Design techniques for continuous deep beams using finite element modelling

Transcription

1 Desgn technques for contnuous deep beams usng fnte element modellng Autor(en): Phllps, D.V. / Green, D.R. / Khaskhel, Ghous B. Objekttyp: Artcle Zetschrft: IABSE reports = Rapports AIPC = IVBH Berchte Band (Jahr): 54 (1987) PDF erstellt am: Persstenter Lnk: Nutzungsbedngungen De ETH-Bblothek st Anbetern der dgtalserten Zetschrften. Se bestzt kene Urheberrechte an den Inhalten der Zetschrften. De Rechte legen n der Regel be den Herausgebern. De auf der Plattform e-perodca veröffentlchten Dokumente stehen für ncht-kommerzelle Zwecke n Lehre und Forschung sowe für de prvate Nutzung fre zur Verfügung. Enzelne Dateen oder Ausdrucke aus desem Angebot können zusammen mt desen Nutzungsbedngungen und den korrekten Herkunftsbezechnungen wetergegeben werden. Das Veröffentlchen von Bldern n Prnt- und Onlne-Publkatonen st nur mt vorherger Genehmgung der Rechtenhaber erlaubt. De systematsche Specherung von Telen des elektronschen Angebots auf anderen Servern bedarf ebenfalls des schrftlchen Enverständnsses der Rechtenhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständgket oder Rchtgket. Es wrd kene Haftung übernommen für Schäden durch de Verwendung von Informatonen aus desem Onlne-Angebot oder durch das Fehlen von Informatonen. Des glt auch für Inhalte Drtter, de über deses Angebot zugänglch snd. En Denst der ETH-Bblothek ETH Zürch, Rämstrasse 101, 8092 Zürch, Schwez,

2 Desgn Technques for Contnuous Deep B Projet de poutres contnues de grande hauteur a Entwurfstechnken für hohe Durchlaufträger D.V. PHILLIPS Lecturer Unversty of Glasgow Glasgow, U.K. D.R.GREEN Head of Depart Unversty of Gl Glasgow, U.K. «jmm Dr. D.V. Phllps obtaned hs Ph.D. from the Unversty of Swansea. Wales n 1972 and then worked at the Unversty workng on shearwa Dr. DR Green has tve n research sn beams and slab st

3 464 DESIGN TECHNIQUES FOR CONTINUOUS DEEP BEAMS 1. INTRODUCTION In recent years varous proposals have been made for the desgn of renforcement for n plane forces based on the lower bound lmt State approach where a stress feld n equlbrum wth the ultmate load s used n conjuncton wth an approprate yeld crteron. Such a stress feld can be obtaned by any sutable procedure such as a lnear elastc fnte element analyss. Renforcement s then provded so that the combned resstance of the steel and concrete at every pont s equal to or greater than the appled stress. In theory, by satsfyng equlbrum and yeld exactly at every pont smultaneously, the entre structure wll become a mechansm at ultmate load. Practcal consderatons, such as renforcement beng provded as dscrete bars, make t mpossble to acheve ths dealsed behavour. Also the theory gves no guarantee that servceablty behavour wll be satsfactory. However, f verfed as an acceptable desgn process, the followng advantages ensue: analyss and desgn becomes one contnuous process whch s suted to automatc computaton, steel s used economcally as the desgn equatons are based on mnmsng steel requrements, although ths wll be affected by the convenence of fabrcaton, excessve ductlty demands are mnmsed by amng for most parts of the structure to yeld smultaneously at a partcular ultmate load. The dfference between the load at whch yeldng Starts and the ultmate load s kept at a reasonable level whch should prevent excessve crackng at workng load. Verfcaton for practcal renforcement detals can be provded by non lnear fnte element modellng of the resultng desgns, backed up by large expermental tests. Although the drect desgn phlosophy s applcable to a wde ränge of contnuum structures, ths paper s concerned wth ts applcaton to the desgn of contnuous deep beams. 2. DIRECT DESIGN EQUATIONS FOR IN-PLANE FORCES The desgn equatons adopted here for n plane actons were orgnally presented by Nelsen [1] usng the yeld crteron for orthogonal tenson renforcement, (Nx* - Nx)(Ny* - Ny) - Nxy2 0 (1) where Nx* and Ny* are yeld ultmate strengths, and Nx, Nv and Nxy are the n plane stress resultants n the concrete at any pont. Clark [2] later developed ths approach to cover compresson steel and skew renforcement. The desgn equatons are derved assumng: 1) the renforcement s placed symmetrcally about the mddle surface of the secton and s n two non orthogonal drectons (Fg.l) and carres only unaxal stress n the orgnal bar drectons so that knkng and dowel actons are neglected, 2) bar spacng s small n comparson wth the overall structure dmensons so that renforcement can be consdered n terms of area per unt length rather than as ndvdual bars, 3) concrete has zero tensle strength, exhbts the Square yeld crteron shown n Fg.2(a) and s perfectly plastc, whlst renforcement exhbts perfect elastc plastc behavour wth a yeld stress of fs n tenson and fs' n compresson, as shown n Fg.2(b), 4) that proper detalng and choce of secton prevents nstablty and bond falures, 5) n plane stresses are ressted by a combnaton of concrete and steel stresses (Fg.3). These assumptons lead to desgn equatons for nne possble combnatons of renforcement whch are shown schematcally n Fg.4. In each drecton tenson, compresson or no renforcement may be requred. When tenson renforcement s provded a\ 0 because the concrete s cracked and when compresson renforcement s provded o^ fc n order to make Optmum use of concrete. Drect Solutons are possble n all cases except when tenson tenson or compresson compresson renforcement s provded. In these cases an addtonal equaton s avalable by mnmsng the total renforcement,.e. d(px + Py) 0 (2) d{tmd) It s also necessary that rxy<0.5fc n order to prevent shear falure.

4 1 1 > D.V. PHILLIPS - D.R. GREEN - G.B. KHASKHELI 465 A x f x A_.,f ot a Fg.l Prncpal concrete stress and drectons of renforcement / / t fc l I r ' 1 w T t \ y x dealsed expermental compressop f' s tenson (a) concrete (b) steel Fg.2 Desgn yeld crtera for concrete and steel er y <s ß+ py fy J xy (O^-Ö^anöcos«? J3** <* Q> pj" on o 1. ^ + a n ' '' APPLIED STRESSES RESISTED STRESSES BY CONCRETE RESISTED STRESS BY STEEL Fg.3 Combnaton of resstant stresses y <«=(o-fc(l+ß)/2)/f6 Py=(oy-fc(l-ß)/2)/fs y=(oy- *xy2)/fs px-(ox^ xxy )/fs py»(oy+kxy )/fs (c,c) Txy] f>x=(xx^-oxf)/f s' py=0 oyf \ px-0 py=0 \ PX=(ax-Txy2)/fB py=o Px=( rxy -ox-fc)/fs' Py=( Txy -oy-fc)/fs' px«0 py-(txy oxf oyf)/fs' px*(ox-fc(1-p)/2)/fs py=(cy-fc(1+p)/2)/fs, a-fc/2 Tx) p»(c2-l)/a oxf=ox-t-fc ayf=cy+fc Fg.4 Desgn equatons for renforcement

5 466 DESIGN TECHNIQUES FOR CONTINUOUS DEEP BEAMS 3. DIRECT DESIGN OF CONTINUOUS DEEP BEAMS Rogowsky and MacGregor [3] and Rcketts and MacGregor [4] have carred out a number of tests on contnuous deep beams whch has provded useful Informaton on ther behavour. They used plastc truss analoges to predct the ultmate loads of ther beams and used a concrete effcency factor to acheve satsfactory agreement between the analytcal and expermental behavour. Lttle other expermental data has been reported and so desgn rules are manly based on a knowledge of Sngle span deep beam behavour. The CIRIA Gude [5], n common wth many Codes of Practce, base deep beam renforcement desgn on ndrect methods whch use moment and shear envelopes and nternal lever arms whch are assumed functons of beam dmensons. Approprate dstrbuton rules are then used for postonng the renforcement n saggng and hoggng zones, and for resstng shear. Ths approach, although smple to use, does not make the most effectve use of the renforcement due to the varyng nature of the stress dstrbuton n the beams. The drect desgn phlosophy attempts to overcome ths by usng an actual equlbrum stress feld. The followng procedure s adopted here. 1. An nternal stress feld n equlbrum wth the desgn ultmate load s obtaned from a lnear elastc fnte element analyss of the unrenforced beam usng elastc materal data for concrete. 2. Renforcement ratos are calculated usng the desgn equatons (Fg.4) wth ntended values of fs, fs' and fc. 3. Dscrete renforcng bars are selected by takng arbtrary levels across and along the beam and calculatng the average renforcement ratos at each level. Renforcng bars are then selected and dstrbuted usng specfed bar dameters and lmts on cover and spacng. Ths procedure was used to desgn the two span contnuous beam shown n Fg.6. The beam was desgned to carry a total ultmate load of 850kN as two concentrated loads actng at the centres of each span. The overall depth and length of the beam was 900 mm and 2000 mm respectvely wth a thckness of 100 mm. The span between the centres of support bearngs was 960 mm. Hot rolled deformed hgh tensle renforcng bars wth a 0.2% proof stress of 501 N/mm2 and concrete wth a cube strength of 45 N/mm^ were specfed. Isometrc vews of the envelopes of the resultng renforcement ratos n the horzontal and vertcal drectons are shown n Fg.5. The fnal desgned beam wth dscrete renforcng bars s shown n Fg LARGE SCALE TEST OF CONTINUOUS DEEP BEAM The füll sze desgned beam was expermentally tested to falure n ncrements of 50 kn under dsplacement control. Appled loads and support reactons, deflectons on the undersde of the beam and steel strans at each renforcement level were automatcally measured usng data loggng facltes. Concrete strans, crack propagaton and crack wdths were also montored. Materal propertes were measured as follows: concrete cube strength 63 N/mm 2, cylnder Splttng strength 3.22 N/mm2, modulus of elastcty 19.5 kn/mm2: steel - 0.2% proof stress for 6 mm and 8 mm bar 513 N/mm2 and 501 N/mm2 respectvely, modulus of elastcty for 6 mm and 8 mm bar 200 kn/mm2 and 195 kn/mm2. The beam faled n shear at a load of 1333 kn, over 50% hgher than the desgn load. The servceablty load correspondng to a 0.3 mm crack wdth was 900 kn. Measurements ndcated that most of the steel had yelded pror to ultmate condtons. Load dsplacement and steel stran curves are shown n Fgs. 11 and 12 whlst Fg.l3(b) llustrates the cracks n the beam at falure. The hgh expermental load can be attrbuted to several causes: the actual concrete strength was 63 N/mm2 compared to the desgn value of 45 N/mm2, the steel exhbted stran hardenng whch was not accounted for n the desgn equatons, sgnfcant dsplacement along shear cracks was observed at hgher loads causng dowel acton and nterface shear transfer effects whch were not accounted for n the desgn equatons, the renforcement provded, partcularly the shear renforcement, was more than the desgn values because of practcal requrements such as mnmum bar sze.

6 D.V. PHILLIPS - D.R. GREEN - G.B. KHASKHELI 467 Notwthstandng, the desgn procedure produced an acceptable desgn wth an adequate reserve of strength and satsfactory behavour at servceablty load. LOAD *** «r ^«v^ support centre Bupport ^ a* edge eupport centre support (a) horzontal steel (b) vertcal steel Fg.5 Isometrc vews of envelopes of renforcement ratos (1/2 beam) 425 _ ' ' T 860 pea r »2TÖ c~ 300.2Y6 15Q-- 2YÖ *t* * ELE7ATI0N ^ 35" SBCTIQH Fg.6 Dmensons and renforcng detals of desgned beam (one span) 5. NON-LINEAR FINITE ELEMENT ANALYSIS The beam was analysed usng a plane stress non lnear fnte element model [6] whch s one of several developed at Glasgow Unversty for analysng the non lnear behavour of structural concrete. A bref descrpton of the analyss follows. 5.1 Dscretsaton and non-lnear Soluton procedure Concrete was modelled by 8 noded soparametrc elements usng Standard Serendpty shape functons and a 3 x 3 Gaussan Integraton rule. Renforcement was modelled by bars embedded anywhere wthn the element along lnes of constant local curvlnear co ordnates [7]. Ths

7 468 DESIGN TECHNIQUES FOR CONTINUOUS DEEP BEAMS representaton assumes the bars carry unaxal stress only and that perfect bond exsts between the steel and concrete. A modfed Newton Raphson ncremental teratve Soluton scheme was used where the stffnesses were upgraded at the frst teraton wthn a load ncrement. Load control was appled and no lne searches were employed. Convergence was checked by a resdual force norm and was assumed to have occured when the convergence factor was less than 2%. 5.2 Materal modes Crackng was smulated by a smeared orthotropc fxed crack approach. A crack was assumed to occur n a plane perpendcular to the maxmum prnepal stress when ths exceeded the tensle strength crteron for concrete. After crack formaton, the materal matrx was modfed to reflect the change from an sotropc to an orthotropc materal and to account for the partal loss of stffness normal and parallel to the crack planes. A second crack could occur at rght angles to the frst crack f the tensle strength was exceeded n ths drecton also. A crack was assumed to close f the stran across the crack became compressve. Post crackng behavour was represented by laws defnng a varable shear retenton factor and a varable tenson stffenng and stran softenng factor, as shown n Fg.7. These laws were used to modfy the modul n the materal matrx for subsequent stffness and stress calculatons. Concrete behavour under baxal stress states was modelled by a short term hyperelastc consttutve law whch uses tangental bulk and shear modul whch vary accordng to the stress nvarants Ij and J2. The relatonshps are shown n Fg.8. A lnear form of the octahedral stress falure crteron was used to predct the baxal ultmate strength of concrete (Fg.9). For renforcng steel a b lnear unaxal stress/stran relatonshp allowng for sotropc stran hardenng and elastc unloadng was used as shown n Fg. 10. p ß G'/G fj /% r« K constant *1 ex f y r?» 1 t 3-Te »» J3"t 2% *5.75 l.cf G /G (a) Shear retenton (b) tenson stffenng Fg.7 Post crackng laws Fg.8 Hyperelastc nvarant law for concrete n baxal compresson f ' G tenson Typeal concrete Ew octahedral law Fg.9 Falure crteron for concrete compresson Fg.l0 Stress-stran law for renforcement

8 D.V. PHILLIPS - D.R. GREEN - G.B. KHASKHELI Results for the contnuous deep beam The load deflecton and load steel stran curves are compared wth the expermental results n Fgs. 11 and 12 where satsfactory agreement s evdent. Fg.l3 shows crack propagaton at varous load levels and compares them wth the expermental crack pattern at falure. Flexural cracks frst ntate at the bottom of the beam at about 300 kn, followed later by flexural crackng at the top of the beam over the central support at a load of 750 kn. At about 825 kn, shear cracks Start to develop whch eventually spread from the load pont to the central support. Falure was by shear and crushng of concrete n the central shear span and close to the appled load. Ths descrpton compares well wth the expermental observatons. The ultmate load predcted by the fnte element analyss was 1275 kn whch was 4% below the test value. A truss analogy approach [8] predcted an ultmate load of 934 kn, some 30% lower than the actual value. + * y.* y^y * +f y +1 t L DEFLECTION IN MH WE - F E cur/e 3H.RTM) 50 Fg.l 1 Comparson of expermental and fnte element load dsplacement curves t eexp. CURVE * F.E CURVE Fg.l2 Comparson between expermental and fnte element load steel stran curves

9 470 DESIGN TECHNIQUES FOR 1 r I 3 / CRACK PATTERN OF BEAM AST AT LOAD KN n CRACK RATTERN OF BEAM A51 AT LOAD KN

10 #% D.V. PHILLIPS - D.R. GREEN - G.B. KHASKHELI CONCLUDING REMARKS By usng the elastc stress feld produced by a fnte element analyss a procedure has been developed whereby renforcement n contnuous deep beams can be drectly desgned to resst n plane forces. The servceablty and ultmate load behavour of the resultng desgn was verfed by a non lnear fnte element analyss and by a large scale expermental test. Other theoretcal and expermental work s contnung on desgns wth dfferent span to depth ratos, and wth other features such as skew renforcement, prestressng and holes wthn the shear spans [9], Although the procedure descrbed uses a lnear elastc stress feld for obtanng the renforcement ratos, any sutable equlbrum feld could be used and studes are also beng made usng elasto plastc fnte element modes wth approprate yeld and flow rules. The non lnear fnte element modes whch have been developed are provdng a better apprecaton of structural behavour at varous stages of loadng whch has proved very useful n developng the most effectve desgn procedures. The fnte element modellng s beng contnuously developed and s backed up by separate expermental tests for verfyng the modes developed and for provdng Informaton on materal behavour. The drect desgn procedures are beng appled to other structural forms, ncludng combned torson and bendng of renforced and prestressed concrete beams [10], perforated deep beams [11], and renforced concrete rght and skew-slabs [12],[13]. The method s provng satsfactory n these areas also. Ultmately, the objectve of ths work s to produce automated technques whch can be used n Computer aded desgn Software. REFERENCES 1. NIELSEN, M.P. "Yeld condtons for renforced concrete Shells n the membrane State". Non classcal shell problems, IASS Symposum, Warsaw, 1963, pp CLARK, L.A. "The provson of tenson and compresson renforcement to resst n plane forces", Mag. Conc.Res., Vol.28, No.94, 1976, pp ROGOWSKY, D.M. & MACGREGOR, J.G. "Shear strength of deep renforced concrete contnuous beams", Structural Engneerng Report No.l 10, Department of Cvl Engneerng, Unversty of Alberta, Canada, RICKETTS, D.R. & MACGREGOR, J.G. "Ultmate behavour of contnuous deep renforced concrete beams", Structural Engneerng Report No.l 26, Department of Cvl Engneerng, Unversty of Alberta, Canada, CIRIA, "The desgn of deep beams n renforced concrete", Gude 2, Constructon Industry Research and Informaton Assocaton, London, PHILLIPS, D.V. "FECON: Nonlnear fnte lement programs for structural concrete", FECON Manual, Mark 2, Unversty of Glasgow, PHILLIPS, D.V. & ZIENKIEWICZ, O.C. "Fnte element non-lnear analyss of concrete structures", Proc.Instt,Cv.Engnrs, Part 2, Vol.61, 1976, pp KUMAR, P. "Collapse load of deep renforced concrete beams", Mag.Conc.Research, Vol.28, No.94, 1976, pp KHASKHELI, G.B. "The drect desgn of renforced concrete transfer grders", Ph.D Thess, Unversty of Glasgow, to be submtted. 10. EBIRERI, J.O. "Drect desgn of beams under combned bendng and torson", Ph.D. Thess, Unversty of Glasgow, MEMON, G.H. "Ultmate strength of perforated deep beams", M.Sc. Thess, Unversty of Glasgow, HAGO, A.W. "Drect desgn of renforced concrete slabs", Ph.D. Thess, Unversty of Glasgow, ABDEL-HAFEZ, L.M. "Drect desgn of renforced concrete skew slabs", Ph.D. Thess, Unversty of Glasgow, 1986