ISBN 978-93-84422-22-6 Procdings of 2015Inrnaional Confrnc on Innovaions in Civil and Srucural Enginring (ICICSE'15) Isanbul (Turky), Jun 3-4, 2015. 206-212 Failur Load of Plan Sl Frams Using h Yild Surfac Mhod Smail Boukloua 1, Mohamd Laid Samai 2, Abdlhadi Tkkouk 2 1 Civil nginring darmn, Univrsiy of Bordj Bou Arréridj, Algria 2 Darmn of Civil Enginring, Univrsiy of Consanin 1, Algria Absrac. In h rsn work, a nonlinar analysis mhod of lan sl frams using yild surfac mhod is roosd. Th yild funcion, considring h inracion of bnding momn and axial forc can b usd o drmin h laso-lasic siffnss marix of bam lmn usd for such srucural. Th collas load and h collas mchanism of lan sl frams ar drmind by a numrical rogram in Malab using h incrmnal dirc mhod (s by s) and h fini lmn for which h yild surfacs of Eurocod3 is adod. Analysis rsuls show ha h roosd mhod is saisfacory. Kywords: nonlinar analysis, sl fram, yilds surfac, collas loads, marial nonlinar. 1. Inroducion Whn h srucur is subjcd o loads ha xcd h roorional limi of h marial, h marial sars o yild, hus h abov mniond assumions bcom inadqua and canno rrsn h ral bhaviour of h srucur. In his cas lasic analysis is rquird. Plasic analysis mhods can b classifid in wo grous: disribud lasiciy mhods ha accoun for srading of lasic zons wihin h whol volum of h srucur (Plasic zon mhods) and lumd lasiciy mhods ha assum lasic zons o b formd wihin small ara sa h nds of fram mmbrs calld lasic hings, whil fram mmbrs xhibi lasic bhaviour bwn lasic hings (Plasic hing mhods). Marial yilding, h ffcs of gomrical non linariis, rsidual srss and h yild funcion includs h ffc of h srss comonns acing in h sysm o rdic h yilding of h marial ar major aramrs ha conrol h load-carrying caaciy of h srucur, and hav bcom a ar of many naional Sandards and Cods (Eurocod3, AISC, Briish Sandards, c). Morovr, fas-sd rsonal comurs dvlod in h las 20 yars mad h us of nonlinar analysis rocdurs mor availabl for racical uross. Howvr, in his cas, h discussion will b limid o h marial nonlinar, h ffcs of rsidual srss and h lasiciy is suosd o b concnrad only in h cross scion of h nds of h bams (lasic hing mhod) and i is in lasic sa by h combinaion of srss ha saisfis h yilding condiion, inracion of h bnding momn wih axial forc. Digial rogram in Malab has bn dvlod for h load facor calculaion by adoing h aroach of yild surfac. This rogram uss h fini lmn mhod o succssiv linar analyzs and basd on h s by s mhod. Corrsonding auhor, Docora sudn, E-mail: smail_ing097@yahoo.fr h://dx.doi.org/10.17758/ur.u0615319 206
2. Yild Surfac Th variaion of h bnding momn wih axial forc in a cross scion can b lod in rms of h dimnsionlss quaniis N/N and M/M. Th rsuling curv is calld h yild surfac bcaus any oin on h yild surfac rrsns a sa of h fully yildd cross scion. Th I or H-shad scions ar ofn usd in sl frams, for which h yild surfacs of Eurocod3 [1] is adod in rsn work. Th quaions ar rsnd bllows: - Yild surfac of Eurocod3 [1] Figur 1 : Wih: M 0.9M 1 M 1 = 0.9M 1-0.5a a =min [A w /A, 0.5] A: ara of scion. A w : ara of wb. for N N a/2 N N (2) 1- for a/2 0.8N N (1) M/M P 1.0 Fig. 1: Yild surfacs of sl I-H scions of Eurocod3 In which, n=n/0.8n is raio of h axial forc ovr h squash load, m=m/0.9m is h raios of h majoraxis momns o h corrsonding lasic momns, and h numbrs 0.8 and 0.9 in h dnominaor accoun for rsidual srsss. 2.1. Normaliy Rul Th yild surfacs dscribd for various cross-scional shas can b rsnd using a yild funcion Ф such ha for a scion in a fully yildd sa undr forc inracion Ф =0. Whn h ffcs of bnding momn and axial forc ar akn ino accoun on h yild surfac, h associad gnralizd srains ar h roaion and h axial dislacmn of scion. Th normaliy rul was originally roosd by Von Miss in 1928, i may b alid for his cas as follows: N/N P h://dx.doi.org/10.17758/ur.u0615319 207
U { } N M Or, symbolically: f { dp } (4) Whr d } rrsn h vcor of h lasic dformaion incrmns, λ is h lasic dformaion magniud, { P f is a gradin vcor a a oin of h yild surfac Ф. Whn h lasic loading occurring, h oin forc is on h yild surfac (or subsqunc yild surfac) Ф=0. In aking h drivaiv of his rlaionshi, w obain: (3) N M 0 N M Whr h arial drivaivs mus b akn a h original sa of srss rsulan. Equaion (5) can b wrin in vcorial form as { P} 0 f (6) Whr {P} rrsn h vcor for h incrmns of srss rsulans. Th orhogonal condiion can b alid o h rlaionshi bwn h incrmns of srss rsulans and lasic dformaion as imlid by Pagr s [2] samn ha for lasic-rfcly lasic marial, h srss incrmn dos no work on h incrmn of lasic srain. Whn alying o fram mmbrs, his samn mans ha { d}{ P} 0 (7) For marials in h lasic sa, h lasic flow always occurs in associaion wih a dissiaion of mchanical nrgy. Thus, for an incrmn of lasic dformaion{ d}, h dissiaiv nrgy ΔW is always osiiv and is givn by: W {P} { d} {P} f 0 (8) 2.2. Elasolasic Siffnss Marix For a scion in lasic sa, h incrmnal dformaion vcor, { d}, consiss of boh lasic and lasic dislacmns, dnding on which forc comonns ar aciv in h yild funcion. Hnc: { d} { d} { d} (9) Whr h incrmnal lasic dislacmn vcor K { d} d f K (5) { d} is rlad o h incrmnal forc vcor by: { P} (10) Whr K is h lasic siffnss marix. Using Equaion (10) in Equaion (6), h lasic mulilir λ can b found o b: h://dx.doi.org/10.17758/ur.u0615319 208
f K d f K f Subsiuing Equaion (13) ino Equaion (12), h lasolasic siffnss marix, K K { d} (11), can b found: { P} (12) Whr h lasolasic siffnss marix is: K K K f f K f K f (13) Equaion (12) is a gnral xrssion for a yildd bam lmn. Sinc a bam lmn may b subjcd o diffrn combinaions of yilding sas a is nds, h form of K varis according o h sa of yilding and h yild funcion adod for lasic analysis. 3. Numrical Vrificaion 3.1. Vogl Poral Fram Th oral fram shown in Figur2 was analysd numrically in 1985 by Vogl [3], and his fram has bn usd by svral rsarchrs (Chn 1993[4], Chn and Kim 1997 [5], Kim and Lu 1992[6], Toma and Chn 1992[7]) as a bnchmark soluion for including marial non-linariis including rsidual srsss, gradual yilding and full lasiciy. Th fram siz, marial roris and load informaion ar illusrad in Figur 2, and h fram mmbr sizs ar lisd in Tabl 1. Th horizonal dislacmn of righ ur cornr (nod A) vrsus load facor curv by h roosd aroach is comard wih h lasic zon mhod and wih h lasic hing mhod of Vogl (1985) wih h lasic hing mhod in Figur3. Th ulima load facor obaind by h mhod roosd is 0.9192 whras ha by Vogl s lasic hing analysis is λ=1.017 and Vogl s lasic zon analysis is λ=1.02. V=2800kN V=2800kN H=35kN HEA340 A Δ HEB300 HEB300 5m 4m Fig. 2: Gomric configuraions and loading arn of Vogl s fram h://dx.doi.org/10.17758/ur.u0615319 209
1.2 1 0.9192 1.017 1.02 0.8 Load facor 0.6 0.4 0.2 Vogl s lasic zon mhod Vogl s lasic hing mhod Prsn analysis (yild surfac Eurocod3) 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Laral dislacmns(m) Fig. 3: Load-dislacmn curv a h o of Vogl s oral fram TABLE I: Mmbr sizs and scional roris of h Vogl oral fram Scion A(mm 2 ) I(x10 6 mm4) h(mm) b(mm) roris W f W l (x10 3 mm 3 HEA340 330 300 9.5 16.5 13300 276.9 1850 HEB300 300 300 11.0 19.0 14900 251.7 1869 3.2. Six Sory Fram Th fram siz and load informaion ar illusrad in Figur 4 and h fram mmbr sizs ar lisd in Tabl 2. Th marial lasic modulus E of sl is 206kN/mm2 and h yild srngh f y is 235N/mm2. Th horizonal dislacmn of righ-ur cornr (Nod A) vrsus load facor curv by h yild surfac modl roosd in his ar is rsnd in Figur 3. Th ulima load facor λ obaind by h mhod roosd is 1.2888. 1,4 1.2888 1,2 1 Load facor 0,8 0,6 0,4 0,2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Laral dislacmns(m) a nod A Fig. 3: Load-dislacmn curv a h o of six sory fram h://dx.doi.org/10.17758/ur.u0615319 210
TABLE II: Mmbr sizs and scional roris of h six-sory fram Scion A(mm 2 ) I(x10 6 mm4) h(mm) b(mm) roris W (mm) f (mm) W l (x10 3 mm 3 HEA340 330 300 9.5 16.5 13300 276.9 1850 HEB160 160 160 8.0 13.0 5430 24.92 354 HEB200 200 200 9.0 15.0 7810 56.96 643 220 220 9.5 16 9100 80.91 827 HEB240 240 240 10.0 17 10600 112.6 1053 HEB260 260 260 10.0 17.5 11800 149.2 1283 HEB300 300 300 11.0 19.0 14900 251.7 1869 IPE240 240 120 6.2 9.8 3910 38.92 367 IPE300 300 150 7.1 10.7 5380 83.56 628 IPE330 330 160 7.5 11.5 6260 117.7 804 IPE360 360 170 8.0 12.7 7270 162.7 1019 IPE400 400 180 8.6 13.5 8450 231.3 1307 P 2 P 2=63.4kN 2P 2 P 2 P 2 IPE240 HEB200 HEB160 A 19 IPE300 IPE300 HEB200 HEB240 HEB160 IPE330 IPE360 HEB240 HEB260 IPE400 H 2 =10.23kN HEB260 6m 6m =20.44kN Fig. 4: Six-sory sl fram. h://dx.doi.org/10.17758/ur.u0615319 211
4. Conclusion An aroach for nonlinar analysis of sl frams using yild surfac mhod is roosd in his ar. This aroach us h yild funcion includs h ffc of h axial forc and bnding momn acing in h sysm o rdic h yilding of h marial. This aroach also considrs h influncs of h marial nonlinar including h rsidual srss. Th numrical rsuls show ha h roosd aroach is saisfacorily, and is suiabl for h nonlinar analysis of sl frams. 5. Rfrncs [1] Eurocod3, Dsign of Sl Srucurs (2003), Euroan Commi for Sandardisaion. [2] W. Pragr,An inroducion o lasiciy, London. Addison-Wsly Pub. Co, Inc (1959). [3] U.Vogl, Calibraing frams. Dr Sahlbau 1985; 10: 296-301. [4] S. Toma, WF. Chn, Euroan calibraion frams for scond-ordr inlasic analysis.enginring Srucurs 1992; 14(1): 7-14. h://dx.doi.org/10.1016/0141-0296(92)90003-9 [5] ZH.Zhou,SL. Chan, Elasolasic and larg dflcion analysis of sl frams by on lmnr mmbr. I: On hing along mmbr, Journal of Srucural Enginring, ASCE 2004;130(4): 538-544. h://dx.doi.org/10.1061/(asce)0733-9445(2004)130:4(538) [6] W. F. Chn, Advancd Analysis in Sl Frams, (1993) CRC Prss, Boca Raon, FL. [7] W. F. Chn ands-e. Kim, LRFD Sl Dsign Using Advancd Analysis,(1997) CRC Prss, Boca Raon, FL. h://dx.doi.org/10.17758/ur.u0615319 212