TIME-DEPENDENT BEHAVIOR OF COMPOSITE BEAMS AT SERVICE LOAD ACCORDING EUROCODE 4

Transcription

1 11 h Nionl Congess on Theoeil nd Applied ehnis, 2-5 Sep. 29, Booves, Bulgi TIE-DEPENDENT BEHAVIOR OF COPOSITE BEAS AT SERVICE LOAD ACCORDING EUROCODE 4 DONCHO PARTOV VSU L.Kvelov,1373 Sofi, 175 Suhodolsk S. e-mil: pov@vsu.bg VESSELIN KANTCHEV VSU L.Kvelov,1373 Sofi, 175 Suhodolsk S. e-mil: knhev@vsu.bg ABSTRACT. The ppe pesens nlysis of he sess hnges due o eep in silly deemine omposie seel-onee bem. Fo deemining he edisibuion of sesses in bem seion beween onee ple nd seel bem wih espe o ime, Vole inegl equions of he seond kind hve been deived. Numeil mehod, whih mkes use of line ppoximion of he singul kenl funion in he inegl equion. Exmple wih he model poposed is invesiged. The eep funions e suggesed by he CEB-FIP models ode 199. KEY-WORDS: omposie bem, Vole inegl equions, EUROCODE 4 1. Inoduion The ime-vying behviou of omposie seel-onee membes unde susined sevie lods dwn he enion of enginees who wee deling wih he poblems of hei design moe hn 6 yes. I is known h while in he seel bem, unde he effe of he seviebiliy lods, we see only elsi defomions, in he onee ple duing he ime signifin inelsi defomion kes ple s onsequene of eep nd shinkge of onee. These inelsi sins in he onee dek use edisibuion of sess nd signifin ineses in defomion. The fis woks, whih give he nswe o his poblem, e bsed on he Lw of Dishinge [3] (heoy of ging), who hd fis fomuled ime-dependen sesssin diffeenil elionship fo onee, using he following equion: dε σ dϕ 1 dσ (1.1) =, whee ϕ is lled eep funion. d E d E d These books nd ppes onneed wih he nmes of Föhlih[6], Esslinge [6], Klöppel [6], Sonng [6], Kune [6], ülle [6], Dimiov [6], zik [6] nd Bujňák [6] epesen one independen goup fo whih i is heisi h by wiing equilibium nd ompibiliy equions nd he onsiuive lws fo he wo

2 D. Pov, V. Knhev meils, he poblem is govened by sysem of wo simulneous diffeenil equions, whih hve been deived nd solved. As known in his diffeenil equions i exiss goup of noml foes N, ( ), N, ( ) nd bending momens, (),, ( ), whih influene he genel sess ondiions of he silly deemine omposie ple bem is expessed by he deese of he sesses in he onee ple nd in he inese of sesses in he seel bem (fig. 1). In pllel wih he developed nlyil mehods, Blszkowik [6], Bdfod [6] Fiz [6] nd Wippel [6] hve developed ppoxime mehods, whih use Dishinge s ide fo pplying in he lulion he idel (fiiious) modulus of elsiiy [3]: E (1.2) Ei = ; whee ϕ n is he ulime vlue of eep. 1 ϕ n N, () (), N,o, o = = 1237, knm N s, () N s, o s, () s, up Fig.1. Composie bem Fig. 2. Noml sesses σ () in ime Wih Wippel s mehods [6] he fis sge of he developmen of he nlyil mehods is bsed eniely on he woks of Dishinge [3], hs been ompleed. Fuhe developmen of heology s fundmenl siene nd is ppliion o onee [1] s well s ge numbe of invesigions in he field of eep of onee hve led o new fomulions of he ime-dependen behvio of onee [2,8].These new fomulions h give he elionship beween σ ( ) nd ε () e fomuled by inegl equions, whih pesen he bsis of he heoy of line visoelsi bodies. The inegl-ype eep lw, i.e., he supeposiion equion fo unixil pesibed σ, is expessed by: sess hisoy ( ) sh (1.3) ε (, ) = ε ( ) σ ( ) J (, ) dσ ( τ ) J (, τ ) ; Howeve, in ode o void he mhemil poblems in solving of he inegl equions of Vole fo eing he poblem onneed wih he eep of onee suues, Tos [9] nd Zen [9], hve evised he inegl elionship ino new lgebi sess-sin elionship :

3 Time Dependen Behvio of (1.4) ε = [ 1 ϕ ] [ 1 ρϕ ] σ E σ σ E whee ρ is he elxion oeffiien. On he bsis of h lgebi sess-sin elionship, new mehods hve been developed onneed wih he nmes Wppenhns [7], Wolff [7], Tos [9], Heim [7], Amdio [7], Dezi [7] (by peposiion h he onneos e defomionsble) nd Gilbe [7], fo solving he poblem ised by Föhlih [6].In pllel wih he mehods developed by Fuk [7], Kindmn [7], Lpos [7], Phl [7], Pov [6], on he bsis of he heoy of line visoelsi bodies, Sle [6], Henzel [6], nd Pofne [6] hve eenly developed new mehods, whih e bsed on he modified heoy of Dishinge, lled lso he heoy of Rüsh-Jungwi [7]. This heoy is desibed by he following equions: dε d σ ϕ f, v 1 dσ (1.5) = ; d Ev d Ev d E ( ) ϕ [ ( ) ( )] f, K f K f whee Ev =, ϕ f,v = ; 1, 4 1, 4 Diffeen ppoh o he solving of he fomuled poblems is pplying he FE by Heing [7], Cumbo [7] nd Wissmn [7].Sine he heoy of Rüsh-Jungwi [7] hs been subjeed o seious iiism in he woks of Alexndovski- Auyunyn [1] nd Bžn [2], he uho of he pesen ppe mke n emp fo new sep owd deiving moe peise soluion of he poblem. An effo is mde o give n nswe o he dispue beween Bžn nd Rüsh-Jungwi in [2]. The fis woks [ 6], whih give he nswe o his dispue [2], using he inegl equion of Vole, e bsed on he Lw of by Bolzmnn-Vole [1,2] who fis fomuled ime-dependen sess-sin diffeenil elionship fo onee, desibed by he following inegl equion ( ) σ dσ τ 1 (1.6) ε () = [ 1 φ( )] [ ( )] d, E ( ) 1 φ τ τ ; E ( ) τ φ τ = ϕ N K τ f τ ; is he so lled he eep funion nd whee ( ) ( ) ( ) ϕ he ulime vlue of eep oeffiien, ( τ ) N K depends on he ge inese of onee. I is lled he funion of ging, nd i heizes he poess of he ging. The inese of τ mkes K ( τ ) monoonously deese. Howeve he new noms suggesed by EUROCODE-4 [5] in nlysis of omposie seel-onee bems egding heology, equied new CEB-FIP eep models ode 199, whih leds o ompleely diffeen ppoh fo solving of he bove fomuled poblems. In his ppe we y o efomule nd solve hese poblems king ino oun he new mhemil fomuls. ( )

4 D. Pov, V. Knhev 2. Bsi equion of Vole of seond kind Fo onsn elsiiy module of onee, fo ssessmen of noml foes N, () nd bending momen, ( ) wo line inegl Vole equions of he seond kind e deived. d N, () = λn N, ( τ) 1 ϕφβ( fm) β( τ) β( τ) (2.7) λ N φ β f β β λ N β (2.8) ( ) ( ) ( ) ( ) N, m N sh d, () = λ, ( τ) 1 φβ( fm) β( τ) β( τ). EI λ φ β f β β λ N ( ) ( ) ( ) (), m, EI 2 E = 1 A A 1 E N = 1 I E A I EI in whih: λ, λ, 3. Numeil exmple The mehod pesened in he pevious pgph is now pplied o simply E = 3,2.1 P, E = 2,1.1 P, A = 882 m, A = 383,25 m, E 4 4 n = = 6,56, I = 6615 m, I = ,7 m, = 23,39 m, E 2 4 = 8,829 m, = 13,868 m, Ai = 2453,5 m, Ii = ,758 m. = 1237 knm, N = 846,6 kn, = 27,56 knm o, o, o, 2 EA A = 33,13 knm, λn = 1 1 =, , EA I EI λ = 1 =, EI suppoed bem, subjeed o unifom lod, whose oss seion heisis e: 2AC h = = 3 mm ; u 18 8 h β H = = 915, 82 <

5 Time Dependen Behvio of 5. 3 fm 1 β ( f ) = 3 6 ; ( ) m =.. 5 f m = 3 1 β = =, 4223 ; ( ) = φ 1 1 = = 131, φ 1 = 1 = 1, 314; h h = 8,h = 3 = 8,h = 3 φ = φ β ( f m ) β ( ) =1,6817; ( 365 6) =, φ = φ ( 365 6) = 365 β =, β ; 4. Sess hisoies in midspn seion oding he eeived numeil esuls The deese of he sesses in onee slb is ompnied by gdul migion of sesses fom he onee slb o he seel bem. These esuls is illused in figues 2 nd 3. down Fig.3. Noml sesses σ ( ) Fig.4. Defleion of omposie bem δ () 5. The developmen of defleion When he disibuion of he bending momens in seel seion, ( ) =, () N, () is known, i is possible o lule he hnge of he veil defleions in ime. The figue 4 shows he vlues of defleion in midspn seion of omposie bems in ime.

6 D. Pov, V. Knhev 6. Conlusion The mos impon onlusion of ou invesigion is h onsideing he eep effe nd using he inegl equions (2.7) nd (2.8) univesl numeil mehod hs been elboed fo silly deemine bidge omposie ple gide. The esuls obined by his numeil mehod e ompleely ompble wih he esuls deived by Doleiš [4] bsed on E poposed by EUROCODE 4. R E F E R E N C E S [1] ARUTYUNYAN, N. K. AND ALEXANDROVSKII, S. V. Cuen Se of Developmen of Conee Ceep Theoy. Soyizd, oskow, 1976, (in Russin) [2] BAŽANT, Z., P., hemil models fo eep nd shinkge of onee, Chpe 7 in Ceep nd Shinkge in Conee Suues, ed. Z. P. Bžn nd F. H. Wimnn, John Wiley & Sons, London, 1982, pp [3] DISCHINGER, F. ELASTISCHE UND PLASTISCHE, Vefomungen de Eisenbeongweke und insbesondee de Bogenbüken, De Buingenieu, H.5/6, Vol. 2, Velg Wilhelm Enns &Sohn 1939, pp [4] DOLEIŠ,J. Oelove mosy,čvut,ph, 26 [5] ENV :1994;EUROCODE 4- Design of Composie Seel nd Conee Suues, CEN, 1994 [6] PARTOV, D., DIITROV, CH., CHERNOGOROV, V., KALCHEV, P. Spnnungsändeugen infolge von Kiehen und Shwinden bei sish besimm gelegeen Shlvebundägen, De Slbu, No 7, Vol. 54 Velg Wilhelm Enns &Sohn, 1985, pp [7] PARTOV, D. AND KANTCHEV, V. Conibuion o he mehods of nlysis of omposie seel-onee bems, egding heology, Engineeing ehnis, No 5,Vol. 14,ČAV,27, pp [8] ŠERDA, Z. AND KŘISTEK, V. Ceep nd Shinkge of Conee Elemens nd Suues, Elsevie, Amsedm- Oxfod- New Yok Tokyo,1988 [9] TROST, H. Zu Beehnung von Shlvebundägen im Gebuhszusnd uf Gund neuee Ekennnisse des viskoelsishen Vehlens des Beons, De Shlbu, H.11, Vol. 37, Velg Wilhelm Enns &Sohn,1968, pp