Shape Modification with Expansive Cement Concrete for Confinement with FRP Composites

Transcription

1 SP Shape Modcaton wth Expansve Cement Concrete or Connement wth FRP Compostes by Z. Yan, C.P. Panteldes, and L.D. Reaveley Synop nopss: s: To mprove the nnement eectveness o FRP mpostes or square and rectangular mns, shape modcaton s perormed by usng preabrcated FRP shells mbned wth expansve cement ncrete. Chemcal post-tensonng usng expansve cement ncrete s used to change the FRP nnement rom passve to actve. Expermental results are presented demonstratng the eectveness o ths method. An analytcal stress-stran model s developed or shape-moded FRP-nned mns wth expansve cement ncrete whch s based on the moded Wllam-Warnke plastcty model, the Popovcs general stress-stran ncrete model, and the dlatancy behavor obtaned rom the present study. Ths model s mplemented by an ncremental approach whch acunts or the varable FRP nnement durng the loadng process. The analytcal results show satsactory agreement wth the experments. Keywords: chemcal post-tensonng; nnement; expansve cement ncrete; FRP mpostes; stress-stran behavor 1047

2 1048 Yan et al. Zhan Yan s a structural engneer or Nshkan Mennger, San Francs, CA. He receved hs PhD rom Unversty o Utah, Salt Lake Cty, Utah. Hs research nterests ncde strengthenng o ncrete wth ber-renorced polymer mpostes, and structural dagnostcs and rehabltaton o renorced ncrete buldng and nstructon. ACI member Chrs P. Panteldes s a proessor o cvl and envronmental engneerng at the Unversty o Utah. He s a member o ACI Commttee 374, Perormance-Based Sesmc Desgn o Concrete Buldngs. Hs research nterests ncde sesmc desgn, evaaton, and rehabltaton o renorced ncrete buldng and brdge nstructon. ACI member Lawrence D. Reaveley s a proessor o cvl and envronmental engneerng at the Unversty o Utah. He s a member o ACI Commttee 369. Sesmc Repar and Rehabltaton. Hs research nterests ncde the areas o perormance-based structural engneerng and the rehabltaton o exstng structures. INTRODUCTION Fber Renorced Polymer (FRP) mpostes have been used n the retrot o ncrete mns to mprove ther capacty, dsplacement ductlty or both. It s wellknown that FRP mposte ackets can provde eectve lateral nnement or crcular ncrete mns, and substantally enhance ther axal strength and ultmate axal stran; however, FRP nnement s much less eectve or square and rectangular mns mpared to crcular mns. Ths observaton has been vered n tests perormed by Rochette and Labossère (000) and Pessk et al. (001). The reason or ths s that FRP mposte ackets resst axal loads by membrane acton, and are more eectve or crcular sectons as opposed to square or rectangular mn sectons wth rners and long lat sdes; stress ncentratons at the rners and necent nnement at the lat sdes cause loss o membrane acton o the FRP mposte and reducton o nnement. Because o the presence o steel tes, roundng o the rner radus n exstng square/rectangular mns s lmted. In addton, lower FRP nnement eectveness results n sotenng behavor or square and rectangular mns and the FRP mposte ruptures prematurely; thereore, the hgh strength o FRP mposte materals cannot be ully utlzed. A possble approach to ncreasng the eectveness o FRP-nned rectangular mns s to perorm shape-modcaton that s to mody the mn cross-secton nto an ellptcal, oval, or crcular secton. One method or perormng shape-modcaton s to use preabrcated (non-bonded) FRP mposte shells mbned wth expansve cement ncrete. For ths method, a preabrcated ellptcal/oval/crcular FRP shell may be used as stay-n-place ormwork or castng addtonal expansve cement ncrete around the square or rectangular secton to acheve shape modcaton. The advantage o usng expansve cement ncrete s that a post-tensonng eect uld be acheved on the FRP mposte shell through chemcal post-tensonng, whch would n turn mprove the mpressve behavor o ncrete mns. Expansve cement nssts o a Portland

3 FRPRCS cement mponent and a calcum-suloamnate anhydrte mponent; the hydraton o the latter mponent causes expanson. Ths materal has been used snce 1960 or makng chemcally prestressed ncrete. Klen et al. (1961) nvestgated the propertes o expansve cement or chemcal post-tensonng and ound that actors nencng the magntude and rate o the expansve reacton ncde: chemcal mposton o mponents, proportons o the two mponents n the total cementng materal, rchness o mx, ndtons o curng, and degree o restrant. Benuska et al. (1971) urther studed the curng eects on expanson and mechancal behavor o expansve cement ncrete; ther tests showed that expansve cement ncrete was very good or preabrcated elements or structural elements and systems n whch the optmum amount o chemcal prestress requred was relatvely low. The mechansm o expansve cement ncrete s used wth FRP mposte shells or nnement enhancement. When expansve cement ncrete s appled to preabrcated FRP shells, expanson o the cement grout s restraned by the FRP shell; the FRP mposte s stressed n tenson, thus creatng a post-tensonng eect. It s obvous that ths post-tensonng eect would ncrease the nnement behavor o FRP ackets and change the nnement acton rom passve to actve. In ths paper, an expermental program was nducted or studyng shape modcaton o square/rectangular specmens nned wth FRP mpostes. Two FRP mposte systems, a Carbon Fber Renorced Polymer (CFRP) system and a Glass Fber Renorced Polymer (GFRP) system were used. Comn shape modcaton was perormed by usng preabrcated FRP mposte shells wth expansve cement ncrete. Test results are presented regardng are modes and the stress-stran behavor; mparson s made between shape-moded specmens and specmens wthout shape modcaton. Ths paper also presents an analytcal stress-stran model or the shape-moded ncrete mn wth the FRP mposte shell and expansve cement ncrete. As a part o ths research, a plastcty model based on the Wllam-Warnke (1975) plastcty ncrete model was developed to acunt or the axal strength o FRP-nned ncrete Concsons regardng the eectveness o the shape modcaton method wth expansve cement ncrete and ts practcal mplementaton are made. RESEARCH SIGNIFICANCE Connement o ncrete utlzng shape modcaton o square and rectangular mns wth post-tensoned non-bonded FRP mposte shells and expansve cement ncrete s nvestgated. Square and rectangular sectons were moded nto crcular and ellptcal sectons; post-tensonng o the FRP mposte shells reduces rner eects, and enhances membrane acton and nnement eectveness. It s shown that expansve cement ncrete s an ecent method or post-tensonng FRP mposte shells and modyng the FRP nnement rom passve to actve. The proposed analytcal model shows a good agreement wth the expermental results.

4 1050 Yan et al. EXPERIMENTAL PROGRAM Specmens The expermental program nvolved three groups o specmens: S, R and R3; S denotes a seres o square specmens; R and R3 denote a seres o rectangular specmens wth cross-sectonal aspect rato o :1 and 3:1, respectvely. All specmens were 914 mm hgh; no steel renorcement was used nsde the ncrete. Each group ncded an unnned (baselne) specmen, two specmens wth the orgnal crosssecton nned by CFRP or GFRP mpostes, and two shape-moded specmens by usng preabrcated CFRP or GFRP shells and expansve cement ncrete. Table 1 lsts the detals o all specmens; the dentcaton o the specmens, as shown n Table 1, uses a three-de base. The rst part o the de s the shape o the mn (Square or Rectangular), and the aspect rato o the rectangular cross-secton (:1 or 3:1). The send part o the de ndcates the type o FRP mposte (CFRP or GFRP) and the number o layers ( or 6). The thrd part denotes the type o materal used to acheve shape modcaton (E denotes Expansve cement ncrete and 0 denotes no shape modcaton,.e. the specmen has the orgnal square or rectangular geometry). The moded cross-secton o the orgnal square specmen was crcular and the moded cross-secton o the orgnal rectangular specmen was ellptcal; the cross-sectonal aspect rato was close to the orgnal pror to shape modcaton. Materal propertes Two types o ncrete were used n ths study: regular ncrete and expansve cement ncrete. Regular ncrete was used to cast the orgnal specmens; expansve cement ncrete was used to perorm the shape modcaton. The ncrete mpressve strength or the orgnal specmens was 15 MPa. The expansve cement used n ths research was Type-K and Komponent. The two prncpal nsttuents o Komponent are calcum suloamnate and gypsum or calcum sulate. The ormaton o ettrngte crystals, whch result rom hydraton o the two ngredents, s what causes the expanson. When expanson s restraned, or example by a pre-abrcated FRP mposte shell, expansve cement ncrete nduces tensle stresses n the FRP mposte shell that cause chemcal post-tensonng. The mx desgn or the expansve cement ncrete s lsted n Table. The mpressve strength o the expansve cement ncrete ater 8 days was 10 MPa. Two FRP mposte materals were used to nne the ncrete mns. One was SkaWrap Hex 103C whch s a hgh strength, undrectonal carbon ber abrc wth epoxy resn. The other was Aquawrap G-06, whch s a undrectonal pre-mpregnated glass ber abrc wth urethane resn. Both FRP mposte materals were cured at ambent temperature ndtons. The materal propertes determned rom tensle upon tests, per ASTM Standard D3039, are shown n Table 3. For shape-moded specmens, preabrcated FRP mposte shells were made pror to castng o expansve cement ncrete. Stran gauges were nstalled on the FRP mposte at mdheght o the specmens and a data acquston system was used to measure the hoop expanson o the FRP mposte shells durng the curng perod o the expansve cement ncrete. Fgure 1 shows the measured FRP hoop stran versus tme

5 FRPRCS startng at castng o the expansve cement ncrete. The FRP hoop stran approached a nstant vae ater 60 days and ths asymptotc vae s dened as the ntal hoop stran,n, whch reers to the state beore axal load s appled. The ntal hoop stran, n depends on the aspect rato B /D o the preabrcated FRP shells, whch s dened as the length o the maor axs, B, to the length o the mnor axs, D, o the ellptcal crosssecton. In general, crcular ackets acheved the hghest expanson whle the ellptcal shell wth the hghest aspect rato had the smallest expanson. Also, GFRP shells acheved a hgher expanson mpared to CFRP shells. These observatons can be vsualzed rom Fg. or the relatonshp between and the aspect rato. From Fg., the proposed relatonshp between, n and B /D or FRP shells wth CFRP layers s: B, n = (1) D and or FRP shells wth 6 GFRP layers s: B, n = () D Loadng and nstrumentaton All specmens were subected to a monotonc unaxal load untl are under dsplacement ntrol wth a nstant loadng rate o 1.3 mm per mnute. The tests were perormed usng a 9 MN actuator wth a stroke o 0.6 m. Two Lnear Varable Derental Transducers (LVDTs) were nstalled to measure axal mpressve strans; stran gauges were used to measure the transverse strans over the crcumerence o the cross-secton., n EXPERIMENTAL RESULTS AND DISCUSSION Fare modes For FRP-nned square/rectangular specmens wthout shape modcaton, are typcally starts wth ncrete crushng ollowed by racture o the FRP mposte acket. In rectangular and square sectons, FRP breakage appeared at one o the rners, n a small area near the mn mdheght; a ncrete ne was seen ater peelng the broken FRP mposte acket. The are was brttle due to the presence o the rner and lat sde eects, whch elmnate membrane acton o the FRP acket and result n weaker nnement. Fgure 3(a) shows the typcal are mode or the FRP-nned specmen wth the bonded FRP acket. In ntrast to specmens wth bonded FRP ackets, the are o shape-moded specmens wth non-bonded FRP shells and expansve cement ncrete was racture o the FRP ackets rst, ollowed by crackng o the expansve cement ncrete and ncrete re. FRP breakage extended over the entre heght o the mn, showng the extensve partcpaton o the FRP acket n nnement. At the end o the test, most

6 105 Yan et al. specmens remaned n one pece, and shear and mpresson cracks were seen n the expansve cement ncrete. Fgure 3(b) shows the typcal are mode or a shapemoded specmen wth non-bonded FRP shells and expansve cement ncrete. Specmens wth non-bonded FRP shells and expansve cement ncrete had a hgher stran ductlty mpared to specmens wth bonded FRP ackets, demonstratng the hgher eectveness o the post-tensoned FRP shells. The degree o damage o shapemoded specmens wth non-bonded FRP shells and expansve cement ncrete vared wth aspect rato. Specmens wth a smaller aspect rato reached a hgher capacty and a hgher degree o damage. Axal stress-stran response Fgure 4 presents the axal stress versus axal stran response or each group, ncdng the baselne specmens and the specmens nned wth CFRP mposte ackets. The axal dsplacements were measured usng the average o two LVDTs, and the axal stress was mputed by dvdng the axal mpresson load by the crosssectonal area. It s seen that CFRP-nned square specmen S-C-0 showed a lmted hardenng behavor and both CFRP-nned rectangular specmens R-C-0 and R3-C- 0 demonstrated a sotenng behavor; a drop o axal stress was observed ater the ntal axal strength was reached, and the degree o sotenng ncreased as the aspect rato ncreased. For shape-moded specmens, the stress-stran curves show ascendng branches wthout sotenng behavor. In some cases, the ntal slope o the stress-stran curve or shape-moded specmens s less than that o specmens wthout shape modcaton. Ths s because the unnned mpressve strength o expansve cement ncrete was smaller than that o regular ncrete, and the membrane eect rom the FRP mposte shell was not sgncant n the ntal phase o axal loadng. Smlar observatons were made or GFRP-nned specmens, as shown n Fg. 5. It s also noted rom Fg. 5(c) that very lmted ncreases as well as the lmted hardenng behavor were obtaned or shape-moded specmen R3-G6-E mpared to the specmen nned wth the bonded FRP acket, R3-G6-0, showng that the GFRP mposte s more senstve to the cross-sectonal aspect rato. To characterze the stress-stran behavor o FRP-nned ncrete mns, three mportant parameters are dented: axal strength, ultmate axal stran, and ductlty rato µ, as shown n Table 4. The ductlty rato s used to evaate the ductlty perormance o FRP-nned ncrete and s calculated as the rato o the total area under the stress-stran curve to the area bounded by a slope o the ntal elastc stness and the plastc plateau (Rochette and Labossère 000); or specmens wth hardenng behavor the plastc plateau passes through the unnned ncrete strength ; or specmens wth sotenng behavor the plastc plateau passes through the peak axal strength. In Table 4, the vometrc rato o the FRP mposte acket or each specmen ρ FRP s also presented; ρ FRP s the rato o the area o the FRP mposte acket to the cross-sectonal area o the ncrete specmen.

7 FRPRCS It can be seen rom Fgs. 4, 5 and Table 4 that the shape-moded specmens showed sgncant ncreases n axal strength, ultmate axal stran, and ductlty. However, the level o mprovement o the mpressve behavor depends on the aspect rato. As seen rom Fgs. 4 and 5, the mprovement s sgncant or shape-moded square specmens S-C-E and S-G6-E snce ther moded shape was crcular; the mprovement was less sgncant or shape-moded rectangular specmens R3-C-E and R3-G6-E wth the hgher aspect rato; ths means that the eect o shape modcaton s reduced as the secton bemes a latter ellpse. Thereore, to mprove the nnement eectveness, a lower aspect rato ater shape modcaton s preerred, n the orm o an oval shape. However, the practcal use o a preabrcated FRP shell wth a low aspect rato such as an oval shape always ncreases the moded cross-sectonal area by a large amount o ncrete, and possble oundaton ssues resultng n st ncreases. Eectve FRP nnement rato For FRP-nned ncrete crcular or ellptcal mns, the FRP nnng stress can be expressed as: l 1 l = ρfrpe (3) where ρ FRP = FRP vometrc rato; E = elastc modus o FRP mposte, and = FRP hoop stran or crcular cross-sectons or average FRP hoop stran or ellptcal cross-sectons, dened as the average o the hoop stran at the mnor and maor axs (Yan 005). The eectve nnement rato s dened as the rato o the ultmate FRP nnng pressure at rupture o the FRP acket to the unnned ncrete strength ; or shape-moded mns, s obtaned by calculatng the mean unnned mpressve strength over the moded cross-secton. The general orm o the ultmate FRP nnng pressure or crcular or ellptcal mns can be expressed as: 1 = ρ E k (4) FRP where k = FRP acket ecency actor; and u u = ultmate FRP tensle stran obtaned rom materal upon tests. In Eq. (4), k s used to acunt or the reducton actor o the FRP ultmate hoop stran mpared to the materal upon tests and depends on the aspect rato o the cross-secton. Snce the FRP mposte shells are already posttensoned pror to axal loadng, k s smaller than that o the rrespondng bonded FRP ackets. Based on ths study, k was ound to be n the range o 0.30 and 0.50 or crcular non-bonded FRP shells, and a vae o 0.40 s remmended; or ellptcal crosssectons, k s ntrolled by the aspect rato as (Yan 005): 0.65 B B k = (5) D D

8 1054 Yan et al. The eectve nnement rato s also an ndcaton o the trend o the stress-stran behavor; the authors suggest that the FRP-nned mn shows a stran hardenng behavor when s larger than 0. and a stran sotenng behavor when s smaller than 0. (Yan 005). Dlatancy behavor o shape-moded mns In ths study, the dlatancy behavor o FRP-nned ncrete s represented by the vometrc stran versus axal stran relatonshp. Vometrc stran V s dened as the FRP area stran n the two transverse orthogonal drectons mnus the axal stran n the ncrete mn c : V = c (6) where s dened as the FRP hoop stran or crcular cross-sectons or average FRP hoop stran or ellptcal cross-sectons. Fgure 6 shows the vometrc stran versus axal stran relatons or shape-moded specmens wth expansve cement ncrete. Snce the FRP shell was already post-tensoned pror to axal loadng through chemcal posttensonng, the amount o radal expanson was smaller mpared to bonded FRP ackets. Thereore, the axal stran was larger than the hoop area stran, ; ths reveals that the axal stran was domnant n the vometrc stran versus axal stran curve. Ths dlatancy behavor s extremely mportant or shape-moded FRP specmens wth expansve cement ncrete because n ths case the FRP nnement bemes actve nstead o passve. From Fg. 6, the relatonshp between vometrc stran and axal stran s approxmately lnear and α s the slope o a straght lne that s: = (7) V α c where α s determned by the FRP nnement rato and s proposed as (Yan 005) α = (8) ANALYTICAL MODEL Moded Wllam-Warnke (MWW) model or FRP-nned ncrete Predcton o the axal strength s based on the ultmate surace descrbed by Wllam and Warnke (1975), as shown n Fg. 7(a); σ 1, σ, and σ 3 are the three stress mponents. For FRP-nned ncrete, σ 1 s taken as the axal mpressve stress; σ and σ 3 represent the nnng stress provded by the FRP mposte n the two transverse orthogonal drectons. In plastcty theory the sgn nventon s that

9 FRPRCS mpressve stress s negatve. Fgure 7(b) shows the proecton o the ultmate surace on the σ a τ a plane, where σ a = normalzed mean normal stress and τ a = normalzed mean shear stress, as expressed n Eqs. (10) and (11) n terms o σ 1, σ, σ 3, and the unnned ncrete strength. σ 1 + σ + σ 3 σ a = (9) 3 1 [( σ 1 σ ) + ( σ σ 3 ) + ( σ 3 σ 1) τ a = ] (10) 15 As shown n Fg. 7(b), the mpresson merdan s a send-order parabola and can be expressed as (Wllam and Warnke 1975): τ a = b0 + b1σ a + bσ a (11) where b 0, b 1, and b are materal nstants obtaned rom experments. From the FRPnned mn tests perormed by the authors (Yan 005) these vaes are: b = , b 0. 1 = and b = For specmens wth hardenng behavor, the axal strength s acheved when the FRP nnng pressure l reaches ts maxmum. Thereore, σ 1 = ; σ = σ 3 = at the ultmate state. Substtutng the rrespondng terms n Eqs. (9), (10), and (11) and solvng ths equaton system gves the axal strength or the range o 0. : = + + ; 0. (1) For the range o < 0., the authors suggest the axal strength as (Yan 005): = max, ; < 0. (13) ln Analytcal stress-stran relatonshp The Popovcs (1973) model was used or developng the analytcal stress-stran relatonshp or shape-moded specmens wth expansve cement ncrete. Ths model descrbes the ascendng stress-stran relaton up to the peak pont, as shown n Fg. 8 and s thus sutable or the shape-moded specmen wth hardenng behavor. An analytcal model or shape-moded specmens wth sotenng behavor has also been developed and s descrbed elsewhere (Yan 005). The analytcal expresson o Popovcs (1973) s based on the relatonshp between axal stress c and axal stran c expressed as:

10 1056 Yan et al. E0 c c = (14) r c 1+ ( K 1) where E 0 = ntal elastc modus and E s = secant modus at the peak stress gven as: Es = (15) and shown n Fg. 8. The parameters K and r are dened as: E0 K = = E0 (16) Es K r = (17) K 1 Parameter s the ultmate axal stran proposed by Imran (1994) as: = (18) where s the axal stran rrespondng to the axal strength o the unnned ncrete ; or regular ncrete, s set to 0.00 mm/mm. Tradtonal steel-nned ncrete models,.e. the Mander et al. model (1988), assume a nstant nnng pressure whch s based on the assumpton that the nnng devce has yelded and behaves n a perectly plastc manner, thus provdng a nstant nnng pressure. However, ths assumpton s napproprate or FRP nnement because the nnng pressure provded by FRP ackets or shells, l, vares ntnuously and exhbts an approxmately lnear behavor untl are (Moran and Panteldes 00). Thereore, the analytcal FRP-nned ncrete model must be developed based on an ncremental approach to acunt or the varable FRP nnng pressure l. In the proposed procedure, the axal loadng s dvded nto a number o steps. The detaled analytcal approach s descrbed as ollows: Based on the gven normaton such as the cross-sectonal dmensons, materal propertes, and the number o FRP mposte layers, the eectve nnement rato / and the dlatancy parameter α can be obtaned rom Eqs. (4) and (8) respectvely. For each step, an ncremental FRP hoop stran s appled and the current hoop stran s calculated by aumulatng the ncremental hoop stran up to the current load step as: k k = Σ (superscrpt attached to the varables ndcates =1 the load step at whch the varables are updated). Then the current axal stran obtaned rom Eqs. (6) and (7) rrespondng to an FRP hoop stran c s by usng the

11 dlatancy parameter α. The nnng pressure rrespondng to an FRP hoop stran. Thereore, by settng axal stress, at FRPRCS l can be obtaned rom Eq. (3) = l the maxmum can be calculated rom the model o Eq. (1) or 0. or Eq (13) or < 0.. The relatonshp between the current axal stran c and axal stress c, whch rresponds to the current hoop stran can be obtaned by usng the Popovcs model o Eqs. (14) - (18). The ncremental steps are repeated untl the hoop stran reaches ts ultmate state u = k u, as shown n Eq. (4). For crcular FRP mposte shells k s set equal to 0.4; or ellptcal shells, Eq. (5) should be used or calculatng k. The ncremental approach s easly mplemented usng a spreadsheet or any mputer program. Fgure 9 shows mparsons between the analytcal model and expermental results; t can be seen that the analytcal results agree well wth the experments. CONCLUSIONS Shape modcaton by usng expansve cement ncrete and preabrcated FRP mposte shells can restore the membrane eect n FRP mposte nnement o square and rectangular ncrete mns; t can change the FRP nnement rom passve to actve, and thus acheve a hgher axal strength or square and rectangular mns mpared to the orgnal mns wth the same number o FRP mposte layers. For lghtly or moderately FRP-nned square or rectangular mns, the shape modcaton method uld mody the stress-stran behavor rom sotenng to hardenng and thereore acheve a hgher strength and ductlty. The expermental results showed that the eectveness o shape modcaton depends largely on the aspect rato o the moded secton: the optmal mn shape or FRP nnement s the crcular crosssecton; or rectangular mns, especally those wth a large aspect rato, change to a crcular secton requres a large vome o expansve cement ncrete and possble modcatons to the oundaton. Thereore, or strengthenng rectangular mns by shape modcaton, the nence o actors such as vome ncrease, ncrease n surace area, and requred strength, ductlty, and ultmate stran need to be nsdered to obtan an optmal soton. The proposed analytcal stress-stran model or shape-moded FRP-nned mns wth expansve cement ncrete s based on the moded Wllam-Warnke (1975 plastcty model, the Popovcs general stress-stran (1973) ncrete model, and the dlatancy behavor obtaned rom the present study. Ths model s mplemented by an ncremental approach whch acunts or the varable FRP nnement durng the

12 1058 Yan et al. loadng process. The analytcal results show satsactory agreement wth the experments ACKNOWLEDGMENTS The authors would lke to acknowledge the ntrbutons o FRP mposte materals by Ska and Ar Logstcs and the ntrbuton o Type K cement and Komponent rom CTS Company. The nancal support provded by the Utah Department o Transportaton s also acknowledged. REFERENCES Benuska, K. L., Bertero, V. V., and Polvka, M. (1971). Sel-Stressed Concrete or Precast Buldng Unts. PCI Journal, March-Aprl 1971, Imran, I. (1994). Applcaton o Non-Assocated Plastcty n Modelng the Mechancal Response o Concrete. Ph.D. Dssertaton, Unversty o Toronto, Klen, A., Karby, T., and Polvka, M. (1961). Propertes o An Expansve Cement or Chemcal post-tensonng. ACI Journal, Proceedngs, Vol. 58, No. 1, July 1961, Moran, D. A., and Panteldes, C. P. (00). Varable Stran Ductlty Rato or Fber Renorced Polymer-Conned Concrete. Journal o Compostes or Constructon, ASCE, 00, 6(4), 4-3. Pessk, S., Harres, K. A., Kestner, J. T., Sause, R., and Rcles, J. M. (001). "Axal Behavor o Renorced Concrete Comns Conned wth FRP Jackets." Journal o Compostes or Constructon, ASCE, 001, 5(4), Popovcs, S. (1973). "Numercal Approach to the Complete Stress-Stran Relaton or Concrete." Cement Concrete, 1973, 3(5), Rochette, P., and Labossère, P. (000). "Axal Testng o Rectangular Comn Models Conned wth Compostes." Journal o Compostes or Constructon, ASCE, 000, 4(3), Wllam, K. J., and Warnke, E. P. (1975). Consttutve Model or the Traxal Behavor o Concrete. Proceedngs, Internatonal Assocaton or Brdge and Structural Engneerng, Vol. 19, Yan, Z. (005). Shape Modcaton o Rectangular Comns Conned wth FRP Compostes. Ph.D. Dssertaton, Department o Cvl and Envronmental Engneerng, Unversty o Utah, Salt Lake Cty, Utah, May 005.

13 FRPRCS

14 1060 Yan et al.

15 FRPRCS Fgure 1 Expanson hstory or FRP shells wth expansve cement ncrete: (a) Square S; (b) Rectangular R; (c) Rectangular R3. Fgure Relatonshp between aspect rato B D and ntal hoop stran, n.

16 106 Yan et al. Fgure 3 Typcal are modes or FRP-nned specmens: (a) wthout shape modcaton; (b) wth shape modcaton. Fgure 4 Axal stress-stran curves or CFRP-nned mns: (a) Square S; (b) Rectangular R; (c) Rectangular R3.

17 Fgure 5 Axal stress-stran curves or GFRP-nned mns: (a) Square S; (b) Rectangular R; (c) Rectangular R3. FRPRCS

18 1064 Yan et al. Fgure 6 Vometrc versus axal stran relatonshps or shape-moded specmens wth expansve cement ncrete.

19 FRPRCS Fgure 7 Wllam-Warnke plastcty model: (a )3D vew; (b) merdan secton on σ τ plane. a a Fgure 8 Axal stress-stran curve or specmens wth hardenng behavor.

20 1066 Yan et al. Fgure 9 Comparsons between analytcal results and experments: (a) S-C-E; (b) S-G6-E; (c) R-C-E; (d) R-G6-E.