NON-LINEAR INTERACTION OF NORMAL AND TANGENTIAL INTERNAL FORCES ON 3D RC BEAM-COLUMN STRUCTURAL SYSTEMS

Transcription

1 13 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August 1-6, 2004 Paper No. 840 NON-LINEAR INTERACTION OF NORMAL AND TANGENTIAL INTERNAL FORCES ON 3D RC BEAM-COLUMN STRUCTURAL SYSTEMS J. M. BAIRAN GARCIA 1, A. R. MARI BERNAT 2 SUMMARY The accurate behavour assessment of complete structural systems s essental n developng Performance Based Engneerng methodologes. In ths aspect, structural models based on beam-column elements are stll a compettve opton compared to sold-fem models; manly because of the practcal data generaton, results nterpretaton and the drect use of nternal forces n desgn. Nevertheless, BC-elements are developed under plane sectons hypothess and under null or smple mposed shear deformaton along cross sectons allowng for reasonable non-lnear evaluaton of normal nternal forces (axal force and baxal bendng moment) but not for tangental nternal forces (baxal shear and torson) whose nfluence on the behavour of structural systems wth rregular dstrbuton of stffnesses, masses or resstances has proved to be determnant. A NL-fbrewse sectonal model for concrete structures capable of smulatng the total nteracton between all sx beam nternal forces and deformatons wth arbtrary shaped cross secton and longtudnal renforcement and strrups arrangement has been developed. Crack concrete s smulated as a 3D orthotropc materal whch n combnaton wth steel renforcements yelds, n general, a full consttutve matrx couplng all stress and stran components. The model enables an explct expresson of the sectonal stffness matrx consderng the nteracton of the complete set of nternal sectonal forces on a cracked secton. A renforced concrete cross secton s analysed for ncreasng levels of combned bendng and shear loads. Notceable dependences on the concomtant load are shown on the overall performance of both momentcurvature and shearng force-deformaton dagrams. 1 Ph.D. Canddate. Department of Constructon Engneerng. Techncal Unversty of Catalona.Campus Nord, Jord Grona St., Mod.C-1.Barcelona 08034, Span. Fax: +(34) emal: jesus.mguel.baran@upc.es 2 Professor of Concrete Structures. Department of Constructon Engneerng. Techncal Unversty of Catalona, Span

2 INTRODUCTION Accurate understandng of non-lnear behavour of complete structural systems under a broad range of loadng, ncludng ultmate loads and dfferent levels of servce loads, s an essental ssue n the development of Performance Based Engneerng (PBE) methodologes. Ths holds not only for the case of Performance Based Earthquake Engneerng, but also for every desgn phlosophy where the desgner should assure a certan structural behavour and damage states for several lmt states. In spte of the great advance that structural analyss usng sold and 2D fnte elements has experenced, structural models based on frame elements are, stll today, the type of numercal model more employed n engneerng practce, for both lnear and non-lnear analyss of complete structural systems, see Marí [1] and Marí [2]. Now a day, ths s manly because of the more practcal and drect generaton of data and nterpretaton of the analyss results rather than to computatonal lmtatons. Although ths last aspect contnues beng mportant. Nevertheless, tradtonal frame elements are only capable to represent approprately, at non-lnear level, the structural behavour under axal stresses. Ths s due to the absence of a theory that allows ncorporatng the nfluence of shear stresses n ths type of elements wth smlar success as the Naver- Bernoull plane secton hypothess does for axal forces and bendng moments. Shear forces and torson moments effects are then consdered, n general, n a very smplfed and decoupled fashon. The behavour of many structures can be consdered domnated by axal stresses; beng, n such cases a tradtonal frame model acceptable. But very often, frame structures are submtted to a combnaton of forces n whch both axal and shear stresses are of smlar mportance, thus the performance n terms of stffness, yeld load and ultmate load - evaluated from a decoupled frame model shall dffer from the real behavour. Structures presentng ths behavour are not so nfrequent. Usually they are characterzed by an rregular stffness confguraton, Calv and Pnto [3]; ths s the case of most vaducts. Also, curve and skew brdges may exhbt ths strong nteracton snce torson and flexure effects are cnematcally coupled. In ths paper, the non-lnear coupled behavour of renforced concrete (RC) sectons s studed by means of a prevously developed numercal model, Baran [4] whch takes nto consderaton the 3D state of stresses on each concrete fbre, nternal equlbrum and compatblty n the cross secton s doman and materal ansotropy nduced by generated cracks n concrete. In the development of the sectonal model, suffcent generalty has been consdered n order to allow any arbtrary shape of the cross secton as well as dfferent materals and renforcement arrangements. Two man peculartes exhbt ths model compared to others coupled models, for nstance Petrangel et al [5]. Frst, t allows the full three dmensonal nteracton of axal force, skew shear forces, skew bendng and torson moments regardless of the shape of the cross secton. Second, an explct sectonal stffness matrx can be wrtten showng the terms that actually couples all nternal forces on the secton. In the case of a cracked secton, ths matrx has shown to be full. A beam specmen tested under shear forces by Kan [6] has been analysed modellng only one cross secton n order to show the valdty of the model. Further, moment-curvature and shear-deformaton curves of the same cross secton are obtaned under ncreasng smultaneous shear forces and bendng moments respectvely. Results show that when the smultaneous actons are strong enough, nfluence s exhbted on these characterstc curves, not only n terms of ultmate and yeld loads, but also n terms of ther overall shape and pre-yeld stffness.

3 DESCRIPTION OF THE SECTIONAL MODEL A numercal sectonal model, developed by Baran [4], for the analyss of cracked concrete cross sectons under general 3D coupled axal and shear forces wll be brefly descrbed here. The cross secton s consdered as a 2D doman submtted to the tradtonal sectonal deformatons,.e. axal elongaton, skew shear deformaton, skew bendng curvatures and a twstng curvature. From ths doman and mposed deformaton, a 3D stress and strans feld s to be obtaned. In order to be able to reproduce a three-dmensonal stran feld, the sectonal dsplacements are expressed as a combnaton of the tradtonal Naver-Bernoull plane secton hypothess (u ps ), allowng a knematcal relaton between the axal stran parallel to the drectrx of the bar, and a warp (u w ) consdered a three dmensonal vector feld wth three components, so changes n the secton s shape are consdered allowng elongaton on the transversal renforcements, and lateral deformaton on concrete. Warp vector feld s dscretzed on nodal values (d F ) along the cross secton and then nterpolated by means of an nterpolaton matrx (N F ), see equaton (2). = ps w u u + u (1) w u N d (2) F F In addton, the warp feld s consdered a functon of the axal elongaton of the bar s axs, twst curvature, the two bendng curvatures and ther correspondng dervatves respect to drecton of the bar s axs; each of those quanttes defne the components of vector ξ *. Further, the nodal values of the warp dsplacement are wrtten as a multplcaton of matrx A, as shown n the next equaton, tmes vector ξ *. d ξ F = Aξ * * T = ε0, φx, φy, φz, ε' 0, φ' x, φ' y, φ' z (3) Matrx A n equaton (3) has dmensons of [n X 8 ], and relates each component of the cross secton warp to the four quanttes descrbed above and ther dervatves. The expresson for the complete three dmensonal stran feld can be wrtten as shows equaton (4). Vector ε s contans the components of deformaton of the beam. N PS s the typcal operator contanng Naver-Bernoull relatonshps for a plane secton hypothess. Matrx B F contans dervatves of N F and s used as an nterpolaton matrx for stran components due to warp. PS ε= ε + ε PS PS ε = N ε ε B d = B Aξ ε = [ ε, γ, γ, φ, φ, φ ] T 0 w w * F F F s s y z x y z (4) Provded a sutable three dmensonal non-lnear consttutve equaton, expressed n a general fashon as n equaton (5), matrx A s then evaluated as descrbed n Baran [4]. The procedure followed takes nto account the three dmensonal nternal equlbrum equatons at each materal pont and compatblty condtons n order to obtan a warp feld that produces shear and n plane axal stresses that are

4 compatble wth the appled normal axal stresses and the shape of the cross secton. The resultng matrx A result dependent on the shape of the cross secton and on the actual non-lnear state on each materal pont. ( ) = Dε = D ε + ε = + PS W = Dε = Dε PS W PS w PS W (5) In order to be consstent n energy and compatblty terms, shear deformatons of the beam s axs shall be obtaned from the shear stran feld over the cross secton, whch n turn has been computed from the warp feld. The shear deformaton of the beam element evaluated ths way assures a correct computaton of the energy absorbed by shear and torson effects a more accurate estmaton of the frame s deformaton. Nevertheless, n general when a skew shear s appled to an arbtrary shaped cross secton bult from dfferent materals, the resultng shear deformaton does not have the same skew angle as the appled shear force at each ncrement of forces. Ths represents a problem n a non-lnear three dmensonal analyss because two shear components are to be obtaned from only one equaton,.e. the balance of energy equaton. In reference [4] ths s dscussed and a way to handle ths stuaton s deduced. Fnally, by equatng the work performed by the stress feld and the nternal forces on the cross secton, the followng explct expresson for a total nteracton stffness matrx s obtaned. = K ε s s s T T * * K Ξ Ω B DB daωξ s = T (6) Where B * s an operator matrx for evaluatng the full three dmensonal stran vector from the sectonal deformaton vector. B * depends on both matrxes N PS and A. Matrx Ω stands for the consstent evaluaton of beams shear deformaton as commented above, and matrx Ξ condensates the dervatves of axal elongaton and torson curvature (ε 0 and φ x ) takng nto account the actual dstrbuted axal load and torson moment. It can be seen that sectonal stffness matrx presented above s symmetrc. More over, t satsfes nternal equlbrum and compatblty condtons over the cross secton. Ths matrx may be used on any shaped cross secton, composed of dfferent materals wth dfferent consttutve equatons, even consderng consttutve ansotropy CONSTITUTIVE EQUATIONS USED IN THIS STUDY For ths study, a rotatonal smeared crack approach has been consdered for plan concrete n order to take nto account crack-nduced ansotropy and ts nfluence on shear resstance mechansm of renforced concrete sectons. An orthotropc equvalent unaxal model has been used for represent plan concrete. Stress and stran prncpal drectons are consdered concdent and a unaxal stress-stran equaton s stated for each thee drectons. In compresson, each stress-stran curve mantans the same shape, however they are a functon of maxmum lateral stran accordng to the next equaton from Veccho and Collns [7]. Where ε s a prncpal compresson stran, ε max s the maxmum lateral stran and f c s the strength of the concrete. Equaton (7) was orgnally formulated for analyss n plane-stress state; however

5 t has been appled for three dmensonal analyses and has shown reasonable results, Veccho and Selby [8]. 2 ε ε = fcmax 2 ε0 ε 0 fc fcmax = ε max ε 0 f c (7) For concrete n tenson, a lnear elastc behavour s consdered prevous crackng. After tensle strength s reached, a non-lnear softenng curve s consdered (Cervenka [9]) n order to take nto account tensonstffenng effects. In ths equaton, k 2 s a parameter specfyng the shape of the softenng branch, c s a stran parameter specfyng the stran for null tensle stress on concrete. For the present study, values k 2 =0.5 and c=0.004 has been consdered. k2 ε = ft 1 0 c (8) Shear stffness on the prncpal drecton of concrete s specfed n order to mantan the concdence of the stress and stran prncpal drectons for each load ncrement. The followng expresson for the shear stffness assures coaxalty of the numercal soluton, Jrásek and Bazant [10], Bazant [11], Zhu et al [12]. D j 1 j = 2 ε ε j (9) Consttutve matrx D s assembled on a coordnate system concdent wth the prncpal drectons. After rotatng ths matrx to the orgnal coordnate system s a full matrx results. In renforcng steel, the usual elastoplastc unaxal stress-stran relatonshp, wth yeld stress f y, s consdered. NUMERICAL IMPLEMENTATION The numercal model has been mplemented consderng wth the possbltes of some numercal facltes such as load and dsplacement control and lne search, Crsfeld [13]. Non-lnear soluton s reached by means of a Newton-Raphson algorthm that allows usng a tangental or a secant stffness matrx. NUMERICAL STUDY The crtcal cross secton of a beam specmen tested under shear loads by Kan [6] was studed usng the prevously developed non-lnear sectonal model. The dmensons of the cross secton studed s shown on fgure 1 and the expermental set-up s descrbed n fgure 2. The concrete strength was 28.2 MPa, stran at peak compresson stress was and the tensle strength was 1.75 MPa. Longtudnal renforcement was eght bars of 25 mm dameter and yeld stress of 442 MPa. Strrups were of 10 mm dameter spaced at 100 mm and 400 MPa yeld stress.

6 Fgure 1. Cross secton studed Fgure 2. Test set-up used by Kan [6] A comparson of the numercal soluton of shear force versus sectonal shear stress and the expermental results s presented n fgure 3. Good agreement between both curves s observed. Fgure 3. Sectonal model compared to an expermental beam specmen under shear.

7 The nfluence of the shear-bendng nteracton on the performance of force deformaton dagrams of the cross secton s studed by calculatng shear deformaton dagrams under dfferent levels of constant bendng moments, fgure 4. Under low level bendng moment, shear force-dsplacement curves present lttle varaton both on ultmate resstance and overall stffness durng the entre loadng process. Only a reducton on the shear force producng ntal crack s apprecable. When the concomtant moment ncreases, a sgnfcant reduce on both ultmate shear force and stffness durng the non-lnear loadng s observed. Fgure 4. Influence of concomtant moment on shearng force-deformaton curves. The nfluence of concomtant shear forces on moment-curvature dagrams s also studed on the next fgure. It s seen that for ncreasng shearng forces, not only the ultmate bendng moment s reduced but also overall shape of the curve changes resultng on lower flexure stffness for the cross secton. Fgure 5. Influence of concomtant shearng force on moment-curvature curves. CONCLUSIONS A non-lnear model for the analyss of renforced concrete cross sectons under combned three dmensonal loadng has been developed. The sectonal model s general enough to allow a correct smulaton of any shaped cross secton bult of dfferent materals. Partcularly, non-lnear behavour, and crack nduced ansotropy has been taken nto consderaton.

8 An equvalent unaxal model wth rotatng smeared cracks has been used as a consttutve model for concrete n ths study. The nfluence of shear-flexure combnaton on the performance of a partcular cross secton has been nvestgated. A shear test on a beam element havng the same cross secton has been smulated and good agreement wth the expermental results have been found by analysng a sngle cross secton under pure shear. Performance of both moment-curvature and shearng force-deformaton on the cross secton has shown to be very dependent on the concomtant acton when these present for moderate to hgh value. Interacton effects were neglectable for low concomtant actons. Influences of the axal-tangental forces nteracton were present on both the ultmate and stffness of the pre-yeld branch predcted by the sectonal model. It s to be expected that effects of normal and tangental nteracton nfluence the overall performance of structural elements under general loadng and complete constructons, determnng the ultmate load and non-lnear redstrbuton capacty when strong combnatons take place. ACKNOWLEDGMENTS Ths research s beng carred on the framework of Research Project MAT sponsored by the Spansh Mnstry of Scence and Technology. The author also wants to expressed hs grattude to the Departament d Unverstats, Recerca Socetat de la Informacó of the Government of Catalona for ther support. REFERENCES 1. Marí, A. R. Non-Lnear Geometrc, Materal and Tme Dependent Analyss of Three Dmensonal Renforced and Prestressed Concrete Frames. Report Nº UCB/SESM-84/12. Unversty of Calforna. Berkeley Marí, A. R. Numercal Smulaton of Segmental Constructon of Three Dmensonal concrete Frames. Engneerng Structures. No 27, pp Calv, G. M., Pnto, P. E., Eds. Expermental and Numercal Investgatons on the Sesmc Response of Brdges and Recommendatons for Code Provsons. European Consortum of Earthquake Shakng Tables. Prenormatve Research n Support of Eurocode 8. Report No 4, pp November Baran, J. M. Ph.D. Thess (on preparaton). Techncal Unversty of Catalona, Span. 5. Petrangel, M., Pnto, P., Camp, V. Fber element for Cyclc Bendng and Shear of RC Structures. I: Theory. Journal of Eng. Mechancs. September Kan, M. An Expermental Investgaton of Renforced and Prestressed Beams n Shear, M.A.Sc. Thess, Unversty of Toronto, Veccho, F. J., Collns, M. P. The Modfed Compresson-Feld Theory for renforced concrete elements subjected to shear. ACI Journal, V. 83, No 2, March-Aprl Veccho, F. J., Selby, R. G. Toward Compresson-Feld Analyss of Renforced Concrete Solds. Journal of Structural Engneerng, ASCE, V. 117, No 6, pp , June, Cervenka, V. Consttutve Model for Cracked Renforced Concrete. ACI Journal, V. 82, No 6, pp , Nov.Dec Jrasek, M., Bazant, Z. P. Inelastc Analyss of Structures. John Wlwy & Sons, pp. 734, England, Bazant, Z. P. Comment of Orthotropc Models for Concrete and Geomaterals. Journal of Eng. Mechancs, ASCE, V. 109, No 3, pp , June 1983.

9 12. Zhu, R., Hsu, T. C., Lee, J-Y. Ratonal Shear Modulus for Smeared-Crack Analyss of Renforced Concrete. ACI Structural Journal, V. 98,No 4, pp , Jul.-Aug Crsfeld, M. A. Non-Lnear Fnte Element Analyss of Solds and Structures. Vol. 1. John Wley & Sons, pp. 345, England, 1991.