross-secional area, ulilling all he sabili consideraions, is similar o he one olloed b [1], and is explained belo: 1. I ζ < 0,95 he secion provides excess capaci. To reduce he cross-secional area, he idhs o boh langes are reduced unil: 0,95 ζ 1 (9). I ζ > 1 he secion behavior is governed b insabili. To provide suicien srengh, he lange areas mus be increased. The approach o increase one lange or anoher depends on axial orce and bending momen: a. I M = 0 or N = 0, he secion is smmeric rom he iniial proporioning given b Eq. 1. The area o boh langes are increased he same amoun unil he condiion given b Eq. 9 is ulilled. b. I M 0 and N 0, he secion rom Eq. 1, he secion ill iniiall be asmmeric. In his case, one o he lange areas is increased in order o reduce he eccenrici given b Eq. 8 unil he ormula given b Eq. 9 is ulilled: i. I M and N have an equal sign, he op lange idh ill increase. ii. I M and N have a dieren sign, he boom lange idh ill increase. Class 1,, and 3 secions Class 4 secions b, e h G e 0 d N G e G e N h d e 0 N b b, e Figure 7. Values or he eccenriciies e 0 and e N Once Eq. 9 is ulilled or cerain values o d - - c (in his example = = c ), he crosssecion ill be sored i he dimensions o he langes in compression are suicien o preven local buckling in he plane o he eb. According o secion 8 in Par 1-5 o EC3 [13], he olloing crierion should be me:

10 d E d k E A d 3 k bcomp comp (10) The value o k should be aken as ollos: - Plasic roaion uilized k = 0,3 - Plasic momen resisance uilized k = 0,4 - Elasic momen resisance uilized k = 0,55 All he cross-secions ih heir corresponding pairs o d - are sored. These soluions are sored b cross-secional area and he minimum is ideniied as he opimal soluion. I is imporan o noice ha he process ma provide some soluions ih he same opimal crosssecional area. In his case, he inal seleced soluion ill be ha ih he minimum value o ineracion acorζ. Furhermore, he procedure provides an ininie number o soluions (depending on he esablished consrains). The opimum (i.e. minimum cross-secional area) or he smmeric soluion is jus one o he possible cross-secions ha ma be chosen [1-3]. 3. Examples The validi and eeciveness o he process have been esed and can be seen in he olloing hree examples; in order o obain minimum cross-secional soluions or hree combinaions o M and N ih he condiions represened in Figure 1: a simple suppored beam ih end-ork condiions (i.e. pin suppored end and ree arping). The applied load combinaions correspond o hree poins in he ineracion equaion (Figure 8) corresponding o a specimen made o seel Grade 35 ( = 35 N/mm ) ih a cross-secion HEB600 ( d = 540 mm; = 15, 50 mm, = 30 mm; b = b = 300 mm; b A = 7000 mm 6,00 m. The load combinaions are presened in Figure 8. ) and an unbraced lengh, l b, equals o A 100 M (kn.m) C B N (compression) (kn) Figure 8. Ineracion equaion corresponding o HEB600, or = 35 N/mm, l b = 6 m and ψ = 0

11 3.1. Combinaion A: M= kn.m bending momen applied on he righ suppor o he beam- The irs combinaion o loads corresponds o poin A in Figure 8, simple srong axis bending ih a value o M = 1391,60 kn m. Figure 9 shos he obained design soluions or dieren eb dephs, d, ih a range rom 50 mm o 000 mm ih a sep o 5 mm. The adoped range o values or lange hicknesses,, varies rom 4 mm o 40 mm, ih a sep o mm. The HEB 600 eb hickness ( = 15,50 mm ) is adoped or ever soluion. According o Eq. 1, i = = c all he obained soluions are doubl-smmeric (i.e. b = bb ). The resuls rom Eq. 1 are presened as a coninuous line. Dos in Figure 9 correspond o he soluions obained aer he adjusmen process or he our dieren Classes o he cross-secion. To disinguish beeen each class dieren have been used, respecivel. In Figure 9 he soluion corresponding o he HEB secion and he opimal ones obained or each class using he opimizaion procedure have been ideniied. As ma be observed rom Figure 9, he iniiall proporioned dimensions or he elemens o he cross-secion given b Eq. 1 are subsequenl modiied b he adjusmen process. In some cases, hose dimensions have been overesimaed since he conribuion o he eb as ignored in Eq. 1. Hoever, man soluions have crosssecional areas greaer han iniiall esimaed due o he ac ha members urned ou unsable and buckled and hereore dimensions need o be modiied in order o ge suicien srengh o ihsand he applied loads. Class 1 Class Class 3 Class 4 From Eq. 1 Min Max Sep (mm) 4 40 d (mm) Opimal soluion C1 - HEB 600 (C1) Opimal soluion C3 Opimal soluion C4 Opimal soluion C Figure 9. Cross-secional area A o he soluions in erms o eb deph d or srong axis bending momen The soluion ih he loes cross-secional area corresponds o: d = 95 mm; = 15, 50 mm ; = 16 mm; b = b = 335 mm; b A = 5058 mm

12 The eb and op lange Classes are 1 and respecivel, leading o cross-secion Class. The ineracion acor is 0,9989 ζ =. Figure 10 shos he opimal soluion or each Class according o EC3 and compares heir crosssecional area ih he one o HEB 600. The able in Figure 10 provides he dimensions or hese opimal soluions. Class 1,, and 3 secions reduce he lange idh, b, hen increasing eb deph, d, hile in Class 4 b increases since eb is reduced or local buckling. In his case, onl compac soluions (Classes 1 and ) provide less cross-secional area han he sandard HEB600. Figure 9 shos ha a saving o 7, % ih respec o he area o HEB600 can be obained A (mm ) Opimal Soluion Class 1 Class Class 3 Class 4 HEB 600 A (mm ) b (mm) (mm) d (mm) Figure 10. Comparison beeen he dimensions o dieren opimal soluions or each Class and HEB 600, or srong axis bending momen. Scale o dimensions skeches: 1/400 In Figure 11 he obained resuls rom he opimizaion process imposing = 30 mm (lange hickness o HEB600) have been represened or boh elded and rolled secions. This igure shos ha i elded secions are considered insead o rolled secions, areas slighl larger are obained. These dierences are due o he dieren values o he imperecion acors corresponding o he buckling curves ha are dieren or boh elded and rolled secions. For his example no elded soluion exis ih a cross-secional area under 7000 mm - HEB600 cross-secion area- hile i a rolled secion is emploed an area (or d = 805 mm andb = 7 mm ). A = 6098 mm is obained

13 Figure 11 shos ha he curve corresponding o rolled secions almos maches he soluion corresponding o he HEB 600. These small dierences are due o he ac ha, as as explained earlier, in his ork he illes in rolled secions are no aken ino accoun. 5.5 x From Eq. 1 Class 1 Class Class 3 Class 4 Welded Rolled A (mm ) HEB 600 (C1) Opimum or elded secions (C1).5 Opimum or rolled secions (C1) d (mm) Figure 11. Cross-secional area A o he soluions ih = 30 mm in erms o eb deph d or srong axis bending momen emploing elded and rolled secions imperecion acors

14 3.. Combinaion B: N= kn (compression) In his case, he seel secion member is subjec o a pure compression ih a value o N = 4180,80 kn. This load combinaion corresponds o poin B in Figure 8, i.e. he buckling capaci o he considered HEB 600 member. The resuls or he dieren values o d, ih a range rom 50 mm o 800 mm ih a sep o 5 mm, are presened in Figure 1. Again, he HEB 600 eb hickness ( = 15,50 mm ) is adoped or ever soluion. The adoped range o values or lange hicknesses,, sars a 4 mm and inishes a 40 mm, ih a sep o mm. The obained opimal soluion corresponds o d = 15 mm; = 15, 50 mm = 18 mm; b = b = 49 mm; b A = 1045 mm. This soluion saves a,05 % o seel ih respec o he HEB600. The cross-secion Class is 3 due o he slenderness raio or he langes in c compression: 10ε < = 13.3 < 14ε. The ineracion acor or his soluion isζ = 0,999. Class 1 Class Class 3 Class 4 Opimal soluion C1 Opimal soluion C HEB 600 (C) From Eq. 1 Opimal soluion C3 Opimal soluion C4 Figure 1. Cross-secional area A o he soluions in erms o eb deph d or pure compression As in he ormer example, Figure 13 shos he opimal soluion or each Class. In his paricular case, all he opimal soluions have cross-secional areas smaller ha he one corresponding o he sandard HEB600. As in he previous example, Eq. 1 provides smmeric soluions since, he onl applied load here is no is he compressive axial load. Because he areas o he langes are no aeced b eb deph, d, he lange idhs, b, ill be he same or ever ixed value o he lange hickness,. Figure 14 shos he evoluion o he lange idh, b, as uncion o he deph o he eb, d, or a ixed value o he lange hickness, = 30 mm. The obained opimal soluion corresponds o a eb deph d = 145 mm and a lange idhb = b = 357 mm. The corresponding cross-secional area is A = 3668 mm. b

15 A (mm ) Opimal Soluion Class 1 Class Class 3 Class 4 HEB 600 A (mm ) b (mm) (mm) d (mm) Figure 13. Comparison beeen dimensions o dieren opimal soluions or each Class and HEB 600, or pure compression N brd,z > N brd,y Class 1 Class N brd,z < N brd,y Class 3 Class b (mm) Opimal soluion HEB From Eq d (mm) Figure 14. Flange idh, b, or he soluions o lange hickness = 30 mm in erms o eb deph d or pure compression In Figure 14 o regions appear. Region 1 corresponds o soluions here he relevan mode or laeral buckling under compression is lexural buckling (soluions are smmeric) abou - axis. In his Region, soluions need o increase heir preliminar proporioned lange idh b an imporan amoun beore reaching sabili, because he relevan mode is governed b he

16 momen o ineria abou -, hich is proporional o b : I b ( meaning being proporional). Hoever, I d 3, resuling in much less ider soluions as d becomes deeper. On he oher hand, Region corresponds o lexural buckling under z-z axis and soluions ge quick sabili since I z b 3, and soluions need o increase lighl heir preliminar proporioned langes. In his Region, he slope o he curve becomes much laer as d increases since no I z d Combinaion C: M= kn.m & N= kn (compression) This case corresponds o a combinaion o simulaneous compression and bending momen abou he srong axis. Poin C in Figure 8 coincides ih hal compression and bending momen capaci o he sandard HEB 600 adoped as a benchmark problem. Soluions have been obained again or he same range o values or eb deph, d, and lange hickness,, as in he previous example. The value o he eb hickness,, is 15,5 mm. Figure 15 shos he resuls obained and he opimal secion, or hich he dimensions are: d = 585 mm; = 15,50 mm = mm; b = 433 mm; b = 334 mm; b A = 594 mm. For his secion, boh op lange and eb are Class, and he enire cross-secion resuls in ha Class. The soluion saves a 4 % o seel ih regards o he sandard HEB x Class 1 Class Class 3 Class A (mm ) 4 HEB 600 (C) Opimal soluion (C) 3 Opimal soluion (C1) 1 From Eq. 1 Opimal soluion (C4) Opimal soluion (C3) d (mm) Figure 15. Cross-secional area A o he soluions in erms o eb deph d or simulaneous compression and bending momen abou srong - Figure 16 shos he obained opimal resuls or each Class o secions. In his case, as in he ormer example, once again, all o hem have a less cross-secional area han he sandard HEB600. There are o o hem, soluions or Classes and 3, hich are almos he same area (sligh dierences in dimensions o langes and eb resul in jus 1 mm less in cross-secional area or soluion in Class ).

17 A (mm ) Opimal Soluion Class 1 Class Class 3 Class 4 HEB 600 A (mm ) b (mm) b b (mm) (mm) d (mm) Figure 16. Comparison beeen he dimensions o dieren opimal soluions or each Class and HEB 600, or simulaneous compression and bending momen abou srong -. 8 x No soluions ih he resricion b = b c Class 3 Class Aer reinemen Cross-secion Flange A (mm ) From Eq. 1 HEB 600 (C) Opimal (C4) soluion d (mm) Figure 17. Cross-secional and lange area in erms o d, or soluions o = 0 mm and orcing boh langes o be equal (i.e. doubl-smmerical cross-secion) or simulaneous compression and bending momen abou srong -.

18 As he procedure is compleel general, he doubl-smmerical cross-secion ma be exrapolaed ihou loss o generali. I he idh o boh langes are orced o be equal, he opimal soluion corresponds o a lange hickness o = 0 mm and d = 65 mm; = 15,50 mm b = 396 mm; b = 396 mm; b A = 6303 mm. Figure 17 represens boh he cross-secional and lange areas or doubl-smmerical cross-secion ih = 0 mm in uncion o he heigh o he eb. For his igure i is eviden ha soluions onl exis or values o d rom 40 mm, being he secions in Class 3 or 4. The sandard HEB600 is included in he lis o possible soluions in Class Global opimizaion In order o exend he ormer opimizaion procedure o oher values o eb s hickness,, he above process has been applied o several values o beeen 6 mm and 19 mm or he axial compression and bending momen abou he srong axis denoed as combinaion C (see Figure 8). The opimal cross-secion (i.e., ih minimum area) obained or each class o cross-secion [9] or each hickness o he eb can be ideni in Figure 18. This igure shos ha he smalles area ha ulil all he EC3 [9] requiremens corresponds o cross-secion in class 4 ih = 8 mm. This opimal secion needs o be siened because he slenderness o he eb is oo high. The opimum cross-secion in class 3 and in Class 1 and appears or = 13,5 mm and = 14,5 mm, respecivel. In such cases he slenderness o he eb is lo enough ha ransverse sieners are no needed. In Figure 18, he opimal soluions obained or he value o he hickness o he eb adoped in he ormer secions ( o he sandard HEB 600) have also been indicaed. x Class 1 Class Class 3 Class A (mm ) 30 8 = 15.5 mm (mm) Figure 18. Opimal (i.e. minimum) cross-secional area in erms o obained or each class o cross-secion and or simulaneous compression and bending momen abou srong -.

19 4. Conclusions As has been explained and demonsraed in his ork, emploed smmerical cross-secions are usuall no, in mos o he cases, he opimal soluions. This ork presens an ieraive procedure in order o ge he opimal soluion or he I-shaped cross-secion o a seel beamcolumn member subjec o an exernal axial load and bending abou srong axis. The process is based on RSD diagrams or opimizing he longiudinal reinorcing seel in reinorced concree secions and complees he procedure proposed b Gil-Marín e al. [1] or obaining hese opimal soluions ih seel secions in Class 1 according o Eurocode 3. This mehod allos engineers o choose among all he possible soluions: compac, non-compac and slender secions, obaining imporan savings in seel and hence leading o reducions in greenhouse gas emissions. Acknoledgemens The presen ork as inanced b he Spanish Minisr o Educaion. The irs auhor is a Spanish Governmen PhD ello (FPU gran AP ). This suppor is graeull acknoledged. As indicaed, his ork has been published b Journal o Seel Consrucional Research. This publicaion is also graeull acknoledged. Reerences [1] Gil-Marín LM, Aschheim M, Hernández-Mones E. Proporioning o seel beam-column members based on RSD opimizaion mehodolog. Engineering Srucures 008; 30(11): [] Hernández-Mones E, Aschheim M, Gil-Marín LM. Impac o opimal longiudinal reinorcemen on he curvaure ducili capaci o reinorced concree column secions. Magazine o Concree Research 004; 56(9): [3] Hernández-Mones E, Gil-Marín LM, Aschheim M. Design o concree members subjeced o uniaxial bending and compression using reinorcemen sizing diagrams. ACI Srucural Journal 005; 10(1): [4] Hernández-Mones E, Gil-Marín LM. Hormigón armado preensado: concreo reorzado preesorzado. Granada: Universidad de Granada, Grupo de Invesigación TEP-190 Ingeniería e Inraesrucuras; 007. (hp://.ugr.es/~emones/prensa/hormigonesrucural.pd) [5] Comié Europeo de Normalización, Agencia Española de Normalización. UNE EN : Eurocódigo 8: Proeco de esrucuras sismorresisenes. Pare 1: Reglas generales, acciones sísimicas reglas para ediicación. Madrid: Aenor; 011b. [6] Llago Acero R, García Rodríguez P. Composie in-girder bridges: A compeiive soluion or medium span bridges. Revisa de Obras Publicas 010; 157(3516): [7] Brozzei J. Design developmen o seel-concree composie bridges in France. Journal o Consrucional Seel Research 000; 55(1 3): [8] Ru H-K, Youn S-G, Bae D, Lee, Y-K. Bending capaci o composie girders ih Class 3 secion. Journal o Consrucional Seel Research 006; 6(9):

20 [9] Dirección General de carreeras. Obras de paso de nueva consrucción: concepos generales. Madrid: Miniserio de Fomeno; 005. [10] Comié Europeo de Normalización, Agencia Española de Normalización. UNE EN : Eurocódigo 3: Proeco de esrucuras de acero. Pare 1-1: Reglas generales reglas para ediicios. Madrid: Aenor; 008. [11] Rugarli P. Classiicaion o I- or H-shaped cross-secions under mixed inernal acions. Journal o Consrucional Seel Research 009; 65(8-9): [1] Seel Consrucion Manual (AISC 35-11). 14h ed. Chicago: AISC American Insiue o Seel Consrucion; 009. [13] Comié Europeo de Normalización, Agencia Española de Normalización. UNE EN : Eurocódigo 3: Proeco de esrucuras de acero. Pare 1-5: Placas planas cargadas en su plano. Madrid: Aenor; 011a.