Failure Load of Plane Steel Frames Using the Yield Surface Method

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1 ISBN Procdings of 2015Inrnaional Confrnc on Innovaions in Civil and Srucural Enginring (ICICSE'15) Isanbul (Turky), Jun 3-4, 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, smail_ing097@yahoo.fr h://dx.doi.org/ /ur.u

2 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 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/ /ur.u

3 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/ /ur.u

4 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 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/ /ur.u

5 Load facor Vogl s lasic zon mhod Vogl s lasic hing mhod Prsn analysis (yild surfac Eurocod3) 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 HEA HEB 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 , ,2 1 Load facor 0,8 0,6 0,4 0, Laral dislacmns(m) a nod A Fig. 3: Load-dislacmn curv a h o of six sory fram h://dx.doi.org/ /ur.u

6 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 HEA HEB HEB HEB HEB HEB IPE IPE IPE IPE IPE 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/ /ur.u

7 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: [4] S. Toma, WF. Chn, Euroan calibraion frams for scond-ordr inlasic analysis.enginring Srucurs 1992; 14(1): h://dx.doi.org/ / (92) [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): h://dx.doi.org/ /(asce) (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/ /ur.u