THEORETICAL CRACK ANGLE IN REINFORCED CONCRETE ELEMENTS SUBJECTED TO STRONG EARTHQUAKES

Transcription

1 THEORETICL CRCK NGLE IN REINFORCED CONCRETE ELEMENTS SUBJECTED TO STRONG ERTHQUKES Jang Hoon KIM 1 nd John B MNDER SUMMRY The purpoe of th paper to deelop a mathematal expreon for omputng rak angle baed on renforement olume n the longtudnal and tranere dreton, member end-fxty and length-to-wdth apet rato. For th a renfored onrete beam-olumn element aumed to poe a ere of potental rak plane repreented by a number of dfferental tru element. Dependng on the boundary ondton, a ontant angle tru or a arable angle tru employed to model the raked trutural onrete member. The tru model are then analyzed ung the rtual work method of analy to relate fore and deformaton. Rgorou and mplfed oluton heme are preented. n equaton to etmate the theoretal rak angle dered by onderng the energy mnmzaton on the rtual work done oer both the hear and flexural omponent of tru model. The rak angle n th tudy defned a the teepet one among fan-haped angle meaured from the longtudnal ax of the member to the dagonal rak. The theoretal rak angle predton are aldated agant expermentally obered rak angle reported by preou reearher n the lterature. Good agreement between theory and experment obtaned. INTRODUCTION When renfored onrete truture are ubjeted to large deformaton reeral due to trong earthquake loadng, trutural damage tart wth rakng. The rtal trength mehanm, goerned by ether hear or flexure, determned not only by the renforng teel layout, but alo by the nature of the rak formulaton, partularly the rak pattern and angle. The rak angle n a trutural onrete member ery mportant a t affet the pot-rakng tffne a well a the ultmate trength of the member. Reent reearh n the Unted State, and elewhere, ha propoed analy and degn model for the em hear trength of renfored onrete olumn member, expltly or mpltly, wth referene to a rtal rak angle [CI, 1995; SHTO, 1994; Euroode, 1991; Colln and Mthell, 1991; Hu, 1993; Pretley, et al., 1994a,b]. For example, the teel omponent of hear retane gen, n t mot general form, by jd V = h f yh ot (1) where h = eton area of hear teel, f yh = yeld trength of hear teel, jd = nternal leer arm, = tranere hoop pang and = rak angle. But no gudane gen to degner a to the arablty of th mportant parameter. Rather, engneer are requred to ue prerbed alue for example 30 and 35 degree for analy and degn, repetely regardle of the renforement onfguraton and/or apet rato [Pretley, et al., 1994a,b]. Thee arbtrary alue for rak angle hae been baed on empral oberaton, but not ound theory. Therefore, a mathematal expreon for omputng rak angle deeloped n the preent paper baed on renforement olume n the longtudnal and tranere dreton, member end-fxty and length-to-wdth apet rato. 1 Department of rhteture, jou Unerty, Suwon , South Korea Emal: kmjh@madang.ajou.a.kr Dept of Cl Engneerng, State Unerty of New York at Buffalo, NY 1460, U.S..Emal: jmander@au.buffalo.edu

2 TRUSS MODELING OF CRCKED CONCRETE ELEMENTS It ha long been reognzed that the behaor of renfored onrete beam-olumn after onet of rakng an be analyzed ung an approprate tru model. In tru analogy, longtudnal renforement repreented by longtudnal hord of a tru, whle tranere hoop teel repreented by tranere tenle te. The effet of onrete n flexural ompreon may be ondered a a part of the longtudnal ompreon hord member. The longtudnal hord and the tranere tenle te are aumed to be nternally tablzed by the trut that model onrete regon under ompreon n the dagonal dreton. The nlnaton of the dagonal trut hould onde wth the probable dagonal rak dreton. For mplty, the longtudnal hord, tranere tenle te and dagonal trut are aumed to be joned together through rgd node. Shlah, et al.[1987] defned two tandard regon n trutural onrete element dependng on the omplexty of tre dtrbuton: undturbed (B-) and dturbed (-D) regon. The defnton ued through th paper. Contant ngle Tru The hear tranfer mehanm for undturbed regon n a dagonally raked long beam-olumn member hown n fgure 1. From the oerall member hown n fgure 1(a), a dfferental porton of tru wth prmat member hang fnte depth an be extrated for analy purpoe a hown n fgure 1(b). In th repreentaton, t aumed that the tranere teel unformly dtrbuted oer the length of the member. Now, onder th ngle dfferental tru ubjeted to the dfferental hear fore dv. Member fore of the dfferental tru an be ealy found by the tat equlbrum. The hear deformaton of a dfferental tru an then be alulated ung the prnple of Vrtual Work. Rgd longtudnal hord are aumed n order to negate the effet of flexural deformaton. It noted that under ontant hear, the deformaton of eah dfferental tru the ame oer the entre ontant angle tru. The hear tffne of the entre raked onrete member due to ontant angle tru mehanm obtaned by arryng out the ntegraton oer the length defned by rak angle, jd ot. That, Fgure 1: Contant angle tru model K = dk = dv Θ ρnot = E 4 1+ ρ noe ()

3 where Θ = drft angle due to hear, ρ = olumetr rato of hear teel, n = E / E = modulu rato, E = modulu of elatty of onrete, E = modulu of elatty of teel and = hear area of onrete eton. Varable ngle Tru The arable angle tru an approxmately repreent a dagonally raked hort olumn where dturbed regon preal a hown n fgure (a). Conder a ngle dfferental tru element ubjeted to dv n fgure (b). But dv not unform through the length at th tme. Note that a dfferental tru ont of a teel te wth depth Ldx and two tapered dagonal trut, where x a non-dmenonal parameter aryng from 0 through 1. tapered trut dealzed a a prmat one wth aerage depth. In a manner mlar to the oluton of the ontant angle tru mehanm, the elat hear tffne of a raked onrete olumn obtaned by ntegratng the dfferental tffne oer the entre length, thu K = dk = ρ n ρ ne ot { 1 + x ot α} + { 1 + ( 1 x) ot α} 0 α dx (3) where α = orner-to-orner dagonal angle. Sne a loed-form analytal oluton to th equaton ha not found, an approprate numeral ntegraton heme ued ntead. Then equaton (3) an be expreed a K = N = ρn { 1 + x ot α} + { 1 + ( 1 x ) ot α} ω ρ n ot α E (4) n whh N = number of numeral ntegraton pont, ω = weght fator at th pont, x = normalzed oordnate Fgure : Varable angle tru model of th numeral pont. ny numeral ntegraton heme uh a two-pont and three-pont Gau quadrature, Trapezodal rule, Smpon 1/3 rule and Boole rule may be ued for olng equaton (4). It hould be noted that for quat olumn where hear generally rtal (mall L / jd ), there rtually no dfferene n tffne alulaton between numeral heme a hown n fgure 3. Sne equaton (4) wth numeral parameter alue lengthy, a onenent mplfed oluton an be ued a 3

4 K = ρn ot α 1 + 4ρ n ( ot α ) E (5) The raked elat hear tffne alulated by th approxmaton faorably ompared to the exat one n fgure 4, where Smpon 1/3 rule wth N = 0 regarded a exat K g /0.4E 0.03 nρ K/Kg ExatSoluton NuneralSoluton L / jd Fgure 3: Comparon of hear tffnee between numeral heme K / Kg K g =0.4E nρ ExatSoluton pproxmate Soluton L / jd Fgure 4: pproxmaton of hear tffne due to arable angle tru COMPRISON OF STIFFNESSES Fgure 5 ompare the tffne of ontant angle tru gen by equaton () wth the exat one of the arable angle tru gen by equaton (4). For th purpoe, t neeary to put α = whh denote that the teepet rak angle to the longtudnal ax of the fan-haped rak at dturbed regon of the olumn wll be equal to the ontant rak angle at undturbed regon. Note that there no remarkable dfferene between ontant angle tru and arable angle tru, and any model an be ued for determnng hear tffne oer the length of the member. 4

5 K / Kg K g =0.4E nρ Varable ngle Tru Contantngle Tru L / jd Fgure 5: Comparon of hear tffne between tru model Two-Pont Gau Tru Model Ung equaton (4) wth numeral ntegraton weght, the arable angle tru model n fgure an be phyally mplfed wth reaonable auray. Implementaton of Gau quadrature wth two pont reult n the two-pont Gau tru model a hown n fgure 6. xal rgdte of tru member at th numeral pont are gen by ( E) = T ω E h L (6) ( E) = E d x 0.5ω + tan α ( E) = 0.5E = 0.5E ρ n L t g t (7) (8) where ω 1 = ω = 0. 5, x 1 = , x = , t = eton area of longtudnal teel, g = gro eton area of onrete element and ρ t = t / g. It noted that the trut eton area n equaton (7) an alo be obtaned by meaurng the depth of dagonal trut along the tru enter lne on the aled keth. The hear deformaton of the tru model determned ung the Vrtual Work method of analy on the tranere te and dagonal trut. Thu the hear tffne of the tru model wth fxed-fxed and fxed-pnned end hould be the ame. The flexural deformaton of the tru model an be determned alo ung the Vrtual Work method of analy onderng only the longtudnal hord member. The elat flexural tffne of a raked onrete olumn about drft angle due to the arable angle tru model gen by Fgure 6: Tru model by two-pont Gau quadrature 5

6 E t K f = ς ot α where ς = for fxed-fxed end and ς = for fxed-pnned end. (9) DETERMINTION OF CRCK NGLE It beleed n th tudy that the rak angle n a onrete member depend on both hear and flexural omponent of dplaement and wll our at an orentaton that requre the mnmum amount of energy. Energy Conderaton The external work done on the trutural member due to unt hear fore ( V = 1 ) the ame a the total drft angle, thu ung equaton () and (9) 4 1+ ρnoe ς ot EWD = Θ1 = Θ,1 + Θ f,1 = + (10) E ρnot E g ρtn Mnmzng the external work done by dfferentatng equaton (10) wth repet to lead to the rak angle aung the mnmum energy, that d ( EWD) d = 0 (11) whh ha a oluton = tan 1 ρ ρ n + ς ρt 1 + ρ n 1 4 g (1) Expermental Valdaton Table 1 preent a omparon of twenty expermentally obered rak angle wth thoe omputed ung equaton (1). The omparon alo made n fgure 7 by ualzng the data n table 1. It edent that the theoretal reult ompare ery faorably wth the expermentally obered rak angle. Th learly how the dependene of the rak angle on the quantty of longtudnal a well a tranere renforement. CONCLUSIONS Craked trutural onrete element are modeled by onderng potulated rak plane reultng n ether of a ontant angle tru or a arable angle tru. Both tru model are atfatory for determnng the hear tffne oer the length of the beam-olumn. It hown that numeral ntegraton heme an be mplemented for phyal mplfaton of the tru model. theoretal foundaton for omputng the prnpal rak angle formulated ung energy onderaton. Therefore, t onluded that the rak angle of 45 that ha been tradtonally aumed n CI 318 ode [1995] for many deade a well a the newly uggeted 30 reommended by Pretley, et al. [1994a,b] both lead to a faulty predton of hear trength. 6

7 Table 1: Comparon of rak angle between theory and experment Spemen Boundary n ρt ρ / g theory exp 1/3 Model Per a Prototype a 1/3 Model b Column Column C Column D Crular C1 d Retangular R d Unt 9 e Unt 13 e Unt 14 e Unt 16 e R10-60u f 4R6-65u f 4R10-60u f 0R6-80b f R6-60b f R1 g R3 g R5 g a Per rular olumn wth retroftted beam-olumn jont [Mander, et al., 1996a,b] b Semally degned rular olumn [Mander and Cheng, 1995] Square hollow-ore olumn [Mander, et al., 1984] d Column[Cha, et al., 1990] e Crular olumn [ng, et al., 1989] f Crular olumn [Wong, 1990] g Retangular olumn [Pretley, et al., 1994a,b] : Fxed-fxed end : Fxed-pnned end Experment (deg Theory (degree) Fgure 7: Crak angle omparon between theory and experment 7

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