SIMPLIFIED APPROACH TO THE NON-LINEAR BEHAVIOR OF RC MEMBERS

SIMPLIFIED APPROACH TO THE NON-LINEAR BEHAVIOR OF RC MEMBERS Shahd NASIR 1, Supratc GUPTA 2 And Hdetaka UMEHARA 3 SUMMARY In ths paper, a smplfed one-dmensonal analytcal tool based on fnte dfference technque to analyze renforced concrete members under cyclc deformatons s presented along wth the detals of the algorthm. Relatvely smple assumpton of plan secton remans plan before and after bendng s used. Nonlnear materal propertes wth cyclc stress-stran relatonshp have been adopted for both renforcement and concrete to check the sgnfcance of mplementaton of the varous characterstcs of the stress-stran relatonshp. Fnally ths tool s used to successfully smulate a few experment results to check the valdty of ths developed analytcal tool. INTRODUCTION The numercal analyss of the behavor of the renforced concrete members has been a subect of ntensve research n the past two decades. Several methods for the analyss are already avalable. Dfferent researchers are tryng to formulate more sophstcated analytcal tools for more accurate smulaton of dfferent structural phenomenon. Though stffness matrx based method s used for formulatng the analytcal tools, some researchers have also used flexblty based analytcal methods(e.g. fber model). These methods generally use theory of elastcty, plastcty or progressve damage for concrete usng dscrete crack or smeared crack approaches. These approaches generally attempt to smulate expermental results of renforced concrete members under monotonc and cyclc loadng usng fnte element based technques. Some of these methods nvolve rgorous analyss, requrng consderably hgh amount of computng tme and hardware requrements. However, many of these results can be smulated usng relatvely smpler methods so as to reduce the computatonal expenses. In ths research, a smplfed analytcal tool based on fnte dfference method s developed to analyze the renforced concrete members under cyclc loadng. Stffness matrx based method s adopted n such way that dsplacement-controlled algorthm can be mplemented. Earler, smlar methods have also been used for smulatng the behavor of renforced concrete members. Those methods requred the sectonal property of the moment-curvature relatonshp as nput. The moment curvature relatonshp s generally mplemented usng mult-lnear models wth varyng number of transton ponts and s used to calculate the load dsplacement behavor of the member. For cyclc behavor of moment-curvature relatonshp, varous models exst, e.g. Raufael and Meyer model, Takeda model, etc. However, all these models requre the calculaton of the envelope of moment-curvature relatonshps, whch often depends on the level of axal force, whch s not necessarly constant. The unloadng rules of these models are not dependent on the actual materal stress-stran relaton. In ths proposed method, no such mplementaton of moment-curvature relatonshps s requred as nput. The non-lnearty n the member behavor s taken care by the drect mplementaton of the non-lnear materal 1 2 3 Department of Cvl Engneerng, Nagoya Insttute of Technology, Nagoya, Japan E-mal: shahd@conc2.ace.ntech.ac.p Department of Cvl Engneerng Gunma Unversty, Gunma, Japan Department of Urban Plannng and Envronmental Engneerng, Nagoya Insttute of Technology, Nagoya, Japan

propertes n the calculaton of nternal forces based on the current condton of stress, stran, axal force, moment, etc. Complete stress-stran relatonshp of both concrete and renforcement s mplemented wth full unloadng crtera. One more postve aspects of ths model s that, the effect of mplementaton of varous characterstcs of the stress-stran curves of both concrete and renforcement can be check wth lesser computatonal cost n comparson to the more sophstcated methods, e.g. 3D analyss etc. In ths research, the developed smplfed method for the analyss of the renforced concrete members s presented n bref. The materal propertes wth hysterc stress-stran relatonshp for both renforcement and concrete adopted n the analyss are also presented wth due consderaton to varous phenomenon. In ths formulaton, the structure s dscrtzed nto elements lke parts. The appled load and the dsplacements, slopes and curvatures at the nodes are taken as global varables. Here, the assumpton that a plan secton remans plan before and after bendng s used as the bass of calculaton of nternal forces. The stffness matrx s calculated based on the present condtons of materal stress, stran, etc. In ths paper expermental results of renforced concrete specmen under cyclc loadng have been smulated to fnd out the effectveness of ths model. FORMULATION Here a cantlever renforce concrete column wth horzontal cyclc load appled at the top s consdered. The secondary moment n the column (P- effect) s neglected. At the nodes, t s assumed that plan secton remans plane durng bendng and the deformaton due to shear force s neglgble. The concrete s assumed to be a homogeneous sotropc materal wth perfect bond between concrete and renforcement. Based on these assumptons, sectonal propertes are consdered at the nodes as shown n Fgure 1b. In ths paper, only un-drectonal horzontal loadng at the top of the column s consdered. The column of heght L s frst dvded nto n element wth n+1 node and the secton s subdvded nto m strps and non-lnear materal propertes wth approprate stress-stran relatonshp are consdered at these ponts to calculate the nternal forces. For b-drectonal horzontal loadng, ths method can be appled by dvdng the secton nto square blocks, rather then strps. Lateral load P and axal force P axl are appled at the top of the column. Nodal varables of dsplacement y, P axl P 1 2 n ~ ~ 1 2 3-1 n+1 L h b dh=h/m th strp n total of m Sectons x c y φ ε t φx c ε f s 1 σ c f s 2 f s 3 ε b a) Column Dscretzaton b) Sectonal Propertes at Nodes Fgure 1: Analytcal model 2

rotaton θ, curvature φ and stran at the top of the secton ε t at th node are consdered. Stran at th strp can be defned as ε = ε t y φ (1) Each strp of the secton s further subdvded nto parts of area A m accordng to the dfferent materal propertes. These areas are core concrete, cover concrete and renforcement. Stress σ m s calculated for each of these areas based on ther materal characterstcs. The nternal axal capacty P ax, the external bendng moment ( M ) EX and nternal bendng moment ( M IN ) at the nodes can be calculated as follows Pax = σ A (2) M EX = l P (3a) M IN = σ A z (3b) Here, the A represent the summatons for all materal parts A m and z = h / 2 y. Wrtng Eq.(2) and (3) n ncremental form, = δpax δσ A = A ( δε t y δφ ) (4) δm m = lδp = Σ AZ δε = 1 = A Z δε Rearrangng Eq.(4), = δε t ( δpax A δy δφ ) / A Substtutng Eq.(6) n Eq.(5), we wll get, δ M = l P = ( M / φ) δφ + ( M / P ) δp ax ax t A Z y δφ The nodal varables dsplacement y, rotaton θ, curvature φ n the ncremental form can be related to each other as follows δθ + 1 - δθ - ( δφ + 1 + δφ ) l/2 = (8) δy+ 1 -δy - ( δθ + 1 + δθ ) l/2 = (9) The stffness matrx n the ncremental form can be formed usng Eq.(7), (8) and (9). There are n +1 varables of dsplacement δy, rotaton δθ, curvature δφ at each node and δp, the appled cyclc load s taken as global varable. There are n +1 equatons from Eq.(7) and n equatons each from Eq.(8) and (9). Snce from boundary condtons δφ n+1 = and l n+1 = at the top, Eq.7 wll not be relevant at the n+1 th node. Therefore 3n equatons have been taken wth 3n +4 unknowns and 4 boundary condtons (at the support δ y 1 =, δθ 1 = and at the top δφ n+1 =, appled δ y n+ 1 ). Therefore, system of equatons can be solved. Even though the stffness matrx was not symmetrc, t dd not create any problem n the analyss. After solvng the nodal varables of dsplacement y, rotaton θ and curvature φ are calculated. At each node, based on the calculated curvature φ, stran at the top of the secton ε t s calculated n an teratve manner such that nternal axal force balances wth the external axal force. The gradent of the stran dagram φ s used n calculatng the moment at the secton. After the convergence of nternal forces, the global convergence at each node s checked. The unbalanced moments due to nternal and external forces s taken as the convergence crtera and terated untl the convergence crtera has been satsfed. (5) (6) (7) Materal model for concrete MATERIAL MODELS The stress-stran curves for monotonc case serves as the envelope for the cyclc stress-stran relatonshp. In compresson, the stress-stran envelope wth parabolc behavor before the peak and lnear softenng after the peak s smlar to the model proposed by Kent and Park [1] s adopted. Confnement effect due to tes or lateral renforcement s consdered by adoptng more ductle post-peak softenng curve. 3

Stress f c Confned Unconfned ε ε cp α 1 f c f t ε ε cf1 ε cf2 Stran E c -f c Turnng pont More cases of Reloadng and Unloadng σ = f c 2 ( 2ε / ε ( ε / ε ) ) σ = m ( ε ε 1 1 σ = β m ( ε ε ) + f cf c ε ) + α f c ε ε < ε ε ε > ε cf cf Where ε =.2 s the stran at the peak stress, m1 =.8 f c /( ε cf ε ) s the post peak slope controllng the softenng/ductlty of the concrete and stran ε cf s the controllng parameter for ths post peak slope where =1 for unconfned concrete and =2 for confned concrete. Ths factor ε cf s dependent on the amount and the spacng of transverse renforcement and the strength of concrete and s calculated as mentoned n Kent and Park [1]. In tenson, lnear behavor untl peak has been taken and gradual degradaton s adopted n post peak behavor and s shown below: σ = Ecε ε ε t.4 (11) σ = Ecε t ( ε t /( ε ε cp )) ε < ε t where f t and E c are calculated on the bass of CEB and ε t = f t / Ec. In varous expermental works, t s shown that plastc stran s accumulated for cyclc cases n both tenson and compresson. However, the accumulaton of plastc stran n tenson s not taken n ths analyss for smplcty. The plastc stran accumulated n compresson ε cp s taken as the orgn of the stress-stran curve n tenson as f the whole stress-stran curve has shfted to compresson sde. Lnear unloadng n the tenson sde s assumed. For the cyclc behavor of concrete n compresson, model proposed by Darwn and Pecknold [2] based on expermental results from Karson and Jrsa [3] s adopted. The strategy of unloadng and reloadng s shown n Fgure 4. Unlke Darwn and Pecknold model, the unloadng branch from the turnng pont s curtaled at 9% stress level and connected to the ε cp pont n order to avod zero slope for better convergence. The plastc stran ε cp s calculated by the method adopted n focal pont model [4] for better numercal convergence. Here ε cp s the ntersecton of stran axs wth lne onng from the unloadng pont at the compresson envelope curve to the pont wth stress f c and stran f c / Ec. Materal model for renforcement Fgure 2: Concrete model For the renforcement, the cyclc stress-stran model adopted n the numercal analyss s shown n Fgure 3. Ths s a mult-lnear model that deals wth one-dmensonal stress-stran relatonshp for renforcement. Ths model takes care of renforcement hardenng and blnear unloadng s mplemented. In tenson after yeldng at ( f yt, ε yt ), there s a yeld plateau to the pont ( f yt1, ε yt1 ) wth a nomnal slope of E s1 (=.1E s ). After ths for smplcty, the hardenng s assumed to start to reach the ultmate strength n a lnear manner to ( f yt2, ε yt2 ) wth (1) 4

Stress Envelope Unloadng/Reloadng case Ref. Lne n Tenson Ref. Lne n Compresson (ε yt,f y ) (Ε yt2, f yt2 ) (Ε yt1, f yt1 ) E ES1 S2 E S Stran E S4 E S3 (ε yc, f yc ) a slope of E s2 n monotonc stress-stran case. Ths model s qute smlar to multple surface models that are used for renforcement n mult-dmenson. Always, one lne n tenson and one n compresson are used as reference lnes wth a reference pont on each of them. When the stran s between these reference ponts, t s consdered to be unloadng. Unloadng s assumed to occur elastcally (E s ) ntally and assumed to follow slope E s4 after reachng the surface mdway between the two reference surfaces (or lnes). Ths s assumed to take care of varous dfferent phenomena ncludng Bauschnger effect, etc. Typcal envelope lne and a typcal case of unloadng and reloadng are presented n Fgure 3. There are two reference lnes n tenson part and one n the compresson part and they are shown clearly. When the stress-stran curve travels along the tensle part, the plastc stran s assumed to accumulate untl t reaches the stran of ε yt ε yt1. After that, the stress-stran behavor s assumed to harden wth a slope E s2. The reference lne n tenson moves up as the stress hardens. The hardenng contnues, tll the stress-stran reaches another reference lne n tenson. A typcal case of unloadng-reloadng case s also shown n Fgure 3. The reference lne n compresson does not shft lke the reference lne n tenson. It passes through compresson yeld pont ( ε yc, f yc ) and then takes the slope of E s3 (=.1E s ) to avod the zero slope problem n analyss. The dfferent parameters adopted here n the analyses are E s = 2.1x1 5 MPa, ε yt1 =.7, ε yt2 =.5, E s4 =.1E s, f =. yc f yt Fgure 3: Renforcng renforcement stress-stran model APPLICATION TO EXPERIMENTAL RESULTS Case No f c (Mpa) Table 2: Detals of Specmen and Materal Propertes of Concrete Specmen Rngs ε cf1 ε cf2 Crosssecton spacng (Unconfned) (Confned) Axal Load (N/mm 2 ) T1 2. 3x15 D1@15 c/c 3.16ε 6.65ε. T2 2. -do- -do- 3.16ε 15.ε 3. Table 1: Materal Propertes of Renforcements Case Type f yc (MPa) f yt (MPa) f yt1 (MPa) E S (MPa) T1 & T2 D1-42. 42. 5. 21.E5 5

P axl P 172.5 cm Core concrete Cover concrete Unconfned concrete Confned concrete a) Specmen T1 & T2 b) Secton of T1 & T2 c) Analytcal model of Secton Fgure 4: Specmen detals 6 4 Dsplacement (cm) 2-2 -4-6 5 1 15 2 Fgure 5: Appled cyclc dsplacement In ths paper two experments are consdered to understand the capabltes of ths developed analytcal method. Experments were carred out on a renforced concrete cantlever column under dsplacement controlled cyclc loadng[5]. The sectonal detals and dmensons of the specmens are shown n Fgure 4. Both the specmens T1 and T2 are of same dmensons. T1 s the specmen wthout axal load whereas T2 s the specmen wth axal load. The axal load of 3. N/mm 2 was appled at the top of the column as shown n Fgure 4. Each cycle of a partcular magntude was appled once and shown n Fgure 5. The secton detals, materal propertes of concrete and values of dfferent parameters of the concrete materal model used n the analyss are presented n Table 1. The materal propertes of the renforcement bars used are mentoned n Table 2. The descretsaton of the specmen nto elements and the cross-secton nto layer s shown n Fgure 4. The specmen s dvded nto n=5 parts and the cross-secton of the specmen s dvded nto the m=3 layers as shown n Fgure 1. In order to smulate member behavor and to see the capabltes of ths analytcal tool, the materal models are also studed. It s well-known fact that the analytcal results are the reflecton of the materal models adopted hence some parameters, whch are very mportant n descrbng the structural behavor under cyclc loadng wll be dscussed and appled n ths secton. 6

GENERAL DISCUSSION In ths secton, the RC cantlever column specmen descrbed n the prevous secton s analyzed and dscussed n detal to check the valdty of the analytcal method. The analytcal results of both the specmens showed stffer behavor n the begnnng n comparson to the expermental results. Ths s the draw back and generally notced n most of the analytcal tools and s outsde the scope of ths paper. The area bounded by the hysterc curve s actually the representaton of the amount of the hysterc energy released. Ths hysterc energy s taken here to understand the cyclc behavor of the structural members. Snce the analytcal results are the reflecton of the materal models adopted hence specal emphass s gven to the dfferent parameters of the materal models. 3 2 Wth blnear unloadng Wth elastc unloadong 3 2 Wth blnear unloadng Wth elastc unloadong 1 1 Load (kn) -1 Load (kn) -1-2 -2-3 -4-2 2 4 Dsplacement (cm) -3-4 -2 2 4 Dsplacement (cm) Fgure 6: Reloadng/unloadng, T1 Fgure 7: Reloadng/unloadng, T2 2 1 Load (kn) -1-2 Analyss -4-2 2 4 Dsplacement (cm) (a) Fgure 8: Specmen T1 (b) 2 Load (kn) 1-1 -2 Analyss -4-2 2 4 Dsplacement (cm) (a) Fgure 9: Specmen T2 (b) 7

For the case of cyclc loadng, renforcement unloadng/reloadng branches are havng maor effect on the shape and sze of the area bounded by the hysterc curve. The amount of hysterc energy released can be controlled by the applcaton of the proper slopes of those branches of renforcement materal model and s studed here. Frstly elastc unloadng/reloadng has been mplemented for the smulaton of specmen T1 and T2. It was found that analyses showed much hgher hysterc energy as compare to experments. Varous parametrc studes were conducted usng dfferent characterstc of the envelope curve and wth dfferent slopes of the unloadng branches of the materal model. It was realzed that characterstc stress-stran curve of concrete has lesser effect. It was also realzed that adoptng blnear unloadng/reloadng n the renforcement materal model n such a way that the slope of unloadng or reloadng decreased from E s to.1e s from mdway between the tenson and compresson envelope shows nce smulaton of the expermental results. Ths s bascally to take care of the Bauchnger effect and other phenomenon. Fgure 6 and Fgure 7 show the analytcal results of the mplementaton of blnear unloadng curve (E s4 =.1E s ) n comparson to the elastc unloadng (E s4 = E s ). Fgure 8 and Fgure 9 show the analytcal results adoptng the blnear model (E s4 =.1E s ) n comparson to the expermental results. Fgure 8(a) and Fgure 9(a) show the analytcal results whereas Fgure 8(b) and Fgure 9(b) show the expermental results along wth the outsde envelope of the analytcal results. It s notced that the adopton of modfed blnear model (E s4 =.1E s ), the shape of the unloadng branches and the hystercs energy matched wth the expermental results. The shape of the load dsplacement dagram envelope after the yeldng of renforcement and the peak load s matchng wth that of the expermental results for specmen T1 whereas n case of T2 only the peak load matched. Ths mples that further attenton s necessary for the T2 where axal load s present and hgher confnement s expected for the concrete stress-stran curve. CONCLUSIONS A smple method for analyss of renforced concrete members has been proposed based on fnte dfference technque. In ths paper, the algorthm of ths analytcal tool s presented. Nonlnear materal propertes are adopted and the effect of few parameters of the materal models on the analytcal results s studed. Fnally usng the approprate materal parameter values, the analyss s done. The followng conclusons can be made: 1. Ths analytcal method can predct the peak strength qute approprately 2. The characterstc of unloadng/reloadng branch of renforcement plays promnent role n member hysterc behavor. Nce matchng between the reloadng/unloadng branches was observed when blnear unloadng (E s4 =.1E s ) s adopted. Hence t s realzed that adoptng a lower slope from some where n the mddle s reasonable. Ths phenomenon s also observed n varous other cases that are not reported here. Fnally, ths method s very fast and effcent method of analyss of renforced concrete members. Ths method s numercally very effcent and can be used to smulate expermental results for the better understandng of the phenomenon occurrng n the damage process. Ths analytcal tool can help to study the effect of mplementaton of varous parameters of the materal model. Hence, ths analytcal tool can also be used n a supportng role to the fnte element analyss wth three-dmensonal mplementaton, as t s dffcult to check the effect of varous characterstcs of the materal stress-stran curve usng the sophstcated methods. REFERENCES 1) Kent D. C. and Park. R. (1971), Flexural members wth confned concrete, Journal of Structural Dvson, ASCE, ST7, Vol. 97, pp. 1969-199 2) Darwn D. and Pecknold D. A., (1977), Analyss of Cyclc Loadng of Plane R/C Structures, Computers & Structures, Vol. 7, pp137-147. Pregamon Press 3) Karson I. D., Jrsa J. O. (1969), Behavor of Concrete under Compressve Loadng, Journal of Structural Dvson, ASCE, Vol. 95, No. ST12, pp2543-2563 4) Yankelevsky. D. Z. and Renhardt. H. W. Unaxal Behavor of Concrete n Cyclc tenson, ASCE, Journal of Structural Engneerng Vol. 115, No. 1, pp. 166-182 Umehara H. (1978), "Structural model of the RC members", Master dssertaton, Cvl Engneerng Department, Tokyo Unversty 8