Torsional resistance of high-strength concrete beams

Transcription

1 Torional reitane of high-trength onrete beam T. Hoain & P. Mendi Univerity of Melbourne, Vitoria, Autralia T. Aravinthan & G. Baker Univerity of Southern Queenland, Queenland, Autralia ABSTRACT: Thi paper aim to explore theoretial model for evaluating the torional behaviour of reinfored onrete beam. Thi paper aim to explore theoretial model for evaluating the torional behaviour of reinfored onrete beam. The paper invetigate the appliability and extenion of exiting analytial model and ontitutive relationhip developed for normal trength onrete to high-trength onrete. The verifiation model inlude the pae tru model, kew-bending theory and oftening tru model theory. The paper preent an aement of the preent Autralian Standard AS3600 for the torional apaity of hightrength reinfored onrete beam in order to make uggetion on how to extend the ode to over the HSC range. Reommendation for the torional deign of HSC beam baed on lattie model are alo preented. INTRODUCTION In reent year, many theorie have been developed for alulating the torional trength of member with both longitudinal teel and tirrup. The mot reognized theoretial model are the pae tru model and the kew-bending theory. It i found that the kew-bending theory overetimate the raking trength of the torion of RC beam by onidering the ontribution of onrete (Fang & Shiau 004). Collin & Mithell (974, 980) extended the tru model to inlude the apaity of onrete and the orientation of the ompreion onrete member, and named thi the diagonal ompreion field theory. One key aumption of their theory wa that for a olid etion ubjeted to torion, the onrete ore doe not ontribute to the torional reitane. Aordingly, thi model doe not onider the advantage of the ue of HSC. Hu & Mo (985) further developed the pae tru model to the oftened tru model, whih aount for the oftening of the raked onrete. For a low-trength onrete, their model give a good deription of the aending branh and the firt part of the deending branh of the uniaxial tre-train urve (Ramuen & Baker 995b). However, the uniaxial tretrain urve for onrete hange dramatially when omparing a NSC to a HSC. Hene, it i not obviou that the model an be extrapolated from NSC to the HSC range. Similarly, aggregate interlok i not inluded in any of thee above model, whih i a very important phenomenon in HSC. Thi i beaue high-trength onrete are muh more brittle than normal trength onrete, with le load arrying apaity one the peak load i reahed. Due to the imilarity of the trength in aggregate and that of the urrounding onrete, the failure urfae are mooth and devoid of aggregate interlok, whih redue the hear reitane in the ae of HSC (Attard & Mendi 993). While HSC ha been reently and inreaingly ued in bridge and building, the available theoretial model regarding the behaviour and deign ode proviion of torion in AS are primarily baed on limited experimental work on low trength onrete beam. Therefore, more work i required to afeguard the related deign when the available information i extrapolated to the deign of torional member made of higher trength onrete. Verifiation of the appliability of the available theoretial model to higher-trength onrete i alo required. Following review of the three theoretial model, oniting of kew-bending theory, pae tru model, and oftening tru model, and other experimental work on pure torion of reinfored onrete beam, thi paper preent an aement of the preent AS3600 for the torional deign of member made of higher-trength onrete. An advaned method of analyi baed on the lattie model will alo be diued in thi paper. The tehnique ha been uefully applied for both normal and hightrength onrete beam ubjeted to hear, and an inherently take into aount ome major influential fator; the redued aggregate interloking hear

2 tranfer mehanim in HSC; the oftening of onrete in ompreion; tenion; and hear. REVIEW OF THEORETICAL MODELS. Skew-bending theory The kew-bending theory wa initially propoed by Leig in 958 (959), and had ubequent ontribution from everal reearher in the field. Hu (968) made a major ontribution to the development of thi theory a it preently tand. He preented the expreion for evaluating the torional trength of retangular etion aording to the PCA tet. The bai approah of the kew-bending theory i that the failure of a retangular etion in torion our by bending about an axi whih i parallel to the wider fae of the etion and inlined at about 45 to the longitudinal axi of the beam. The torional trength of reinfored onrete member i ontributed to by both onrete, T and torional reinforement, T. But, from the PCA tet by Hu (968) on hollow and olid retangular beam, it wa oberved that the onrete ore doe not ontribute to the ultimate torional trength. Later he onluded that the onrete ontribution T wa mainly due to the hear reitane of the diagonal onrete trut. Baed on thi approah, the torional reitane propoed by Hu (979) inlude the ontribution of hear reitane of diagonal onrete trut, axial fore of tirrup, and the dowel fore of the longitudinal bar, and an be evaluated a follow: x yat f y Tn = T + α t () x y T = (. 4 f ) (in lb-in.) 3 α t f = ly y m + 0. f x y.6 x = horter entre-to-entre dimenion of loed tirrup; y = longer entre-to-entre dimenion of loed tirrup A t = area of one leg of a torional loed tirrup = paing of tirrup f ly = yield trength of longitudinal bar f y = yield trength of loed tirrup m = ratio of volume of longitudinal bar to volume of tirrup Aˆ l = At ( x + y ) Âl = ro-etional area of longitudinal bar within the hear-ompreion zone. Spae tru model with palling of onrete over The mot reognized theoretial mode of pure torion in reinfored onrete i the pae tru model, whih wa firt developed by Rauh (99), and onit of diagonal trut in the onrete retrained by tenion in the tirrup and longitudinal reinforement. After the raking of a reinfored onrete member ubjeted to pure torion, the diagonal rak eparate the onrete into a erie of onrete trut. In the pae tru model the torion i reited by ompreion diagonal, whih onit of the onrete between rak that piral around the beam at a ontant angle. Baed on an aumed train ditribution in the beam, Collin & Mithell (974, 980) extended the tru model and developed a theory to deribe the torque-twit repone. Mithell & Collin (974) oberved that the inreae in the thikne of the onrete over aued palling and a redution in the torional trength. Aording to thi aumption of onrete over palling, the ultimate torional apaity of the member i baed on the trength of the palled etion. In their theory, baed on the knowledge of the uni-axial tre-train urve, a paraboli tre-train urve of the onrete wa given by the following equation: ε ε f = f () ε 0 ε 0 f = tre in onrete orreponding to the train ε f' = ompreive trength of onrete ε = train in diagonal onrete trut ε 0 = train at maximum onrete ompreive tre The detailed derivation of the equation and the olution tehnique for the ultimate torional apaity an be found ele (Mithell and Collin 974)..3 Softening tru model The oftening tru model, developed by Hu & Mo (985), i imilar to the pae tru model deribed above, exept that it utilize the full onrete ro-etion and take the oftening of the onrete into onideration. The oftening of onrete i baed on an aumed effetive tranvere ompreive tre omponent, whih i ued to predit the torional behaviour of reinfored onrete. The model wa developed aording to the fundamental priniple of the mehani of material, tre equilibrium, train ompatibility, and the ontitutive law of material. In thi model, the ontitutive law of material i given in term of the tre-train urve of the oftened onrete hown in Figure. The equation for the aending portion of the tretrain urve of normal-trength onrete i modelled a:

3 ε ε f = f ε o ζ ε o (3) f = average ompreive tre in the diagonal onrete trut ε = ompreive train in the diagonal onrete trut ε o = train at maximum tre of non-oftened onrete taken a 0.00 for NSC ζ = oftening oeffiient = ε l + ε + ε d 0.3 ε ε l = train in ordinary longitudinal teel ε = train in tirrup The equation of the deending portion of the tre-train urve i given a: ε ε k f = f k (4) ε o ε k f k = ζ f' = peak oftened ompreive trength ε k = ζε o = oftened train orreponding to peak oftened ompreive trength f Equation 3 Figure. Stre-train urve for oftened onrete The torque i obtained from equilibrium equation. The detailed derivation of the equation and the olution tehnique for the ultimate torional apaity an be found ele (Hu & Mo 985, Hu 988). 3 AS TORSION DESIGN CRITERIA d Equation 4 The deign of reinfored onrete beam ubjeted to pure torion aording to the AS3600 i baed on the pae tru model, and doe not operate with a variable wall-thikne dependent on the onrete trength and the degree of reinforement. Aording to the torion proviion of AS3600, the ultimate trength of a reinfored onrete beam i derived from the thin-walled tube analogy, auming it i diretly proportional to the amount of reinforement a follow: A f y Tp = Ao otθ 0.08 f' b h (5) A o = area enloed by hear flow path = x o y o A = area of one leg of loed tirrup f y = yield trength of loed tranvere torional reinforement = paing of tirrup θ = angle of ompreion diagonal b, h = maller and larger outer dimenion of the ro-etion repetively x o, y o = horizontal and vertial ditane between the entre line of the longitudinal orner bar repetively Thu the onrete ontribution i ignored after torional raking in AS3600, whih make no ditintion between the behaviour of normal and hightrength onrete; i.e., there i no advantage in uing higher onrete trength in reiting ultimate torion. 4 REVIEW OF PAST WORK ON TORSION OF RC BEAMS In the pat, there have been ome invetigation onduted whih tudy the behaviour of RC beam under pure torion. Reid & Bridge (990) have arried out theoretial and experimental invetigation to ae the behaviour of pre-treed onrete edge beam heavily loaded in hear and torion. Their theoretial invetigation indiated that torion would greatly redue the load-arrying apaity of the beam, and the ombined torion and hear would reult in brittle failure. The effet of onrete ompreive trength on the torional behaviour of reinfored onrete beam wa invetigated by Ramuen & Baker (994, 995a, 995b). The reult of that invetigation howed a ignifiant effet of onrete trength on rak width, torional tiffne, and torional apaity of the tet beam. Kouthoukali & Belarbi (00) tudied the effet of high-trength onrete on the torional behaviour of RC beam under pure torion. They found from their tudy that the torional apaity of under-reinfored beam i independent of onrete trength, and the amount of longitudinal reinforement wa more effetive in ontrolling rak width than the amount of tranvere reinforement. Fang & Shiau (004) experimentally tudied the torional behaviour of normal and high-trength onrete beam under pure torion. Their tet reult howed that the HSC beam had higher torional trength and raked tiffne

4 than the NSC beam whih were deigned with the ame amount of reinforement. Reult of everal invetigation on the torional behaviour of normaltrength onrete beam have alo been reported (Hu 968; Mithell & Collin 974; Hu & Mo 985). However, no rational method to inorporate the onrete ontribution to ultimate torional trength i propoed in thee pat work. 5 PARAMETRIC STUDY ON PURE TORSION To examine the appliability of the three theoretial model for high-trength onrete, the experiment arried out by Ramuen & Baker (995a) and Fang & Shiau (004) were analyzed. The tet peimen were of olid retangular beam failing in pure torion, and repreent all poible reinforement ituation and a ignifiant variation in geometri and material parameter. Their experimental reult are onidered in the orrelation between the tet and the predited torional apaity from the theoretial model, and the effet of onrete trength on ultimate torional trength. Table, Figure and Figure 3 ompare the experimental reult with the predited torional trength of the teted beam from the literature, uing the aforementioned theorie. On the bai of obervation and reult reported in thi paper, it an be een from Figure and Figure 3 that AS 3600 alway overetimate the experimental torional apaity. Due to the palling of the onrete over, the area enloed by the entre line of the thikne redue, reulting in a redued torional apaity of the etion. But the preent AS3600 doe not operate with thi variable wall thikne, whih may reult in an unonervative Beam f MPa Table. Comparion of experimental and alulated torional trength ρ l % ρ t % T u(exp) AS Skewbending 3600 kn-m theory Ramuen & Baker (995a) T u(al) /T u(exp) Spae tru model Softened tru model B B B B B B B B B B B B Fang & Shiau (004) H H H H H H H H N N N N N N N N

5 l)/t(exp) T(a.5 and unafe predition of the ultimate torional apaity. 0. 5Alo, the onrete ontribution, i ignored after torional raking in AS3600, and i diretly 0 proportional to the amount of tranvere reinforement. The angle of ompreion diagonal i alo kept ontant with the hear f' formula. T(al)/T(exp) f' AS Skew-bending theory Spae tru model Softening tru model Figure. T (al) /T (exp) veru onrete ompreive trength (Ramuen & Baker 995b) T(al)/T(exp) AS Skew-bending theory Spae tru model Softening tru model f' Figure 3. T (al) /T (exp) veru onrete ompreive trength (Fang & Shiau 004) It an be een from Table that the pae tru model with palling of onrete, and the oftening tru model, give the bet predition for the ultimate torional apaity when ompared with the experimental reult, with a mean and a tandard deviation of 0.9 and 5.8% for the oftened tru model, and.05 and 6.8% for the pae tru model. On the other hand, kew-bending theory give reaonable predition for beam teted by Fang & Shiau (004), but i unonervative for the beam teted by Ramuen & Baker (995a). Thi i beaue all the beam teted by Ramuen & Baker (995a) were over-reinfored the failure of the beam reulted from the onrete failure. The overetimate of the ultimate trength from the kew-bending theory i beaue the theory inlude the volume of the longitudinal and tranvere reinforement and the onrete ontribution. It an be een from Table that for beam with the ame reinforement ratio, the ultimate torional trength of HSC beam inreae approximately. time than that of NSC beam when the ompreive trength of onrete inreae from approximately 35 to 78 MPa. Thi further verified the effet of f on the ultimate torional apaity. It exhibit that the ultimate torional trength of RC beam inreae a the ompreive trength of onrete inreae. Table. Comparion of alulated and experimental raking trength Beam f T r(exp) AS3600 Skewbending theory MPa kn-m T r(al) T r(al) / T r(al) T r(al) / T r(exp) T r(exp) Ramuen & Baker (995) B B B B B B B B B B B B Fang & Shiau (004) H H H H H H H H N N N N N N N N Table provide the omparion of the alulated raking torque to the experimental data. Craking our when the maximum tenile tre due to torion reahe the tenile trength of the onrete. It an be een from Table that the rak load inreae a the onrete trength inreae, whih i apparently due to the higher onrete trength. The mean value of the ratio T r(al) /T r(exp) in the HSC

6 erie are 0.8 for the AS3600 expreion, and.3 for the kew-bending theory; while for thoe in the NSC erie, the value are 0.88, and.3, repetively, from Ramuen & Baker (995a) beam. In the ae of Fang & Shiau (004) beam, the mean value of the ratio T r(al) /T r(exp) in the HSC erie are 0.89 for the AS3600 expreion, and.3 for the kew-bending theory; while for thoe in the NSC erie, the value are 0.93, and.5, repetively. So in all ae, whether over-reinfored or under-reinfored, the AS3600 expreion underetimate the raking trength, while kew-bending theory overetimate the raking torque of reinfored onrete beam. Alo, the ratio of ultimate and raking torional trength, T u /T r, from Ramuen & Baker (995a) beam i.6 for NSC and.6 for HSC, while the repetive value from Fang & Shiau (004) beam are.5 and.08. Thi prove that after raking, the inreae in torional trength for HSC i le than in NSC. Thi i probably due to the well-known redued aggregate interloking hear reitane mehanim in HSC. adopted. A value of π/4 wa propoed by Ngo (005) for the roughne angle. The tehnique of the lattie model ha been uefully applied for both normal and high trength onrete beam ubjeted to hear, by inluding a frature energy baed damage model a developed by Hoain (006). The frature energy baed damage model developed in thi tudy i baed on the MCFT, redued frature energy and variable hear retention fator, whih repreent the hange of material propertie due to the higher trength of onrete. A linear dereaing funtion related to the tenile train that i normal to the rak plane i propoed in thi tudy, whih take into aount the fat that frition due to the roughne of the rak urfae dereae a the opening of the rak inreae. Thi improve the damage model, allowing for the orret proviion of the eondary mehanim of hear reitane of a truture and aounting for the well-known experimental obervation of the redued aggregate interlok hear reitane in the ae of HSC. 6 LATTICE MODEL The lattie model wa initially developed by Niwa et al. (994), and an be expreed uing a finite element formulation by mearing out onrete and reinforement lattie into a ontinuum. The main harateriti of the lattie model i the poibility to hange the diretion of the loal oordinate (or the inlination angle of the lattie omponent) in proportion to the progre of frature in the onnetion between the tre field of the global and loal oordinate ytem. Thi model provide the freedom to hange the inlination angle of longitudinal reinforement lattie omponent. Thu it i poible to evaluate the dowel effet by ontrolling the inlination angle of the reinforement lattie omponent. In addition, it i poible to evaluate the reiting mehanim of tranvere reinforement, whih may not be vertial. With regard to the reiting mehanim of onrete, the hear tranfer of the rak urfae an be evaluated not by uing the hear tretrain relationhip, but by uing the hear lattie. For eah rak urfae, two hear lattie S and S are provided. The hear fore along a rak i arried by thee hear lattie. The effet of aggregate interlok i dependent on the relative movement of onrete on two ide of the rak and thu S and S will beome ative a hown in Figure 4. In thi hear tranfer mode, the roughne of the rak urfae i repreented by the roughne angle θ of the rak urfae. Itoh et al. (000) uggeted a value θ = π/3 for NSC baed on a omprehenive tudy omparing the lattie model and the panel tet reult of Vehio & Collin (986). For HSC with moother rak urfae, a maller value of θ hould be Figure 4. Modelling hear ontat area with hear lattie 7 CONCLUSIONS AND RECOMMENDATIONS Baed on the aement of the theoretial model and AS3600 for retangular beam under pure torion, the following an be onluded:. A higher rak load and higher torional apaity for a given ro-etion are obtained uing hightrength onrete both from experimental and theoretial model.. Among the three theoretial model ued, the pae tru model with palling of the onrete over and oftened tru model give the bet etimate of the ultimate torional trength of tet beam. 3. The AS give an unonervative predition for the ultimate torional trength of both normal and high-trength RC beam.

7 4. HSC provided higher torional trength than NSC for beam deigned with the ame amount of reinforement. Further reearh effort remain to be implemented to aurately predit the torional behaviour of high-trength reinfored onrete beam, by taking into aount the effet of redued aggregate interlok hear tranfer mehanim due to the mooth rak urfae in HSC. An advaned method of analyi baed on the lattie model an be developed to arry out a rational analyi for torion up to the ultimate range, to inlude the important apet of aggregate interlok and dowel ation, whih have been ignored in the pat for torional reitane. REFERENCES AS Conrete Struture, Standard Aoiation of Autralia, Autralia. Attard, M. M., & Mendi, P. A Dutility of High- Strength Conrete Column. Autralian Civil Engineering Tranation, The Intitute of Engineer, Autralia, Vol. CE35, No. 4, Otober 993, pp , Collin, M. P., & Mithell, D Shear and Torion Deign of Pretreed and Non-Pretreed Conrete Beam. PCI Journal, 5(5), Fang, I. K., & Shiau, J. K Torional Behavior of Normal- and High-Strength Conrete Beam. ACI Strutural Journal, 0(3), Hoain, T Shear Behaviour of High-Strength Conrete Beam at Low and High Strain-Rate. PhD Thei, Department of Civil and Environmental Engineering, Univerity of Melbourne. Autralia, 89 pp. Hu, T. C Torion of Strutural Conrete- Behaviour of Reinfored Conrete Retangular Member. Torion of Strutural Conrete, SP-8, Amerian Conrete Intitute, Farmington Hill, Mih., Hu, T. C Diuion of Pure Torion in Retangular Setion-A Re-examination by MMullen, A. E. and Rangan, B. V. Journal of the Amerian Conrete Intitute, Pro., 76(6), Hu, T. C Torion of Reinfored of Conrete. Van Notrand Reinhold, New York, USA, -56. Hu, T. C., & Mo, Y. L Softening of Conrete in Torional Member-Theory and Tet. ACI Journal, 8(3), Hu, T. C., and Mo, Y. L. (985). Softening of Conrete in Torional Member-Deign Reommendation. ACI Journal, 8(4), Hu, T. C Softened Tru Model Theory for Shear and Torion. ACI Strutural Journal, 85(6), Itoh, A., Ihtiaq, A. S., & Tanabe, T Expreion of the Compreion Field Theory Baed on the Lattie Equivalent Continua Model. Tranation of the Japan Conrete Intitute,, Kouthoukali, N. E., & Belarbi, G. 00. Torion of High- Strength Reinfored Conrete Beam and Minimum Reinforement Requirement. ACI Strutural Journal, 98(4), Leig, N. N Determination of Load-Carrying Capaity of Retangular Reinfored Conrete Element ubjeted to Flexure and Torion. Conrete and Reinfored Conrete Intitute, Trudy No. 5, Moow, 5-8. Mithell, D., & Collin, M. P Diagonal Compreion Field Theory-A rational Model for Strutural Conrete in Pure Torion. ACI Journal, Pro., 7(8), Ngo, T. D Behaviour of High Strength Conrete ubjet to Impulive Loading. PhD Thei, Department of Civil and Environmental Engineering, Univerity of Melbourne. Autralia, 33 pp. Niwa, J., Choi, I. C., & Tanabe, T Analytial Study for Shear Reiting Mehanim uing Lattie Model. JCI International workhop on Shear in Conrete Struture, Japan, Ramuen, L. J., & Baker, G Aement of Torional Strength in Reinfored Normal and High-Strength Conrete Beam. Autralian Civil Engineering Tranation, IEAut., CE36(), Ramuen, L. J., & Baker, G. 995a. Torion in Reinfored Normal and High-Strength Conrete Beam-Part : Experimental Reult. ACI Strutural Journal, 9(), Ramuen, L. J., & Baker, G. 995b. Torion in Reinfored Normal and High-Strength Conrete Beam-Part : Theory and Deign. ACI Strutural Journal, 9(), Rauh, E. 99. Deign of Reinfored Conrete in Torion. PhD Thei, Tehnihe Hohhule, Berlin, 53 pp. Reid, S. G., & Bridge, R. C Shear and Torion in Reinfored Conrete Beam. National Conferene Publiation, Intitute of Engineer, Autralia, 90(0), Vehio, F. J., & Collin, M. P Modified Compreion Field Theory for Conrete Element Subjeted to Shear. ACI Journal, Proeeding, 83(), 9-3.