ULTIMATE FLEXURAL BEHAVIOUR OF EXTERNALLY PRESTRESSED NEW BEAMS AND DISTRESSED BEAMS

Transcription

1 Journal o Engineering Siene and Tehnology Vol. 10, No. 4 (2015) Shool o Engineering, Taylor s University ULTIMATE FLEXURAL BEHAVIOUR OF EXTERNALLY PRESTRESSED NEW BEAMS AND DISTRESSED BEAMS R. SENTHIL 1, R. MANISEKAR 2, * 1 Strutural Engineering Division, Anna University, Chennai , India 2 CSIR-Strutural Engineering Researh Centre, Taramani, Chennai, India *Corresonding Author: rmanisekar17@yahoo.o.in Abstrat External restressing has beome a oular tehnique in retroitting o distressed onrete strutures, mainly bridges. It is being suessully used in segmental onstrution o new bridges, whih is evident in metro onstrutions and other tyes o bridges. Tendons used in external restressing are unbonded tendons and thereore they are analytially treated as internal unbonded tendons. Stress at ultimate in unbonded tendons is the arameter used to evaluate the ultimate lexural aaity o a onrete member restressed with unbonded tendons. As ar as new members are onerned, suiient researh works were arried out on rediting the ultimate lexural behavior and develoing equations or inding. Thereore, an in deth review has been done on the subjet and the behavioural mehanism regarding evaluating ultimate lexural aaity are disussed. As ar as distressed onrete members are onerned, an analytial model has been develoed or a raked RC beams strengthened by external restressing and validated with exerimental data, and the results are resented and disussed. The dierene in behavior o externally restressed new beams and distressed beams strengthened by external restressing are disussed. Keywords: Analysis, RC beams, Flexure, Strengthening, External restressing. 1. Introdution External restressing is being used widely or new onstrutions and retroitting o distressed onrete strutures. Moreover, rovisions or external tendons (in terms o deviators and anhorages monolithially) are given in internally restressed members to retroit, i any distress our in uture. As ar as retroitting is 461

2 462 R. Senthil and R. Manisekar Nomenlatures C C sht dlt d n e e new ak k e u ys t y I I rak I tr I tr L b L t L t1 L t2 M M rak M r M P S bot, S to T t w w d w dk w a y y n y Resultant omression Shit in osition o resultant omression Change in length o draed tendons due to deletion Deth o rak ti Eentriity New eentriity Stress in external tendons at deomression Cylinder omressive strength Eetive restress in external tendons Stress in external tendons at ultimate Ultimate strength o restressing steel Stress in external tendons at yielding o untensioned steel Tensile strength o onrete Yield stress o steel Moment o inertia o unraked setion Moment o inertia o raked setion Moment o inertia o transormed setion or RC member Moment o inertia o transormed setion or externally restressed member Total beam length Total tendon length o draed tendons in between anhoring ends (onsidering straight) Total tendon length without deletion (onsidering single drae at entral deviator) Total tendon length with deletion (onsidering single drae at entral deviator) Moment o RC member Craking moment o RC member Craking moment o externally restressed member Moment o externally restressed member Eetive restressing ore Stress in externally restressed member at bottom and at to iber Tensile ore in tendon beore alying load Thikness o end late Alied load Load due to sel weight Load at deomression Load at urther distress Deletion o RC member Deth o neutral axis Deletion o externally restressed member Greek Symbols χ Curvature at mid san setion o RC member ε 1, ε 2 Prinial tensile and omressive strains resetively ε Strain in onrete at bottom most iber o RC member ε rak Craking strain in onrete Strain in tendon beyond eetive restressing stage ε T Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

3 Ultimate Flexural Behaviour o Externally Prestressed New Beams and onerned, external restressing is one o the major strengthening tehniques, being alied or distressed onrete bridges. Aart rom bridge ield, it is being used or retroitting o industrial strutures also. However, onrete members strengthened by external restressing behave in a dierent manner omared to new members artiularly with regard to ultimate lexural behaviour. Formation o lasti hinge, stress-inrease in external tendons beyond eetive restress, omatibility issues in the analysis and other issues need detailed investigation, whih is observed rom the literature review, resented in Setion 2.0. Analytial solution has been identiied or externally restressed new onrete members. In the resent study, an analytial model has been develoed or rediting ultimate lexural behavior o distressed onrete beams strengthened by external restressing, and the results are disussed. 2. Literature Review 2.1. Stress in unbonded tendons o externally restressed new onrete members Tendons used in externally restressed onrete members are analytially being treated as unbonded tendons, and thereore analytial solutions o members restressed with internal unbonded ost-tensioning tendons are alied or externally restressed members. Various studies were arried out or the ast seven deades on rediting stress at ultimate in unbonded tendons,, as it is used or evaluating ultimate lexural behavior o a onrete member restressed with unbonded tendons. Naaman and Alkhairi [1] erormed a state o-the-art review and summarised that stress at ultimate in unbonded tendons is bounded by eetive restress e and yield strength o the restressing tendons y. Subsequently, Naaman and Alkhairi [2] roosed an equation to redit in unbonded tendons, whih is untion o the bond redution oeiient Ω u and the ratio o the neutral axis at ultimate to the deth o the restressing steel /d. Aariio and Ramos [3] erormed an analytial study on 74 externally restressed onrete bridges using non-linear Finite Element Model, and suggested onstant values o or inororation in odes. Ng [4] roosed a modiied bond redution oeiient to redit, whih aounts or tye o loading and seond-order eets but indeendent o san-deth ratio. Die et al. [5] develoed an equation o able strain on the basis o deormation omatibility, whih inororates the ritional resistane at deviators. Die et al. [6] redited the strutural behaviour o externally restressed onrete beams uto the ultimate state using Finite Element algorithm. Harajli et al. [7] evaluated seond order eets or externally restressed onrete new members and suggested that the seond order eet is mainly inluened by the oniguration o deviators, roile o the tendons and the magnitude o the inelasti deletion under ailure load. Rao and Mathew [8] resented an analytial method or externally restressed onrete beams, whih takes into aount the seond order eets and rition at deviators. Manisekar and Rao [9] resented a review on behaviour o various tyes o deviators, whih are emloyed in beams and box girders inluding reinorement details, and suggested that tendon sli, ritional resistane, Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

4 464 R. Senthil and R. Manisekar seond order eets and ultimate strength are inter related. Aravinthan et al. [10] erormed an exerimental study, in whih they used high eentriity or external tendons or simle and ontinuous girders and observed that the ontinuous girders with linearly transormed tendon roile exhibit the same lexural behaviour irresetive o tendon layout, and the resene o oninement reinorement enhanes the dutility behaviour. Manisekar and Senthil [11] arried out a state o-the-art review and arametri study on stress in internal unbonded tendons and external tendons, and suggested that is diretly related to ormation o lasti hinge, rovided equivalent lasti hinge length should be deined aurately. Harajli [12] roosed an exression or evaluating the equivalent lasti hinge length and roosed methods to redit or modiying equations (18-4) and (18-5) o ACI Building Code [13]. Ng and Tan [14] resented a eudo-setion analysis method or analysing a simly suorted externally restressed beams subjeted to two-oint load, based on the bond redution oeiient in strain omatibility by taking into aount o seond order eets. Kwak and Son [15] develoed an analytial method on the basis o relative dislaement between the deviators and the onrete matrix, and onluded that the additional rimary moment develoed by the laement o external tendons auses an inrease in ultimate resisting aaity. Au et al. [16] onduted an exerimental investigation to study the ost-eak behaviour o artially restressed beams with external tendons using either steel or Aramid FRP (Fibre Reinored Plasti) tendons as external tendons, and suggested that the reinoring index ω is a better indiator than PPR (Partial Prestressing Ratio). Pisani [17] develoed a numerial model to simulate the behaviour u to ollae o ontinuous onrete beams restressed with bonded and external tendons, by adoting a disretization rule to evaluate the rotation o lasti hinges. Based on his model, he suggested that ater yielding o the bonded tension reinorement, urvature is onstant along a disrete element o the onrete struture whose length is equal to the develoment length o that reinorement. He and Lao [18] roosed a design equation or stress at ultimate in external and internal unbonded tendons using linear relationshi between stress inrease in unbonded tendons at ultimate and mid san deletion. Du et al. [19] tested onrete beams artially restressed with CFRP (Carbon Fibre Reinored Plasti) external tendons and onluded that roer ombination o external CFRP tendons with internal reinoring steel an imart dutility to the beams. Bennitz et al. [20] tested externally restressed RC beams with either steel or CFRP tendons by two oint monotoni load. They omared their test data with the model o Tan and Ng [21] and Ng [4] and onluded that beams restressed with CFRP tendons have inreased the strength, stiness and ailure load, but have dereased the dutility relative to unstrengthened beams. Researhers, working on external restressing have adoted the internal unbonded tendon mehanism or analysis and redited the behaviour. The aroahes they adoted are moment-urvature relationshi, emirial methods, strain redution oeiient method, non-linear Finite Element Method and lasti hinge length aroah. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

5 Ultimate Flexural Behaviour o Externally Prestressed New Beams and The works are Naaman [22], Tan and Ng [21], Harajli et al. [7], Tan et al. [23], Die et al. [5], Die et al. [6], Ng [4], and Ghallab and Beeby [24]. However, all the equations dislayed unsatisatory erormane in rediting stress in unbonded tendons when they omared with exerimental results. Lak o auray in evaluating equivalent lasti hinge length is the reason or the unsatisatory erormane o the equations. The equivalent lasti hinge length has been diretly or indiretly involved in all the analytial methods mentioned. Thereore, the redition equations develoed using lasti hinge length aroah is identiied or reviewing, and disussed in the Setion Distressed onrete members strengthened by external restressing Harajli [25] tested onrete beam seimens whih were earlier raked and then strengthened by external restressing, and observed that lexural strength o the strengthened members were inreased by 146% and indued deletions were redued by 75%. Ghallab and Beeby [26] tested twelve restressed onrete beams strengthened by external tendons using G Parail roe, out o whih three were raked rior to strengthening. They suggested that i the internal reinoring steel does not yield at reraked stage, then the strengthened member at ultimate an be analysed as same as unraked strengthened beams. Another investigation o the same kind is done by Elreai et al. [27], in whih RC beams were reraked at stages: overloaded (internal steel is yielded) and non overloaded (internal steel is not yielded), then they were strengthened by external restressing using Carbon-Fibre Reinored Polymer Tendon. They observed that beams overloaded rior to strengthening did not have disernable eet on the beam atigue lie in omarison with that o non overloaded beam. Antony and Mohankumar [28] investigated the reliability o external restressing on strengthened beams using only straight tendon roile, whih were earlier damaged by orrosion, and observed that stiness inreased in strengthened beams omared to undamaged beam. Sirimontree and Teerawong [29] tested a ull-sale restressed onrete highway bridge girder with RC dek slab, whih was earlier raked u to the level o inelasti raked stage or two times and then strengthened by external restressing, and ound that the required external restressing ore to reover strutural erormane o a damaged girder deends diretly on the damage index, whih is the ratio o ermanent deormation to the rak deormation o the reerene undamaged girder. From the review, it an be understood that distressed onrete members strengthened by external restressing behave dierently in ultimate lexural behaviour eseially in ourrene o lasti hinge ater yielding o reinoring steel. Comatibility between deormation o onrete and strain in external tendons needs to be introdued, and ultimate lexural behavior o them needs to be redited. In view o this, an analytial model has been develoed to redit the ultimate lexural behaviour o distressed RC beams strengthened by external restressing and validated using ublished data, whih is resented in Setion 4. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

6 466 R. Senthil and R. Manisekar 3. Plasti Hinge Length Aroah It is well known that ultimate lexural behaviour o onrete members restressed with unbonded tendons are assessed by the stress at ultimate in unbonded tendons, or whih researhers have been using ACI orm as ollow: = + (1) e Pannell [30] onduted analytial and exerimental studies on beams without nonrestressing steel and roosed the Eq. (3) or the stress in the restressing steel at ultimate, by onsidering strain omatibility and equilibrium. He made ollowing assumtions: 1) the beam remains in the elasti range uto ailure, exet in lasti zone; 2) setions lane beore bending remain lane during bending; and 3) the strain in the onrete at the steel level is negligible in the elasti zone. He has onsidered the onrete strain in the lasti zone and assumed the ollowing: l ε = (2) L and the roosed equation is: q u = ' MPa (3) ρ q + λ where, q = e u and q 1+ 2λ e A e ψρ ε u E d =, λ =, ψ = 10, L = L o =10, bd ' L where, ε is hange in strain in the onrete at the level o the restressing steel, L is length o the lasti zone at ultimate, l is onrete elongation at the level o the restressing steel that measured within the length o the lasti zone, is deth o the neutral axis at ultimate, and ε u is strain in the onrete to iber at ultimate. Tam and Pannell [31] have made an exerimental study on artially restressed onrete beams with the san-to-deth ratio ranged rom 20 to 45. The main arameters taken or study were, the initial eetive restress in the tendon, amount o restressing and nonrestressing steel, the san-to-deth ratio and the initial eetive restress. They observed that all beams develoed ine raks similar to those ontaining bonded reinorement. Based on their observations, they modiied the Eq. (3) and roosed the Eq. (4) or omuting or retangular setion, by taking into aount the eet o sulementary nonrestressing reinorement. qe + λ qsλ λ 1+ α + λ α = ρ where, q e A e =, q bd ' s MPa (4) As y =, λ = ψr ε uesd / L u, L = L o =10.5, bd ' Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

7 Ultimate Flexural Behaviour o Externally Prestressed New Beams and where α = 0.85β (based on ylinder omressive strength) or α = 0.68β 1 (based on ube omressive strength), β 1 is the stress blok redution ator deined in the ACI Building ode. Gauvreau [32] and Gilliland [33] demonstrated that deormation in the onrete due to shear at the level o the tendon is small in omarison to the deormation due to lexure. Thereore, the ontribution o deormation to the inrease in tendon stress at eah o the high moment zones, ould redit a reasonable total stress inrease in tendon. Lee et al. [34] desribed a omutational method o the unbonded tendon stress at the lexural ailure o a member. They roosed a new equation or, with the onsideration o the strain omatibility-moment equilibrium and the lasti hinge length. They derived the equation in suh a way that: 1) by obtaining main arameters and their ombinations with theoretial study; and 2) determining the oeiients o arameters by regression, using revious test results. They deined the lasti hinge length, as a untion o loading tye and San/deth ratio together, in the suort o strain omatibility. Finally they roosed the ollowing: = 10, se ( A A ) s A s y d s + 80 d 1 + ρ L Psi (5) d 1 / in the limit o se + 10, 000 y Au and Du [35] reinvestigated the equivalent lasti hinge length (L o ) o Pannell s equation and roosed the value o the arameter ϕ as 9.3 ater doing data analysis. They urther suggested that the value ϕ = 9.3 ould ensure, about 84% o (redited) values to be on sae side. Finally they adoted the Pannell s aroah and roosed the ollowing: = e where e E + l A = e 0.85β1 ( d ) e ' s + A b y e y y MPa (6) Based on the above works, Manisekar and Senthil [11] arried out a arametri study, relating the arameter with equivalent lasti hinge length L o on the assumtion: Prestressed onrete members do not exeriene raks beore the yielding o untensioned reinorement. Beore ormation o raks, tendon stress also does not inrease. Further, ormation o lasti hinge starts ater yielding o the untensioned reinorement till the onrete rushing at the extreme omressive iber o the member. The work is given below: In Fig. 1 Pannell s [30] lasti hinge length ϕ (i.e., L o = ϕ) and the simle arameter o lasti hinge length 1.5 d (i.e., L o = 1.5 d ) is shown. Seven test results omrise o retangular setions and T-beam oniguartions used are shown in the legend with the variation o loading tye (see Fig. 1). Data o single oint load is mentioned as =10, and the data o two oint load is mentioned as =3. Equivalent lasti hinge length, ϕ and 1.5 d were alulated based on the data, shown in the legend. It is shown in Fig. 1 that three values are orrelated exatly. Further, the ϕ and (Data) by the exerimental tests (shown in the Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

8 468 R. Senthil and R. Manisekar legend) were omared, as shown in Fig. 2. Here, exatly the same trend o orrelation is obtained as in Fig. 1. It is reorted that the lasti hinge length an be diretly related to the. Then, Chakrabarti s [36] data (two oint loads) was used to see the erormane o all the redition equations, using L o o Lee et al. [35], and L o o Harajli and Hijazi [37], whih are shown in Figs. 3 and 4 resetively. It is seen that no orrelation is ossible. 1.5 d-in φ-in Tam and Pannell (1976) (=10) Pannell (1969) (=10) Du and Tao (1985) (=3) Harajli and Kanj (1991) (=3) Harajli and Kanj (1991) (=10) Chakrabarti (1995) (=3) Cooke et al. (1981) (=3) Fig. 1. Comarison o 1.5 d with ϕ o Pannell [30]. (Data)-MPa φ-mm Tam and Pannell (1976) (=10) Pannell (1969) (=10) Du and Tao (1985) (=3) Harajli and Kanj (1991) (=3) Harajli and Kanj (1991) (=10) Chakrabarti (1995) (=3) Cooke et al. ( 1981) (=3) Fig. 2. Comarison o (data) with ϕ o Pannell [30]. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

9 Ultimate Flexural Behaviour o Externally Prestressed New Beams and Tam and Pannel (1976) Pannell (1969) Du and Tao (1985) (redited)-mpa Chakrabarti (1995) 800 Harajli and Kanj (1991) Harajli (1990) 400 Harajli and Hijazi (1991) ACI Lee et al. (1999) Au and Du (2004) -400 Lo-mm Fig. 3. Comarison o (Predited) with L0 o Lee et al. [34] (using data o Chakrabarti) Tam and Pannell (1976) (redited)-mpa Pannell (1969) 1200 Du and Tao (1985) Chakrabarti (1995) 800 Harajli and Kanj (1991) Harajli (1990) 400 Harajli and Hijazi (1991) ACI Lee et al. (1999) Au and Du (2004) -400 Lo-mm Fig. 4. Comarison o (redited) with L0 o Harajli and Hijazi [37] (using data o Chakrabarti). Seondly, the data o Tam and Pannell [31] (subjeted to single oint load) was taken and made omarisons with L o o both Lee et al. [34] and Pannell s [30] ϕ, are shown in Figs. 5 and 6 resetively. Here a big imrovement is there in orrelation when omared to Figs. 3 and 4. From these omarisons, it an be observed that the inauray in evaluating equivalent lasti hinge length L o is the reason or the unsatisatory erormane o all the redition equations, whih were used or omarison. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

10 470 R. Senthil and R. Manisekar (redited)-mpa 800 Tam and Pannell (1976) Pannell (1969) Du and Tao (1985) 600 Chakrabarti (1995) Harajli and Kanj (1991) 400 Harajli (1990) Harajli and Hijazi (1991) ACI Naaman and Alkhairi (1991) Lee et al. (1999) Au and Du (2004) φ.-mm Fig. 5. Comarison o (Predited) with L0 o Lee et al. [34] (Using data o Tam and Pannell). (redited)-mpa 800 Tam and Pannell (1976) Pannell (1969) 600 Du and Tao (1985) Chakrabarti (1995) Harajli and Kanj (1991) 400 Harajli (1990) Harajli and Hijazi (1991) ACI Naaman and Alkhairi (1991) Lee et al. (1999) Au and Du (2004) Lo-mm Fig. 6. Comarison o (redited) with L0 o Pannell [30] (Using data o Tam and Pannell). L o Harajli [12] evaluated an equation or equivalent lasti hinge length L o as 20.7 = and roosed his equation or as ollows = e ρ sd s y + K 0E ε u d 0.85β 1 K 0E ε u ρ d β 1 (7) Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

11 Ultimate Flexural Behaviour o Externally Prestressed New Beams and Thereore, it is observed rom the works o Pannell [30], Tam and Pannell [31], Gauvreau [32] and Gilliland [33], Lee et al. [34], Au and Du [35] and Manisekar and Senthil [11] (rom Figs. 1 to 6), that stress inrease in external tendons or externally restressed new beams,, is diretly related to the lasti hinge length, whih in the region: rom yielding o untensioned reinorement to onrete rushing in the extreme omressive iber. However, this onlusion ignored the arameters suh as seond order eets and sli onditions at deviators o externally restressed members. 4. Modeling on Strengthened RC Beams by External Prestressing In the resent study, distressed beams onsist o retangular RC beams were onsidered or modeling. They were raked and strengthened by external restressing using single draed tendons. RC beams were raked to a limit that strain in onrete at extreme iber varies rom to , and the strain in reinoring steel in the limit varies rom to That means the reinoring steel was not yielded. An analytial model was develoed and validated with seimens B4D and B5D o exerimental data o Harajli [25] Comression sotening The deterioration in omression resistane exhibited by the onrete, due to raking is generally alled as omression sotening. Sine the RC member was raked earlier, the strength o onrete in omression region would have redued due to raking, and thereore, strength redution o onrete in omression was evaluated. Based on the modiied Thoreneldt base urve (Fig. 7), Vehio and Collins [38] suggested the omression sotening oeiient β as ollow: 1 β = (8) 1 + where K ε 1 K 0.27 = 0.37 (9) ε 0 ' ε 0 = (10) E where ε 0 is strain in onrete ylinder at eak stress ', and modulus o onrete. E is elasti ε 1 is rinial tensile strain, omuted by reerring Modiied Comression Field Theory (Vehio and Collins [39, 40] as ollows: 2 ε 1 ε ε ε v = (tanθ + otθ )( ε 1) ot θ (11) x x Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

12 472 R. Senthil and R. Manisekar where ε x = longitudinal strain at mid deth o the member (web) in shear region, whih was taken as 0.002, ermitted by Canadian ode [41]. ' ε = strain in onrete ylinder at eak stress ' where shear stress ratio V V v = or v b jd = λφ ' b d (12) wl W V = d = lever arm, not less than 0.9 d v φ = material resistane ator w λ = ator to aount or density o onrete or normal density o onrete λ =1.00 or semi-low onrete λ =0.85 or low onrete λ =0.75 θ = inlination o the rinial omressive stresses whih was alulated using the relation, shear stress ratio (ν)-angle o rinial omressive stresses (θ) (Colins and Mithell, [42]), based on longitudinal strain at mid deth, ε x. Conrete strength redution arameter β are alulated as 0.88 and 0.82 or B4D and B5D seimens resetively. v v - 2 β = n(ε 1 ) Thoreneldt Base Curve β ε 0 -ε 2 Fig. 7. Modiied thoreneldt base urve-soure o omression sotening oeiient Analysis o strengthened RC beams by external restressing (ost-strengthening) Seimen B4D and B5D were having single draed tendons, and deviator was rovided at the soit o the mid san. Normally, external tendons are also treated analytially as unbonded tendons, and thereore analytial status o any member restressed with external tendons is member deendant and not setion deendant [11]. In this investigation, omatibility between deletion at the Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

13 Ultimate Flexural Behaviour o Externally Prestressed New Beams and entral deviator loation and strain inrease in external tendons beyond eetive restress, was alied. Analysis o externally restressed raked RC beams was erormed by alying ore onet [43]. At the stage o eetive restress, tensile ore oered by Tendon, P, and the omressive ore oered by the resultant omression are in same osition. When the moment due to sel weight and live load ating uon the member, M, the resultant omression shit rom the osition o tensile ore oered by tendons, M /P times. Based on the shit in osition o resultant omression, eentriity will go on varying. Shit in the osition o resultant omression = C sht When there was no any load and no inrease in tendon stress, the resultant omressive ore and tendon ore were same, whih was equal to eetive restressing ore. Thereore, there was no shit in resultant omression line, whih is as ollows: When C = 0 sht C = T (13) when the osition o the resultant omression lies above the tendon line and below the original eentriity, i.e., 0 C sht < e the distane o the loation o resultant omression rom the enter line o the member (h/2), i.e., e new was omuted as e = e (14) new C sht stress in onrete member at to iber due to external restressing omuted as S to t S to was P P( e Csht ) = (15) A Z Similarly, stress in onrete member at bottom iber due to external restressing S was omuted as S bot bot P P( e Csht ) = + (16) A Z b when the osition o the resultant omression is exatly at enter line o the member (h/2), i.e., when C sht = e, then the stress in onrete member at to iber due to external restressing S to was omuted as P S to = (17) A Sine the C sht is exatly at the entre o the member e new has beome zero. Thereore, the seond term o Eq. (15) also has beome zero. Similarly, stress in onrete member at bottom iber due to external restressing S bot was omuted as Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

14 474 R. Senthil and R. Manisekar P S bot = (18) A It was assumed on the basis o exerimental data (Harajli [25], whih was used or validation) that the raks losed due to external restressing will oen only ater deomression taking lae, and the deletion will start ouring only ater raks oening (i.e., ater deomression). Thereore, stressinrease will our only ater deomression takes lae sine stress-inrease stritly bounds by deletion o the member (as deletion omatibility ontrols the analysis). Moment o inertia or original setion I was used or the analysis o externally restressed (strengthened) member at stage beore deomression, as there is no rak oening beore deomression. Whereas Moment o Inertia or transormed setion I tr was used or the analysis o externally restressed (strengthened) member at stage ater deomression, as there is rak oening ater deomression. The ollowing analysis was done based on the above assumtion: when the osition o resultant omression is above the enter line o the member, i.e., 0 e C sht, the deomression used to haen, and thereore tension reates at bottom iber o the onrete member and omression reates at to iber o the onrete member. S Thereore, P P( C e) y sht t to = + (19) A I tr Similarly, stress in onrete member at bottom iber due to external restressing S bot was omuted as S bot P P( Csht e) yb = (20) A I tr Deletion o strengthened raked RC beams When the alied load is suh that the member does not ome to the deomression stage, i.e., 0 < w w y d 2 PenewL = (21) 12E I When the alied load is suh that the member reahed the deomression stage, but not reahed the urther distressing stage, i.e., w > w w (Reason or using the term urther distress is exlained in setion 4.2.3) Deletion o the externally restressed raked RC beams y was omuted as y Pe L M M M w L 2 4 new 2 d d d = + kl + + (22) 12Eφ1I tr EI 0.85Eφ1I tr 384 Eφ1I tr 5 d d Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

15 Ultimate Flexural Behaviour o Externally Prestressed New Beams and When the alied load is suh that the member is reahed deomression stage and also the urther distressing stage, i.e., w w d > w then the deletion o the externally restressed raked RC beams y was omuted as y Pe L M M M w L 2 4 new 2 d d d = + kl + + (23) 12Eφ2I tr EI 0.85Eφ2I tr 384 Eφ2I tr where I tr M = M d 3 M I + 1 M 3 n d 3 I rak b. d 2 I rak = + m. Ast ( d d n ) 3 Φ 1 = redution ator or moment o inertia o transormed setion or the stage rom load at deomression to load at urther distress. Φ 2 = redution ator or moment o inertia o transormed setion or the stage rom load at urther distress to the load at ultimate. 5 d Stress in external tendons For seimen B4D and B5D, deletion omatibility was alied or omuting stress in external tendons at ultimate as ollow: Total length o tendon in between anhorages (onsidering straight), as shown in Fig. 8(a) L = L + 2t (24) t b Total length o tendon without deletion (onsidering single drae at entral deviator), as shown in Fig. 8(a) 2 2 Lt t1 = 2 e + L (25) 2 when t 2 t1 w < wd L = L (26) Total length o tendon with deletion (onsidering single drae at entral deviator), shown in Fig. 8(b) is alulated as ollows: when w wd 2 2 Lt t 2 = 2 ( e + y ) + L (27) 2 t Then the hange in length o tendon is omuted as: dl = L L (28) t 2 t1 Strain in tendons is omuted as: Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

16 476 R. Senthil and R. Manisekar dl t ε t = (29) Lt1 Stress-inrease in external tendons is omuted as: = ε E (30) where t ε t is omuted using equations rom Eqs. (24) to (29) Thereore, stress at ultimate in external tendons is omuted, whih is in ACI orm (ACI ), as ollow: = + (31) e w w L t e L t 1 2 h L b (a) Without deletion w w y L t 2 2 (b) With deletion Fig. 8. Strengthened beam with draed tendons. Stress-inrease in external tendons or externally restressed raked RC beams was observed at the stage rom deomression stage to yielding o untensioned reinorement. Thereore, stress in external tendons at ultimate state was omuted using the Eq. (31). Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

17 Ultimate Flexural Behaviour o Externally Prestressed New Beams and Validation with exerimental data Seimens B4D and B5D o data o Harajli [25] were used or validation. Material roerties or seimens B4D and B5D are given in Table 1. Cross setional details or seimens B4D and B5D are shown in Figs. 9 and 10 resetively. For details o longitudinal view, Fig. 8 an be reerred. 127mm 229 mm 203 mm d e HTS 2-7 mm ϕ HTS Wire Fig. 9. Cross-setional details o the seimen B4D. 127 mm 203 mm d 229 mm e HTS 2-5 mm ϕ HTS Wire Seimens Fig. 10. Cross-setional details o the seimen B5D. Table 1. Material roerties or seimens B4D and B5D. No. o Bars B4D 2-10 mm B5D 3-12 mm Reinoring steel Area o steel mm 2 Yield stress o steel MPa No. o wires mm mm External restressing steel Area o restre Eetive ssing restress steel e A Comressive strength o onrete MPa Ultimate strength o restressing steel MPa u mm 2 MPa The tendon roile is a draed one and the deth o external tendons at mid san, d e, is 273 mm. The thikness o end late is 19 mm, whih is a taered one. In both the seimens, untensioned steel was not allowed to yield, beore strengthening. Based on the exerimental data, external restressing was done Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

18 478 R. Senthil and R. Manisekar when the beams were subjeted to 30% o the alulated ultimated load. Further the seimens were subjeted to monotoni load, ater strengthening by external restressing. Shit o neutral axis towards bottom due to external restressing, whih is named here as deth o rak ti rom extreme omressive iber, d n, was omuted by regression method. The d n is 70 mm and 105 mm or seimens B4D and B5D resetively. Comarison o results with exerimental data or seimen B4D (ext. re) and B5D (ext. re) are urnished in Tables 2 and 3 resetively. It was observed that deomression in externally restressed raked beams ourred at 24% and 24.9% o ultimate load or B4D and B5D resetively, and yielding o untensioned reinorement ourred at ultimate load, or both seimens. It was ound generally or both seimens (B4D and B5D) that the stress-inrease in external tendons,, is at stage rom deomression to yielding o untensioned reinorement. Seondly, It was also observed that strengthened member exhibited three stages o behaviour viz., i) rom the eetive restressing stage to the load at deomression, w d ; ii) rom the load at deomression w d to load at urther distress w d ; and iii) rom the load at urther distress w d to load at ultimate w u. Amount in deletion was divided into two stages ater deomression, and thereore the stage ater deomression was named urther distress as it showed large deletion. Table 2. Comarison o results with exerimental data o strengthened raked RC beams by external restressing or seimen B4D (ext.re). Alied load w kn Stressinrease in external tendons Stressinrease in external tendons (ex) Ratio (ex) (model) Deletion y (model) mm Deletion y (ex) mm Ratio y (ex) y (model) (model) MPa MPa Mean Standard deviation Coeiient o variation 16.77% 16.89% It was sensed rom the iterations and results that there was neessity to ontrol the deletion in terms o moment o inertia or transormed setion, I tr, so as to redit the stress in external tendons, sine deletion omatibility ruled the analysis. Aordingly, redution ators or I tr or seond stage and third stage were introdued as ϕ 1 and ϕ 2 resetively. For seimen B4D, the redution ators ϕ 1 and ϕ 2 or I tr, were 0.82 and 0.72 resetively. For seimen B5D, the redution ators ϕ 1 and ϕ 2 were 1.00 and 0.76 resetively. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

19 KNm - Moment Ultimate Flexural Behaviour o Externally Prestressed New Beams and Moment-deletion urve and Moment- urve, are shown in Figs. 11 and 12 resetively or seimen B4D. Figures 13 and 14 show the Momentdeletion urve and Moment- urve or seimen B5D resetively. They show good agreement with exeriment results. Table 3. Comarison o results with exerimental data o strengthened raked RC beams by external restressing or seimen B5D (ext. re). Alied load w kn Stressinrease in external tendons (model) MPa Stressinrease in external tendons (ex) MPa Ratio (ex) (model) Deletion y (model) mm Deletion y (ex) Mm Ratio y (ex) y (model) Mean Standard deviation Coeiient o variation % 6.88% Moment-Deletion B4D (ext. re) Moment-kNm Ex Model Deletion-mm Fig. 11. Moment-deletion urve or seimen B4D (ext. re). Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

20 KN m - KNm - Momt knm R. Senthil and R. Manisekar 25 Moment- B4D (ext.re) Moment-kNm Ex Model Ma Fig. 12. Moment- urve or seimen B4D (ext. re). 60 Moment-Deletion B5D (ext.re) 50 Moment-kNm Ex Model Deletion-mm Fig. 13. Moment-deletion urve or seimen B5D (ext. re). Moment - B5D ( ext.re) Moment-kNm Ex Model MPa Fig. 14. Moment- urve or seimen B5D (ext.re) Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

21 Ultimate Flexural Behaviour o Externally Prestressed New Beams and Conlusions In summary the ollowing onlusions have been drawn: Stress inrease in external tendons or externally restressed new beams,, is diretly related to equivalent lasti hinge length, whih is in the stage: rom yielding o untensioned reinorement to onrete rushing in the extreme omressive iber. Stress-inrease in external tendons or strengthened RC beams,, is at stage rom deomression to yielding o untensioned reinorement, rovided the extent o the damage is limited that untensioned steel is not yielded. Externally restressed new beams ould be analysed using internal unbonded tendon mehanism, exet arameters assoiated with deviators. Aordingly, an be alulated using ACI orm (Eq. 1), in whih, is diretly related to equivalent lasti hinge length. Distressed beams strengthened using external restressing annot be analysed using internal unbonded tendon mehanism. Thereore, an be alulated using Eq. (30), in whih stress-inrease in tendons,, is at stage rom deomression to yielding o untensioned reinorement. Distressed beams strengthened using external restressing exosed three stages o behaviour ater attaining deomression. While externally restressed new beams dislay lasti hinge beore ailure, distressed beams strengthened using external restressing dislay very little lasti hinge and raise doubts or dutility. Aknowledgements Authors are grateul to Dr. Ing. N. Rajagoalan Chie Tehnial Advisor (Bridges) o L & T Ramboll Consulting Engineers Ltd, Chennai, India, and Dr. D. S. Ramahandramurthy, Prinial, St. Peter s College o Engineering and Tehnology, Chennai-54, India, or their valuable suggestions on this researh. Authors are thankul to Dr. Nagesh R Iyer, Diretor, CSIR-SERC, Chennai, or ermitting the aer or ubliation. Reerenes 1. Naaman, A.E.; and Alkhairi, F.M. (1991). Stress at ultimate in unbonded ost-tensioning tendons: Part 1 - Evaluation o the state-o-the-art. ACI Strutural Journal, 88(5), Naaman, A.E.; and Alkhairi, F.M. (1991). Stress at ultimate in unbonded ost-tensioning tendons: Part 2 - Proosed methodology. ACI Strutural Journal, 88(5), Aariio, A.C.; and Ramos, G. (1996). Flexural strength o externally restressed onrete bridges. ACI Strutural Journal, 93(5), Ng, C.K. (2003). Tendon stress and lexural strength o externally restressed beams. ACI Strutural Journal, 100(5), Die, B.K.; Tanabe, T.; and Umehara, H. (2001). Study on behaviour o externally restressed onrete beams using the deormation omatibility o Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

22 482 R. Senthil and R. Manisekar able. Journal o Materials, Conrete Strutures, Pavements, JSCE, 51(676), Die, B.K.; Umehara, H.; and Tanabe, T. (2002). Numerial analysis o externally restressed onrete beams onsidering rition at deviators. Journal o Materials, Conrete Strutures, Pavements, JSCE, 57(718), Harajli, M.H.; Khairallah. N.; and Nassi, H. (1999). Externally restressed members: evaluation o seond order eets. Journal o Strutural Engineering, ASCE, 125(10), Rao, P.S.; and Mathew, G. (1996). Behaviour o externally restressed onrete beams with multile deviators. ACI Strutural Journal, 93(4) Manisekar, R.; and Rao, M.V.B. (2003). Behaviour o deviators in external restressing. Indian Conrete Journal, 77(9), Aravinthan, T.; Withukreangkrai, E.; and Mutsuyoshi, H. (2005). Flexural behaviour o two-san ontinuous restressed onrete girders with highly eentri external tendons. ACI Strutural Journal, 102(3), Manisekar, R.; and Senthil, R. (2006). Stress at ultimate in unbonded osttensioning tendons or simly suorted beams- A state o-the-art-review. Advanes in Strutural Engineering, 9(3), Harajli, M.H. (2006). On the stress in unbonded tendons at ultimate: Critial assessment and roosed hanges. ACI Strutural Journal, 103(6), ACI Committee 318 (2008). Building ode requirements or reinored onrete. Amerian Conrete Institute, Detroit, Mihigan. 14. Ng, C.K.; and Tan, K.H. (2006). Flexural behaviour o externally restressed beams. art I: analytial model. Engineering Strutures, 28(4), Kwak, H.G.; and Son, J.K. (2010). Inelasti behavior o PSC beams with unbonded external tendons. Magazine o Conrete Researh, 62(5), Au, F.T.K.; Su, R.K.L.; Tso, K.; and Chan, K.H.E. (2008). Behaviour o artially restressed beams with external tendons. Magazine o Conrete Researh, 60(6) Pisani, M.A. (2009). Numerial analysis o ontinuous beams restressed with external tendons. Journal o Bridge Engineering, ASCE, 14(2), He, Z.; and Liu, Z. (2010). Stresses in external and internal unbonded tendons: uniied methodology and design equations. Journal o Strutural Engineering, ASCE, 136(9), Du, J.S.; Dong, Y.; Ng, P.L.; and Au, F.T.K. (2011). Resonse o onrete beams artially restressed with external unbonded arbon iber-reinored olymer tendons. Advaned Materials Researh Volumes, Trans Teh Publiations, Switzerland, Bennitz, A.; Jaob, W.S.; Jonny, N.; Björn, T.; Per, G.; and Dorthe, L.R. (2012). Reinored onrete T-beams externally restressed with unbonded arbon iber-reinored olymer tendons. ACI Strutural Journal, 109(4), Tan, K.H.; and Ng, C.K. (1997). Eets o deviators and tendon oniguration on behaviour o externally restressed beams. ACI Strutural Journal, 94(1), Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

23 Ultimate Flexural Behaviour o Externally Prestressed New Beams and Naaman, A.E. (1990). A new methodology or the analysis o beams restressed with external or unbonded tendons. External restressing in bridges, SP-120, Amerian Conrete Institute, Detroit, Tan, K.H.; Al Farook, M.A.; and Ng, C.K. (2001). Behaviour o simle-san reinored onrete beams loally strengthened with external tendons. ACI Strutural Journal, 98(2), Ghallab, A.; and Beeby, A.W. (2004). Calulating stress o external restressing tendons. Proeedings o the Institution o Civil Engineers, Strutures & Buildings, 157(SB 4), Harajli, M.H. (1993). Strengthening o onrete beams by external restressing. PCI Journal, 38(6), Ghallab, A.; and Beeby, A.W. (2002). Ultimate strength o externally strengthened restressed beams. Proeedings o the Institution o Civil Engineers, Strutures and Buildings, 152(4) Elreai, A.; West, J.; and Soudki, K. (2008). Eet o overloading on atigue erormane o reinored onrete-beams strengthened with externally osttensioned arbon-ibre-reinored olymer tendons. Canadian Journal o Civil Engineering, 35(11), Antony, J.C.; and Mohankumar, G. (2008). Strengthening by external restressing. Conrete International, 30(10), Sirimontree, S.; and Teerawong, J. (2010). Flexural behaviour o damaged ull-sale highway bridge girder strengthened by external ost tension. Amerian Journal o Engineering and Alied Sienes, 3(4), Pannell, F.N. (1969). The ultimate moment o resistane o unbonded restressed onrete beams. Magazine o Conrete Researh, 21(66), Tam, A.; and Pannell, F.N. (1976). The ultimate moment o resistane o unbonded artially restressed reinored onrete beams. Magazine o Conrete Researh, 28(97), Gaureau, D.P. (1993). Ultimate limit state o onrete girders restressed with unbonded tendons. Beriht nr.198, Institutur Baustatik und Konstruktion, ETH, Zurih, Switzerland Gilliland, J.A. (1994). Truss model or rediting tendon stress at ultimate in unbonded artially restressed onrete beams. M.S. Thesis. Queen s University, Kingston, Ontario, Canada. 34. Lee, L.H.; Moon, J.H.; and Lim, J.H. (1999). Proosed methodology or omuting o unbonded tendon stress at lexural ailure. ACI Strutural Journal, 96(6), Au, F.T.K.; and Du, J.S. (2004). Predition o ultimate stress in unbonded restressed tendons. Magazine o Conrete Researh, 56(1), Chakrabarti, P.R.; Whang, T.P.; Brown, W.; Arsad, K.M.; and Amezeua, E. (1994). Unbonded ost-tensioning tendons and artially restressed beams. ACI Strutural Journal, 91(5), Harajli, M.H.; and Hijazi, S.A. (1991). Evaluation o the ultimate steel stress in artially restressed onrete members. PCI Journal, 36 (1), Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)

24 484 R. Senthil and R. Manisekar 38. Vehio, F.J.; and Collins, M.P. (1993). Comression resonse o raked reinored onrete. ASCE Journal o Strutural Engineering, 119(12), Vehio, F.J.; and Collins, M.P. (1986). The modiied omression ield theory or reinored onrete elements subjeted to shear. ACI Journal, 83(2), Vehio, F.J.; and Collins, M.P. (1988). Prediting the resonse o reinored onrete subjeted to shear using modiied omression ield theory. ACI Strutural Journal, 85(3), CSA (Canadian Standard Assoiation) (1994). A Design o onrete strutures. Rexdale, Ontario, Canada. 42. Collins, M.P.; and Mithell, D. (1986). A rational aroah to shear design- The 1984 anadian ode rovisions. ACI Journal, 83(6), Rajagoalan, N. (2005). Prestressed onrete (2 nd Ed.). New Delhi: Narosa Publishing House. Journal o Engineering Siene and Tehnology Aril 2015, Vol. 10(4)