STRUCTURAL ANALYSIS OF MONO-SYMMETRIC PLATE GIRDERS IN COMPOSITE BRIDGES

Transcription

1 ACTA TEHNICA CORVINIENSIS Bullin of Enginring Tom VIII [05] Fascicul [Januar March] ISSN: Mohamd A. A. El-SHAER,. Ghada M. El-MAHDY STRUCTURAL ANALYSIS OF MONO-SYMMETRIC PLATE GIRDERS IN COMPOSITE BRIDGES. Civil and Consrucion Enginring Dparmn, Highr Tchnological Insiu, 0 h of Ramadan Ci, EGYPT Sl Srucurs, Srucurs & Mallic Consrucion Rsarch Insiu, Housing &Building Naional Rsarch Cnr, Giza, EGYPT Asrac: Mono-smmric pla girdrs ar ofn usd in simpl suppord composi ridgs o limina local pla uckling in h comprssion flang during consrucion. This causs h nural axis of h pla girdr o shif downwards sujcing mor of h w o comprssiv srsss du o nding. In slndr ws his inass h possiili of local uckling in h comprssion par of h w during consrucion. Howvr, dpnding on h slndrnss (widh-o-hicknss raio of h w, h pos-uckling rsrv capaci ma accommoda his local uckling wihin h lasic limi of h w for during consrucion loads. Hnc, his would allow for h us of mor slndr ws in composi pla girdr consrucion wihou h nd for longiudinal w siffnrs or h rducion of h ovrall composi scion du o local pla uckling in h w. Rcommndd valus of srss lvl ar givn for mono-smmric pla girdrs in h non-composi sag asd on h rsuls of a non-linar fini lmn analsis. Kwords: composi ridgs, ffciv widh, FEA, local pla uckling, mono-smmric scions, pla girdrs, srss gradin, srss lvl INTRODUCTION Pla girdrs in conjuncion wih a rinforcd con sla ar ofn usd as composi pla girdr ridgs in posiiv nding. This has h advanag of rh possiili of local pla uckling in h pla girdr s w and comprssion flang as a rsul of composi acion undr srvic loads. In addiion o his, h during consrucion loads acing on h pla girdr scion alon, DL, ma dsignd o allow for local uckling in h w whil kping h srsss wihin h lasic limi of h pla girdr scion during consrucion > > -.0 Gross sl scion Effciv sl scion Srss disriuion (DL (a Un-shord non-composi scion wih inffciv con sla (DL onl > Composi scion Srss disriuion (DL +LL Final srss disriuion ( Un-shord composi scion afr hardning of con sla (DL +DL +LL (.0.7 > upwards wih h ons of composi acion, as shown in Figur (. Hnc, prvning rsidual srains du o h ilding of h scion during h consrucion non-composi sag and h afr consrucion composi sag. Th srss disriuion for oh h non-composi and composi sags for un-shord consrucion is shown in Figur. Ohr rsarchrs o sud composi I-girdrs ar Gupa [], Gupa al. [], Baskr al. [3], and Yakl and Azizinamini [4]. Rcn rsarch was also conducd on I-scion flxural ams Shokouhian and Shi [5] and L al. []. LOCAL PLATE BUCKLING Local pla uckling occurs in slndr pla lmns whn h comprssiv srss in h pla lmn xcds h iical pla uckling srss of h pla lmn, as shown in Figur. Afr h ons of pla uckling, a wav-lik propagaion of ou-of-plan dformaions raks ou inasing in ampliud wih h inas in loading. This causs h comprssiv srsss o rdisriu in h pla lmn, concnraing in h rgions suppord sal oundar condiions. Du o h loss of in-plan siffnss of h unsuppord rgions, comprssiv and nsil nding srsss Figur : Srss disriuion in composi consrucion For pla girdrs wih ws having a widh-o-hicknss raio in h vr slndr rang, h iical srss is wll low h ild srss, and wih h inas in loading afr h ons of local uckling up o h undr h ffc of consrucion loads onl, Figur (a, h posuckling sa of h slndr w ma caus a nonlinar srss ild srss,. Onc h dg srss has rachd h ild srss, h disriuion, u h srsss in h w ma sill in h lasic rang. This would allow for h us of mor slndr ws for rachd h ild srss. Whras, h unsuppord unsal inrnal composi pla girdrs kping in mind ha h nural axis will shif par of h pla lmn is assumd o inffciv. Hnc, h dvlop hrough h hicknss of h pla, flucuaing along h lngh of h pla. Th srss a h sal dgs graduall inass plasicizaion of h pla lmn propagas in h nar rgions ill h suppord pars of h pla lmn ar assumd o hav coprigh Facul of Enginring - Hundoara, Univrsi POLITEHNICA Timisoara

2 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] pla lmn dos hav a pos-uckling rsrv capaci which can Assuming ha h avrag uniform srss of h nonlinar posuckling srss disriuion is av as shown in Figur, h ffciv wihin h lasic limi of h lmn if h dg srss dos no rach h ild srss capaci. widh is assumd o h widh sujc o a srss qual o h dg srss,, of h nonlinar srss disriuion such ha i dvlops a srngh qual o h avrag srss acing on h whol widh. Hnc, = av (5 giving > Acual srss disriuion. Rgion assumd no o ransmi srss caus of uckling. Figur : Concp of ffciv widh Th lasic uckling srss,, of slndr plas as drivd von Kàrmàn al. [7] is kπ E = ( ν ( which is invrsl proporional o h squar of h widh-ohicknss raio, /, of h pla lmn. Th pla uckling facor, k, dpnds on h longiudinal oundar condiions of h pla lmn and h normal srss disriuion in h pla, shown in Figur. Exprssions for k for diffrn oundar condiions can found in h Eurocod EC3 EN 993--:003 [8] or h Egpian Cod of Pracic for Sl Consrucion and Bridgs ECOP-ASD [9]. Th modulus of lasici, E, can akn as 0,000 MPa and Poisson s raio, can akn as 0.3. From h xprssion for h uniform lasic iical srss, givn in Eq. (, acing on a pla wih a widh-o-hicknss raio of / w g = kπ E ( ν Assuming an ffciv widh of and a uniform srss acing on i of, which can hav a valu anwhr from h iical srss o h ild srss as shown in Figur, hn analog w g = kπ E ( ν Hnc, h raio of h ffciv widh, o h original widh, known as h ffciv widh paramr ρ, is = ρ = av av > (4 av = = Taking h non-dimnsional slndrnss paramr λ n as ( ν = = kπ E and susiuing his ino Eq. ( givs av = (8 To accoun for h ffc of rsidual srsss in h modral slndr and h non-compac slndrnss rangs, h Amrican Iron and Sl Insiu (AISI [0] suggss h following xprssion for h avrag srss. λ 0. n av = Boh Eqs. (8 and (9 ar plod in Figur 3. Figur 3: Normalizd pla uckling curvs EFFECT OF STRESS GRADIENT ON LOCAL PLATE BUCKLING To includ h ffc of srss gradin in h pla lmn du o comind comprssiv and flxural srsss in h mmr, as shown ( in Figur 4, h ffciv widh paramr, ρ, is assumd o ak h form av x ψ (0 whr x and can drmind from h limis of / for siffnd (3 slndr pla lmns in pur comprssion, ψ = +.0, and pur nding, ψ = -.0. This is h sam mhod usd El-Mahd and Au-Hamd [,, and 3] o driv h currn quaion for h ffciv widh of siffnd slndr pla lmns sujc o a srss gradin in h Egpian Cod of Pracic for h Dsign of Sl Consrucion and Bridgs [9]. ( (7 (9 48

3 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] = comprssion flang and h nsion flang. This causs a sligh das of h srss gradin in h w du o a minor chang in = +.0 h posiion of h modl s nural axis, u his das is = 0.5 = 0.5 ngligil. Boh flangs wr larall suppord as shown in Figur = +0.5 = +0.5 (a o prvn an ou-of-plan laral orsional-flxural uckling occurring in h comprssion flang. Elasic-plasic shll lmns = 0.0 = 0.0 wr usd o modl h flang and w pla lmns, howvr, h = 0.0 = = 0.0 nd pars of h op and oom flangs wr siffnd inasing hir hicknss and akn as lasic shll lmns o ovrcom local dformaions du o loads applid o hs flangs. Th marial of h modl was akn as lasic-prfcl plasic wih a modulus of = -0.5 = -0.5 lasici of 0 GPa and a ild srss of 350 MPa. = -0.5 = -.0 = - = -0.5 Figur 4: Schmaic rprsnaion of srss gradin, ψ, du o comind comprssiv and flxural srsss For xampl, using h limis for pur comprssion and pur nding givn in h Eurocod EC3 [], = 4ε for ψ = +.0 = 4ε for ψ = -.0 whr ε = 35/ ( and aking k = 4.0 for h cas of pur comprssion and k = 3.9 for h cas of pur nding and assuming = givs h valus 0.44 and for x and, rspcivl. Hnc, according o h limis of h EC3 [8]: av ψ ( Facorizing and approximaing his lads o h xprssion av 0. 05(3 ψ + (3 which is clos o h xprssion givn in h EC3 EN :00 [4] av (3 ψ + (4 Th normalizd avrag srss for h cass of ψ = +.0 and ψ = -.0, according o Eq. (, ar plod in Figur 3. FINITE ELEMENT ANALYSIS A fini lmn paramric analsis, using COSMOS. sofwar, was conducd on modls of pla girdrs having a w dph of 000 mm and varing h w hicknss, w, from 5 mm o mm giving a widh-o-hicknss raio for h w varing from 00 o 9. Th comprssion flang was kp consan in h non-compac rang having a siz of 00 x mm, whras, h siz of h nsion flang was inasd o achiv a srss gradin in h w of ψ = -.0, - 0.8, -0., and -0.4, as shown in Figur 5. A modl wih a simpl suppord span of lngh L = 0 m was usd. In h acual fini lmn modl h high of h w was modld having a dph of 000 mm plus half h hicknss of oh h 49 = -.0 = -0.8 = -0. = -0.4 Figur 5: Schmaic rprsnaion of paramric pla girdr ross-scions Figur : Tpical fini lmn modl and normal srss disriuion of linar analsis; a Fini lmn modl; Dformd shap and srss disriuion Th comprssion and nsion flangs of ach modl wr loadd o caus a momn qual o h ild capaci of h scion. This was achivd appling quivaln nd comprssion and nsion forcs a

4 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] in h op and oom flangs, rspcivl, according o h DISCUSSION OF RESULTS following formulas: Figur 8 illusras h in-plan mmran normal srss disriuion ( = Aw and h ou-of-plan local pla uckling of h w in flxural F Af F + comprssion for h modl wih a w slndrnss of 00 (i.., w = 5 Af 3 ψ (5 mm and a srss raio of ψ = -0.. ( Aw ψ F = ψ + ( Af F Af 3 ψ ( This givs a ild momn, M, of 3 A + ψ ( ( ( f Af Aw + ψ M = hwf + ψ 3 ψ (7 whr F and F ar h quivaln comprssion and nsion forcs assumd o concnrad a h cnroids of h flangs ha caus a momn qual o h ild momn capaci of h scion, a rspcivl; A f and A f ar h aras of h op and oom flang plas, rspcivl; and h w and A w ar h dph and ara of h modl s w pla, rspcivl. Th rsuls of h linar analsis conducd on h fini lmn modls vrif ha h gross siffnss of h modl, calculad from h midpoin dflcion,, compar accural wih h analic xprssion for a simpl suppord am sujc o a uniform momn, M, viz., I = ML /8E. Th posiion of h nural axis can also drmind from h normal srss disriuion in h dflcd modl. Excssiv srsss wr nod in h flangs nar h loadd dgs. Figur ( shows h normal srss disriuion in h dformd modl wih a w hicknss of 5 mm or a w slndrnss of 00 and a oom flang sizd o giv a srss gradin of ψ = -0.. A nonlinar analsis which follows h Nwon-Raphson inmnaliraiv procdur was usd o dc h propagaion of local pla uckling in h slndr w. An iniial mm ou-of-flanss a h cnr-poin of h w was usd in h modl o iniia local w uckling. Finall, a fini lmn analsis of h composi scion, shown in Figur 7, using a sla of 000 x 00 mm wih a con cu c srngh, f c, of 40 MPa and uniforml loadd aov h sla gav approxima valus of h rsidual capaci of h composi scion in h afr consrucion phas. Figur 7: Normal srss disriuion in composi scion d Figur 8: Dformd shap and normal mmran srss disriuion for modl wih w slndrnss 00 and ψ= -0.; a Srss lvl.7 Srss lvl c Srss disriuion along h w a diffrn srss lvls d Load-dflcion curvs A a srss lvl of.7, shown in Figur 8(a, i can sn ha a noal amoun of local uckling in h comprssion par of h w 50

5 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] occurrd wihou causing an disorional uckling in h noncompac adjacn comprssion flang, and wihou xcding h considral nonlinari in h srss disriuion occurs nar h uckling has iniiad a h iical srss lvl, howvr, lasic limi as shown h maximum comprssiv srss of 4 nd of h nonlinar analsis as h srss lvl approachs h ild MPa. Whras, for a srss lvl of = 350 MPa, shown in Figur srss. This is also dmonsrad h load-dflcion curvs shown 8(, h local uckling of h w in comprssion is gral magnifid in Figur 8(d. I can also nod ha h nural axis nds o shif causing disorional uckling in h adjacn comprssion flang. upwards wih h occurrnc of local pla uckling in h w and h nonlinar srss disriuion. a a c c Figur 9: Dformd shap and normal mmran srss disriuion for modl wih w slndrnss 5 and ψ = -0.; a Srss lvl.47 ; Srss lvl ; c Srss disriuion along h w a diffrn srss lvls; dload-dflcion curvs Figur 8(c shows h srss disriuion along h w for h sam modl a diffrn srss lvls. I can sn ha h srss disriuion along h w rmains rlaivl linar vn afr local 5 d Figur 0: Dformd shap and normal mmran srss disriuion for modl wih w slndrnss 00 and ψ = -0.; Srss lvl.09 ; csrss disriuion along h w a diffrn srss lvls; dloaddflcion curvs Figur 9 shows h dformd shap and normal mmran srss disriuion for h modl wih a w slndrnss of 5 (i.., w = 8 mm and a oom flang proporiond o giv a srss raio of ψ = -

6 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] 0.. A sligh amoun of local w uckling can dcd a a srss formulas givn in Eq. (8 ar also compard o h curvs oaind lvl of.47 whr h maximum comprssiv srss is 337 MPa for h nonlinar fini lmn analsis in Figur. Th maximum and hnc is sill low h ild srss, as shown in Figur 9(a. rror wn hs wo curvs is lss han % and is consrvaiv for Howvr, a a srss lvl of h local uckling in h w is all valus of /. magnifid causing disorional uckling in h comprssion flang, as.8x0 for ψ = -.0 = shown in Figur 9(. Figur 9(c shows h srss disriuion along ( hw w h w for his modl a diffrn srss lvls. I can sn ha.x0 for ψ = -0.8 =.30.0 (8 h srss disriuion along h w rmains rlaivl linar vn ( hw w afr local uckling has iniiad a h iical srss lvl, howvr, a.74x0 for ψ = -0. =.45.0 sligh nonlinari in h srss disriuion occurs nar h nd of h ( hw w nonlinar analsis as h srss lvl approachs h ild srss. Th 5.58x0 sligh nonlinari is again shown in h load-dflcion curv in for ψ = -0.4 =.5.0 ( hw w Figur 9(d. Finall, Figur 0 shows h dformd shap and normal srss disriuion for h modl wih a w slndrnss of 00 (i.., w = 0 mm and a lowr flang proporiond o giv a srss raio of ψ = -0.. A a srss lvl of h maximum srss is clos o and vr lil local w uckling has occurrd, as shown in Figur 0(a. In fac, a a srss lvl of.09 h local uckling is sill hard o dc alhough h scion has ildd, as shown in Figur 0(. Figur 0(c shows h srss disriuion along h w for his modl a diffrn srss lvls. I can sn ha h srss disriuion along Figur : Comparison of Eq. (8 wih h FEA rsuls h w rmains linar up o h nd of h nonlinar analsis as h From h fini lmn rsuls of h composi scions, i can srss lvl approachs h ild srss. This linari is also dpicd nod ha inasing h hicknss of h w rsuls in a minor in h load-dflcion curvs shown in Figur 0(d. inas in composi capaci. Whras using a grar valu of Th rcommndd valus of srss lvl wih rspc o h iical srss for ohr srss gradins as drmind h nonlinar fini lmn analsis ar lisd in Tal and ar plod in Figur. Tal : Rcommndd valus of srss lvl wih rspc o iical srss / h w / w ψ Figur : Rcommndd valus of srss lvl for mono-smmric girdrs A procss of curv fiing has ld o h drivaion of h formulas givn in Eq. (8 for h rcommndd valus of srss lvl as a funcion of h w slndrnss raio for ach valu of ψ. Th srss gradin (i.., ψ = -0.4 gral inass h composi capaci 3 ims, and an loss in h capaci of h non-composi scion du o h uckling of h slndr w can compnsad for using shoring during consrucion. CONCLUSION Th fini lmn paramric analsis shows ha mono-smmric non-composi pla girdrs wih slndr ws can srssd ond h iical srss, iniiaing h ons of local uckling of h w in flxural comprssion, wihou xcding h lasic limi. Howvr, du o h occurrnc of xcssiv local uckling dformaions in slndr ws causing disorional uckling in h comprssion flang h following srss limis ar rcommndd dpnding on h srss raio in h w;.7 for ws in h vr slndr rang dasing o.0 for ws in h lss slndr rang. Smols A f = ara of oom flang pla A f = ara of op flang pla A w = ara of w pla = pla widh = ffciv widh E = modulus of lasici f c = con cu srngh F = quivaln nsil forc in oom flang F = quivaln comprssiv forc in op flang h w = dph of w pla k = pla uckling facor 5

7 ACTA TEHNICA CORVINIENSIS Fascicul [Januar March] Bullin of Enginring Tom VIII [05] L = span of girdr M = nding momn M = ild momn of girdr = pla hicknss w = hicknss of w pla x, = varials = midpoin dflcion ε = 35 / λ n =non-dimnsional slndrnss paramr ν = Poisson s raio ρ = ffciv widh paramr = srss = largr dg comprssiv srss = smallr dg comprssiv srss or nsil srss av = avrag srss = iical uckling srss = dg srss = ild srss ψ = srss gradin Rfrncs [.] Gupa, V.K., Dvlopmn of scion classificaion irion and ulima flxural quaion for composi I-girdrs, Docoral Dissraion, Saiama Univrsi, Japan, 00. [.] Gupa, V.K., Yoshiaki, O., and Nagai, M., Dvlopmn of w slndrnss limis for composi I-girdrs accouning for iniial nding momn, Dooku Gakkai Ronunshuu A (JSCE Journal of Srucural and Earhquak Enginring, Vol., No. 4, pp , 00. [3.] Baskr, K., Shanmugan, N.E., and Thvndran, V., Fini-lmn analsis of sl-con composi pla girdr, J. Sruc. Eng., ASCE, Vol. 8, No. 9, pp. 58-8, 00. [4.] Yakl, A.J. and Azizinamini, A., Improvd momn srngh prdicion of composi sl pla girdrs in posiiv nding, J. Bridg Eng., ASCE, Vol. 0, No., pp. 8-38, 005. [5.] Shokouhian, M. and Shi, Y., Classificaion of I-scion flxural mmrs asd on mmr ducili, Journal of Consrucional Sl Rsarch, Vol. 95, April, pp. 98-0, 04. [.] L, C.H., Han, K.H., Uang, C.M., Kim, D.K., Park, C.H., and Kim, J.H., Flxural srngh and roaion capaci of I-shapd ams faricad from 800 MPa sl, J. Sruc. Eng., ASCE, Vol. 39, No., pp , 03. [7.] von Kármán, T., Schlr, E.E., and Donll, L.H., Srngh of hin plas in comprssion, Transacions of h Amrican Soci of Mchanical Enginrs, Vol. 54, No. APM-54-5, p. 53, 93. [8.] Eurocod 3, Dsign of sl srucurs Par. Gnral srucural ruls (EN 993--:003, Europan Commi for Sandardizaion (CEN, Brussls, Blgium, 005. [9.] ECOP-ASD, Egpian Cod of Pracic for Sl Consrucion and Bridgs Allowal Srss Dsign, Egp, 00. [0.] Amrican Iron and Sl Insiu, Spcificaion for h dsign of coldformd sl srucural mmrs, AISI, Washingon DC, 007. [.] Au-Hamd, M.H. and Elmahd, G.M., Th ffciv widh of slndr pla lmns in pla girdrs, Journal of Enginring and Applid Scinc, Facul of Enginring - Cairo Univrsi, 50(: 59-78, 003. [.] Elmahd, G. and Au-Hamd, M., Nw formula for h ffciv widh of slndr pla lmns, Procdings of h CSCE Annual Confrnc - h Srucural Spciali Confrnc, Quc Ci, Quc, Canada, 008. [3.] El-Mahd, G. and Au-Hamd, M., Local uckling of slndr pla girdrs in composi ridgs, Procdings of h SSRC Annual Saili Confrnc, Orlando, Florida, USA, 00. [4.] Eurocod 3, Dsign of sl srucurs Par.5 Plad srucural lmns (EN :00, Europan Commi for Sandardizaion (CEN, Brussls, Blgium, 00. coprigh Univrsi POLITEHNICA Timisoara, Facul of Enginring Hundoara, 5, Rvoluii, 338, Hundoara, ROMANIA hp://aca.fih.up.ro 53