Analysis of Reinforced Concrete Columns Subjected to Combined Axial, Flexure, Shear, and Torsional Loads

einde

Joodaemun-gu, Seoul 130-743, Republic of Korea712-, Chu

Keywords:

retehe m

force equilibrium and strain compatibility in a three-dimensional space should be satised. Accordingly,

n con

substantial consideration of torsion during structural design[19,20], the torsional behavior of SFRC is an important research

of a torsional member are under biaxial stresses, and thus, a clearbiaxial tensile behavior model is required to accurately predict thetorsional behavior of SFRC members. However, most of the tensilebehavior models of SFRC are based on the results of uniaxial ten-sion tests. Accordingly, a constitutive relationship of SFRC in ten-sion is proposed in this paper based on the results of shear paneltests subjected to biaxial stresses, and a torsional behavior modelof SFRC member that adopts the proposed tensile constitutive rela-tionship is presented. Moreover, the presented torsional behaviormodel reects the difference of the angle between principal stres-ses and crack direction, which has been ignored in the xed-angle

Corresponding author. Tel.: +82 2 2210 5707; fax: +82 2 2248 0382.E-mail addresses: [email protected] (H. Ju), [email protected] (D.H. Lee), asorange

@hanmail.net (J.-H. Hwang), [email protected] (J.-W. Kang), [email protected](K.S. Kim), [email protected] (Y.-H. Oh).

1 Tel.: +82 2 2210 5375; fax: +82 2 2248 0382.2 Tel.: +82 2 2210 5354; fax: +82 2 2248 0382.3 Tel.: +82 53 810 2429; fax: +82 53 810 4625.

Composites: Part B 45 (2013) 215231

Contents lists available at

Composite

journal homepage: www.elsev4 Tel.: +82 41 730 5615; fax: +82 41 730 5615.steel-ber-reinforced concrete (SFRC), which compensate for thebrittle material characteristics of conventional concrete [1,2].Existing studies report that the inclusion of steel bers in concretedrastically improves the crack and drying-shrinkage control capac-ity of concrete as well as tensile strength, exural capacity andshear resistance performance [316]. In addition, SFRC may beapplicable to converting the brittle failure mode of concrete mem-bers to the ductile failure mode [1,4,5]. However, only some lim-ited research on torsional behavior of SFRC members has beenperformed, and most studies focused on investigations of tensileor shear behavior [618]. As the complex and various oor plans,heights and shapes of modern buildings and bridges often require

results of the shear panel test [21,22]. This paper presents a tor-sional behavior model for SFRC members utilizing the proposedtensile constitutive relationship. This analytical model is also vali-dated by comparing to those test results reported in the literature[20,2229].

2. Research signicance

Recent torsional behavior/strength models consider that thetorsional behaviors of SFRC members are heavily inuenced bythe tensile performance of SFRC. In general, the sectional elementsA. FibersA. Polymermatrix composites (PMCs)B. StrengthC. Analytical modelingTorsion

1. Introduction

Since the 1960s, studies have bee1359-8368/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compositesb.2012.09.021many studies proposed empirical evaluation equations for the torsional strength of SFRC members basedon experimental results. Therefore, this study derived a constitutive model of SFRC in tension, whichgreatly inuences the torsional behavior of SFRC, based on the test results of SFRC shear panels underbiaxial stress, and this tensile behavior model was introduced to a xed-angle softened-truss model. Atheoretical evaluation model based on the modied xed-angle model for torsional behavior of SFRCwas developed, and the performance of the analytical model was also evaluated compared to test resultsobtained from literature.

2012 Elsevier Ltd. All rights reserved.

sistently carried out on

theme and should be claried. Thus, in this study, a constitutiverelationship of SFRC in tension, which is very important in analysisof the torsional behavior of SFRC members, was derived based onAccepted 6 September 2012Available online 15 September 2012

formances than conventional concrete. It can improve torsional behavior as well as exural and shearbehavior. However, analysis of the torsional behavior of SFRC members is quite complicated becauseTorsional behavior model of steel-ber-rmodifying xed-angle softened-truss mo

Hyunjin Ju a,1, Deuck Hang Lee a,2, Jin-Ha Hwang a,2,aDepartment of Architectural Engineering, University of Seoul, 90 Jeonnong-dong, Dongb School of Architecture, Yeungnam University, 280 Daehak-Ro, Gyeongsan, GyeongbukcDepartment of Architectural Engineering, Konyang University, 121 Daehak-Ro, Nonsan

a r t i c l e i n f o

Article history:Received 25 January 2012Received in revised form 3 September 2012

a b s t r a c t

Steel-ber-reinforced concsate for the drawbacks of tll rights reserved.749, Republic of Koreangnam 320-711, Republic of Korea

(SFRC) is an efcient cement-based composite material that can compen-aterial properties of conventional concrete and has better structural per-forced concrete membersl

-Won Kang b,3, Kang Su Kim a,, Young-Hun Oh c,4

SciVerse ScienceDirect

s: Part B

ier .com/locate /composi tesb

cr,f

ffu ultimate strength of steel ber

Parfly direct tensile strength of longitudinal directionsfn average yield stress of the embedded steel barsfr modulus of rupturefs average stress of steel barsfsp splitting tensile strength of concreteNomenclature

A0 area enclosed by the centerline of shear ow zoneAc cross-sectional area bounded by the outer perimeter of

the concreteB cross sectional width of memberdf diameter of berEc elastic modulus of concreteEcf elastic modulus of concreteEs elastic modulus of the SFRCF ber factorf 0c specied compressive strength of concretefcr stress in concrete at crackingf stress in concrete or SFRC at cracking

216 H. Ju et al. / Composites:model. The presented model is relatively concise and enables accu-rate evaluations of SFRC torsional behavior compared with theexisting torsional behavior models based on the smeared trussapproach.

3. Review of previous researches

The thin-walled tube theory proposed by Bredt [30], in whichtorsional behavior of a thin and arbitrarily shaped tube is clearlyexplained, has been adopted as the theoretical background forthe torsion design methods suggested in such modern concrete de-sign codes as ACI318 [31], ASHTTO-LRFD [32], CEB-FIP [33], CSA[34], and KCI [35].

Rausch [36] utilized the 45 plane truss analogy concept, pro-posed by Ritter [37] and Mrsch [38], for torsional analysis of con-crete members, which so-called the space truss model or theplastic space truss model [19,39]. Later, Anderson [40] pointedout that the space truss model did not take into account the contri-bution of concrete to the torsional capacity, and proposed the tor-sional strength of RC members with sum of the torsionalcontributions of the concrete and torsional reinforcements. In1959, Lessig [41] proposed the skew-bending theory, which was

ft direct tensile strength of transverse directionsfty direct tensile strengthft direct tensile strengthfy yield stress of bare steel barsH cross sectional height of memberkc ratio of the average compressive stress to the peak com-

pressive stress in the concrete strutskt ratio of the average compressive stress to the tensile

cracking stress in the concrete strutsp0 perimeter of the centerline of shear owph perimeter of the centerline of closed stirrup (2(x0 + y0))pc perimeter of the outer concrete cross sectionq shear ows spacing of transverse hoop barsT torsional momenttd effective thickness of shear ow zoneVf volume fraction of steel bera2 angle of applied principal compressive stress with re-

spect to l axisb deviation angle (a2 a); 2b = tan1(c21/(e2 e1))c21 average shear strain in the 21 coordinateclt average shear strain in the lt coordinateeo strain at specied compressive strength of concretee1 average strain in the 1-directione2 average strain in the 2-directione1s average surface strain in the 1-directione2s average surface strain in the 2-directionecr cracking strained average principal compressive straineds maximum principal compressive strainel average strain in the l-directionen average yield strain of the embedded steel barser average principal tensile strainers maximum principal tensile straines average strain of the steel barset average strain in the t-directioney yield strain of the bare steel barsf softened coefcient of concrete in compressiong reinforcement index, taken as (Atftyph)/(Alflys)

t B 45 (2013) 215231further developed by Hsu and his colleagues [42,43], and wasadopted in the ACI318 Building Code [44] from 1971 to 1989. Cur-rently, many international design codes, such as ACI [31], CSA [34],and CEB-FIP [31], include the 45 space truss models as the designmethod for torsion utilizing the thin-walled tube theory proposedby Lampert [45], Lampert and Thlimann [46], and Lampert andCollins [47].

Truss models have been continuously developed according tobetter understanding on the torsion and/or shear behavior of rein-forced and prestressed concrete members [4856]. In torsionalmodels [4850,5254,5759], the effective thickness of the tubewalls is dened as the shear ow zone determined utilizing Bredtsthin-walled tube theory [30] and the compatibility relationshipsfor torsion with the assumption that the distribution of strain inthe crack direction (or principal tensile stress direction) withinthe effective thickness is linearly distributed from the extreme out-er ber (i.e., surface) of the tube to the depth of the effective tubethickness. Such a linear strain gradient was measured closely fromexperimental researches [60,61]. As torsional analysis modelsusing the softened-truss model are applicable for evaluating thetorsional behavior of reinforced concrete members as well as pre-stressed concrete members, many recent studies have adopted thesoftened-truss model [53,54,6265]. For instance, softened-truss

h angle of twist per unit lengthhcr cracking angle of twist per unit lengthhu ultimate angle of twist per unit lengthq steel ratioqf bond factor that accounts for differing bond characteris-

tics of the berql longitudinal steel ratioqt transverse steel ratiorcd average principal compressive stress in conrete

rfd average principal compressive stress in SFRCrcr average principal tensile stress in concreterc1 average normal stresses of concrete or SFRC in the 1-

directionrc2 average normal stresses of concrete in the 2-directionrl applied normal stresses in the l-directionrt applied normal stresses in the t-directionsc21 applied shear stresses in the 21 coordinateslt applied shear stresses in the lt coordinatewc curvature of the concrete struts along the 2-directionwt curvature of the concrete struts along the 1-direction

models have been utilized to predict the torsional capacity modelof high-strength concrete deep beams [61], FRP strengthened con-crete beams [65] and SFRC beams [20,26] as well as prestressedconcrete or reinforced concrete members. With the progress of tor-sional analysis methods, many cement-based composites havebeen developed since the 1960s to improve the brittle materialcharacteristics of concrete, and SFRC is such advanced high-performance material. There have been many studies on the ten-sile, shear, and exural behavior of SFRC [1,2,516] whereas thereare few studies on the torsional behavior of SFRC in the literature.

Due to the enhanced tensile strength and stiffness resulting

difcult to reect the tensile behavior of SFRC subject to biaxialstress due to the uniaxial-based tension constitutive relationshipused in the FDM model. Gunneswara Rao and Rama Seshu [20] im-proved the Mansur et al.s [26] model that tended to underestimatethe torsional stiffness of SFRC prior to cracking, by applying SaintVenants elastic theory [67] to the tosional behavior of SFRC beforetorsional cracking, which yielded more accurate analysis results.

The aforementioned researches reported a substantial enhance-ment of torsional behavior of SFRC due to the tensile stress transfercapacity of steel bers at crack interfaces, which is believed to beinuenced by the volume fraction of steel ber (Vf), interfacial

a constitutive relationship of SFRC in tension was derived in thisstudy based on the results of SFRC shear panel tests conducted re-

m)

H. Ju et al. / Composites: Part B 45 (2013) 215231 217from the addition of steel ber to concrete, the torsional perfor-mance of a SFRC member is drastically increased [11,12].Narayanan and Kareem-Palanjian [34] reported that the torsionalstrength of SFRC with over 1.5% of the volume fraction of steel -bers (Vf) increases by more than 25% compared to that of RC, andMansur and Paramasivam [23] also reported an about 27% increasein torsional strength depending on the volume fraction of steel -bers and the ber types. Moreover, Craig et al. [24] experimentallyconrmed that effectiveness of steel ber is maximized in SFRCtorsional members reinforced in both the longitudinal and trans-verse directions [24]. Many other studies [20,2629,66] also havereported improved torsional performances of the SFRC membersbased on experimental observations.

As the torsional behavior of SFRC members is heavily inuencedby the material performances of the concrete under tension, manyresearchers proposed constitutive relationships of SFRC in tension,which were reected on their torsional strength or behavior mod-els [47]. Mansur and Paramasivam [23] proposed three types oftorsional strength equations for SFRC based on the elasticity, plas-ticity, and skew-bending theories, and Craig et al. [24] proposed anultimate torsional strength model using the enhanced splittingtensile strength (or modulus of rupture) of SFRC as a key parame-ter. Other torsional strength equations for SFRC were also proposedby El-Niema [66] and Narayanan and Kareem-Palanjian [25].

Mansur et al. [26] proposed their initial model on the torsionalbehavior of SFRC, in which the tensile constitutive relationship ofRA-STM [50,67] is modied for SFRC using Lim et al.s uniaxial ten-sile behavior model [12]. This model showed good estimation forthe post-cracking behaviors of SFRC specimens subjected to puretorsion. However, since they assumed a fully cracked section statusthrough the overall behavior of a member, the initial torsional stiff-ness was signicantly under-estimated compared to the test re-sults [26]. Karayannis [68] proposed two types of uniaxial tensilebehavior models of SFRC after cracking based on the model pro-posed by Lim et al. [12] with the critical volume fraction of steelber (Vf,cr) as the main variable, and presented a torsional behaviormodel of SFRC using the nite difference method (FDM). Moreover,they carried out torsional experiments on SFRC members with awide range of ber types and section shapes [27]. Although theirtorsional behavior model was somewhat complex, it accuratelyestimated the experiment results. In this approach, however, it is

Table 1Summary of Toronto SFRC panel specimens [21,22].

Specimen names Concrete Steel ber

f 0c MPa e0cu 103 Vf (%) lf (m

C1F1V1 51.4 2.150 0.5 50C1F1V2 53.4 2.670 1.0 50C1F1V3 49.7 2.500 1.5 50C1F2V3 59.7 3.280 1.5 30C1F3V3 45.5 2.340 1.5 35C2F1V3 79.4 2.770 1.5 50

C2F2V3 76.5 2.220 1.5 30C2F3V3 62.0 2.030 1.5 35cently at the University of Toronto [21,22]. The size of each shearpanel specimens was 890 mm 890 mm 70 mm, and as shownin Table 1, the key variables of the experimental program werethe specied compressive strength of concrete (f 0c), the volumefraction of steel ber (Vf), and the types of steel ber. The reinforc-ing bars were uniformly placed only in the longitudinal direction,and the total area of the reinforcements was 2063 mm2, which cor-responded to 3.31% of the reinforcement ratio (qs). In the test pro-gram, the concrete compressive strengths (f 0c) were 50 MPa and80 MPa for series C1 and C2, respectively; the aspect ratios of ber(lf/df) were 81, 79, and 64 for series F1, F2 and F3, respectively; andthe volume fractions of steel ber were 0.5%, 1.0%, and 1.5% for ser-ies V1, V2 and V3, respectively.

Fig. 1 shows the tensile stressstrain relations obtained fromthe SFRC shear panel tests and those estimated by existing models[1216] shown in Table 2 for comparison. The panel specimens

Reinforcement

df (mm) F fy (MPa) Asx (mm2) qsx (%)

0.62 0.40 552 2063 3.310.62 0.81 552 2063 3.310.62 1.21 552 2063 3.310.38 1.18 552 2063 3.310.55 0.95 552 2063 3.310.62 1.21 552 2063 3.31bond strength (su), and ber directionality at the crack interfaces.The softened-truss models, which were adopted by the existingtorsional behavior models [39,48,50,52,53,60,67], are based onthe test results of shear panels subject to biaxial stress; however,the constitutive models of SFRC in tension are based on the resultsof the uniaxial direct tension tests. Therefore, shear panel tests arerequired to more accurately estimate the torsional behavior ofSFRC members [1216]. The estimation of the torsional behaviorafter cracking of the SFRC members reinforced asymmetrically inthe transverse and longitudinal directions also requires a consider-ation on the difference in angles between the principle stress andthe crack direction [56,69]. Therefore, this paper presents a consti-tutive relationship for SFRC in tension based on the results of re-cently conducted SFRC shear panel tests [21,22], and proposes atorsional behavior model for SFRC modifying the original xed-angle softened-truss model (FA-STM) [56,69].

4. The constitutive model for SFRC in tension

As mentioned in the previous section, the shear and torsionalbehaviors of SFRC members rely heavily on the tensile perfor-mances of the materials subject to biaxial stress. In this respect,0.38 1.18 552 2063 3.310.55 0.95 552 2063 3.31

Par3

4Pa

)C1F1V1 test result Proposed modelVecchio and Collins (RC) Lim et al.Tan and Mansur Voo and FosterBischoff

218 H. Ju et al. / Composites:with high ber factors (F, say greater than 0.95) or a high volumefraction of steel ber (Vf) of 1.5 tend to show some strain-harden-ing behaviors, which can also be found in Chao et al.s test observa-tions [70]. Moreover, the shear cracking strength appeared to besimilar to the cracking strength of conventional reinforced con-crete, 0:33

f 0c

p[39,51], but the tensile behavior of SFRC in the

post-cracking region showed a drastic increase compared to the

0

1

2

Ten

sile

stre

ss,

1(M

Ten

sile

stre

ss,

1(M

Pa)

Ten

sile

stre

ss,

1(M

Pa)

Tensile strain, 1 (x10-3mm/mm)



100 mm crack spacing200 mm crack spacing

0

1

2

3

4

5

6C1F1V3 test result Proposed model

Vecchio and Collins (RC) Lim et al.Tan and Mansur Voo and Foster

Bischoff100 mm crack spacing

200 mm crack spacing

0

1

2

3

4

5



Bischoff

100 mm crack spacing200 mm crack spacing

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6

(a) C1F1V1 panel

(e) C1F3V3 panel

(c) C1F1V3 panel

Fig. 1. Tensile stressstrain behavior of SFRC tPa) 4



Bischoff 200 mm crack spacing

t B 45 (2013) 215231RC models (i.e., the tensile stressstrain relationship proposed byVecchio and Collins [39,51]). The tensile behavior model proposedby Lim et al. [12], which was adopted by most of the existing shearand torsional behavior models, considerably under-estimated thetensile behavior of the normal-strength specimens with low vol-ume fractions of steel ber (Vf), while it accurately estimated thetensile behaviors of the high-strength specimens. It is shown that

Ten

sile

stre

ss,

1(M

Ten

sile

stre

ss,

1(M

Pa)

Ten

sile

stre

ss,

1(M

Pa)




0

1

2


0

1

2

3

4

5

6C1F2V3 test result Proposed modelVecchio and Collins (RC) Lim et al.Tan and Mansur Voo and Foster

Bischoff 100 mm crack spacing200 mm crack spacing

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7

C2F1V3 test result Proposed model



(d) C1F2V3 panel

(b) C1F1V2 panel

(f) C2F1V3 panelest panels and various prediction models.

(con

Par0

1

2

3

4

5

6

7

0 1 2 3 4 5 6

C2F2V3 test result Proposed model



(g) C2F2V3 panel

Ten

sile

stre

ss,

1(M

Pa)


Fig. 1.

H. Ju et al. / Composites:this model actually cannot capture the hardening behavior ob-served in the specimens with high ber factors (F). The model pro-posed by Abrishami and Mitchell [14] considers the contribution ofbers only after yielding of reinforcement, thus, no direct compar-ison is made in Fig. 1. The tensile behavior model proposed byBischoff [15] was derived semi-empirically introducing a bond fac-tor based on the uniaxial tensile responses of the SFRC prisms. Thismodel estimated the tensile behavior of SFRC relatively similar tothe experimental results before reinforcement yielding, but un-der-estimated the tensile behavior SFRC after reinforcement yield-ing compared to the other models. The model proposed by Tan andMansur [13], greatly underestimated the tensile behavior of SFRCwhen the volume fraction of steel ber (Vf) was low, but yielded re-sults similar to those of Bischoff [15] when the volume fraction ofsteel ber (Vf) was high. The variable-engagement model (VEM)proposed by Voo and Foster [16] is also shown in Fig. 1, for whichthe crack width was calculated by multiplying the tensile strain (er)by crack spacing (Sr), and the cracking spacing (St) of 100 mm and200 mmwere presented. The VEM showed relatively good analysisresults, and in particular, a considerably good estimation was pro-vided for the normal strength SFRC panels (C1 series). However,the tensile behaviors of the high-strength SFRC specimens (C2)

Most of the existing tensile behavior models presented in Table

Table 2Constitutive models of SFRC in tension reported in literature.

Researcher(s) Constitutive equation (descending branch) Ref.

Vecchio and Collinsa rr fcr1

500er

p for er > ecr [51]

Lim et al. b rfr 2glg0suVf lfdf for er > ecr [12]Tan and Mansurc rfr fcr ftu erecrecretf

h i fcr for ecr < er < etf [13]

rfr ftu for er > etfAbrishami and

Mitchellrfr 16Vf Ef er ecr;y 6 16Vf f yf for er > ecr;y [14]

Bischoff d rfr bf f cr for er > ecr [15]Voo and Fostere rfr rr Kf Kd lfdf

Vf sb for er > ecr [16]

a fcr 0:33f 0c

p.

b gl 0:5;g0 0:405; su 2:5f ct 2:5 0:33f 0c

p .

c ftu 2glg0suf Vf lfdf ;gl 0:33;g0 0:5; suf 4:12 MPa;ecr fcrEc; etf ecr 1 10:48

0:39 fcr.

d bf bc 0:4ff =fcr; ff 3 MPa for er < ecr;y; bf ff =fcr; ff 1:5 MPa forer > ecr;y; bc 1=1

500er

p .e rr fcr

1500er

p ;Kf atanw=ap 1 2wlf 2

;a df =3:5;Kd 1; sb 2:5f ct 2:5

0:33 f 0cp ;w er crack spacing.2 were derived based on the results of the uniaxial tension tests,leading to inappropriate evaluation of the tensile behaviors ofthe SFRC panels subjected to biaxial stresses. Some models are alsoquite complex in their form. Thus, it is necessary to develop a sim-ple constitutive model in tension that can describe the tensilebehavior of SFRC subjected to biaxial stresses. To reect the effectof bers on tensile behavior of concrete concisely, this paper uti-lizes the ber factor (F) [6], which simultaneously considers the -ber length (lf), ber diameter (df), volume fraction of steel ber (Vf),and bond factor (qf) according to ber types. The tensile stressstrain relation of SFRC subject to biaxial stress was derived usingthe Vecchio and Collins model [51] as the basic form of the equa-tion as follows:

rfr Ecf er 6 fcr;sfrc for er < ecr 1a

rfr 0:33

f 0c

p 3:5F1 500er0:51F

for er P ecr 1bwere over-estimated, and the cracking strength was also estimatedvery different from test results.

0

1

2

3

4

5

0 1 2 3 4 5 6

C2F3V3 test result Proposed modelVecchio and Collins (RC) Lim et al.Tan and Mansur Voo and FosterBischoff



(h) C2F3V3 panel

Ten

sile

stre

ss,

1(M

Pa)


tinued)

t B 45 (2013) 215231 219where rfr is the average tensile stress of SFRC, Ecf is the modulus ofelasticity of SFRC (fcr,sfrc/ecr), and fcr,sfrc is the cracking strength ofSFRC, which can be obtained by substituting the cracking strain (ecr)into Eq. (1b) under the assumption that the cracking strain (ecr) issimilar to that of conventional concrete, (that is, 0:33

f 0c

p=Ec) [51].

The ber factor (F) is dened as (ld/df)Vfqf, and qf is the bond factor,which is 1.0 for hooked-type, 0.75 for crimped-type, and 0.5 forstraight-type [8]. As shown in Fig. 1, the proposed model very clo-sely estimated the tensile behaviors of the shear panels subjectedto biaxial stresses. In particular, the tensile strain-hardening behav-iors of the specimens with ber factors (F) greater than 1.0, such asC1F1V3, C1F2V3, C2F1V3, and C2F2V3, were accurately estimated.

Fig. 2a shows the normalized shear cracking strength of SFRCmeasured from SFRC shear panel tests. It is worthwhile to notethat, in the case of the ber factor (F) with greater than about1.0, the cracking strength tended to be lower than 0:33

f 0c

pthat

is generally used for the cracking strength of RC. This shows thatber-reinforced concrete may have a similar or slightly lower levelof shear cracking strength compared to that of RC as reported byHarajli et al. [71] In this study, this phenomenon was inferred tobe due to the decrease in the net concrete cross-sectional areaaccording to the increase in the volume fraction of steel bers(Vf). Thus, the proposed model reected the decrease in cracking

5. Modied xed-angle softened truss model

Smeared-truss models can be classied into the rotating-angle

Part B 45 (2013) 2152311

1.2

ngt

h220 H. Ju et al. / Composites:strength as the ber factor (F) increases, and changes in the tensilebehavior of SFRC due to the volume fraction of the steel ber (Vf)and the ber types, as shown in Fig. 2b and c, respectively. Thatis, the tension stiffening and strain hardening can be described eas-ily by the proposed tensile behavior model as function of the berfactor (F).

model and the xed-angle model according to consideration ofthe crack direction. The former category includes the compressioneld theory (CFT) [39], the modied compression eld theory(MCFT) [51] and the rotating-angle softened-truss model (RA-STM) [50], while the latter category includes the xed-anglesoftened-truss model (FA-STM) [69] and the softened-membranemodel (SMM) [67]. Despite the slight difference between the twotypes of truss models, both theoretical models fully satisfy theequilibrium, strain compatibility, and stressstrain relationshipsof material, which are known as Naviers three fundamental prin-ciples of structural analysis [42,54,67]. For the analysis of SFRC tor-sional behavior, this study proposes a constitutive relationship ofSFRC in tension based on the SFRC panel tests, and FA-STM is mod-ied to make it suitable for torsional analysis under the assump-tion that SFRC members can be treated as plane elementsidealized with the thin tube [41]. The FA-STM [67,69] applies thestressstrain relationships of concrete in the xed-crack direction,which is determined from the stressstrain relation in the principalstress direction. Thus, this model cannot appropriately considerthe difference between the stresses in the crack direction and inthe principal stress direction as the deviated angle (b) betweenthe initial crack (a2) and the principle stress direction (a) increases,as shown in Fig. 3 [72,73]. Later, the deviated angle (b) was

0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Nor

maliz

ed sh

ear

cra

ckin

g st

re(f c

r,te

st/ 0

.33

f ck)

Fiber factor, F

Toronto panel test results

(a) Effect of fiber factor (F) on the shear cracking strength

0

0.5

1

1.5

2

0 1 2 3 4

No

rma

lized

st

ress

,

1/fcr

No

rma

lized

st

ress

,

1/fcr


df = 0.5, lf = 50, f = 1.0

Vf = 1.5 %

Vf = 0.5 %

Vf = 1.0 %

(b) Effect of fiber volume fraction

0

0.5

1

1.5

2

0 1 2 3 4Tensile strain, 1 (x10-3mm/mm)

df = 0.5, lf = 50, Vf = 1.5 %

straight-type (f = 0.5)

hooked-type (f = 1.0)

crimped-type (f = 0.75)

(c) Effect of fiber type Fig. 2. Characteristics of proposed tensile behavior model for SFRC.

d

tr

l

1 2

2

l

t

lt

ltt

l(a) Element coordinatesystem

(b) Element in l - t direction

2

2t1l

2

1

12

21

d

r

r

d

dtrl

(c) Stresses at initial crackdirection

2

( )1 21,c c

222

c

rcd

( ),l lt

21c

21c

( ),t lt ( )2 21,c c

(d) Stresses at principaldirection (e) Mohrs stress circle Fig. 3. Average stresses of an element in thin-walled tube.

incorporated in the softened membrane model for torsion (SMMT)[53] and the more recent text book [70], whereas it was consideredin the concrete in compression only but not in the concrete in ten-sion. In this study, the stresses and strains in the xed-crack direc-tion (21 direction) were calculated by transforming the stressesand strains in the principal stress direction (dr direction) by thedeviated angle, b, rather than modifying the concrete stressstrainrelationship to account for the b. The proposed method for tor-sional behavior analysis of SFRC can be also applied to other ad-vanced softened truss models developed for shear behavioranalysis such as softened membrane model (SMM) and distributedstress eld model (DSFM). In this study, however, the derived SFRCtensile constitutive model is applied to the modied FA-STM be-cause this model is relatively simple and can reect the differenceof the angle between principle stresses and crack direction.

5.1. Equilibrium equations

As shown in Fig. 4, when a SFRC member is subjected to tor-sional moment (T), this external force is resisted by the shear ow(q) in a thin-walled tube with an effective thickness of td [53]. Equi-librium equations in the shear ow zone can be expressed with ref-

qt Atstd

6

where Al is the area of the longitudinal reinforcements p0 is theperimeter of the centerline of shear ow, td is the effective thicknessof the shear ow zone, and s is the spacing of transverse reinforce-ment. According to Bredts thin-walled tube theory [30], the rela-tionship between the shear stress within the shear ow zone andthe torsional moment can be expressed as follows:

T 2A0q 2A0tdslt 7where A0 is the cross-sectional area enclosed by the center line ofthe shear ow zone. In the case of a SFRC member subjected to puretorsion, all normal stresses are assumed to be zero (i.e., rl = rt = 0),and the initial crack angle (a2) becomes 45 [53,69] because the ele-ment A is under a pure shear stress condition, as shown in Fig. 4a.

5.2. Compatibility equations

Fig. 4b shows element A in detail, which is the part of the thin-walled tube shown in Fig. 4a. The compatibility equation of thiselement is calculated using Mohrs strain circle shown in Fig. 5

H. Ju et al. / Composites: Part B 45 (2013) 215231 221erence to Fig. 3 as follows:

rl rc2 cos2 a2 rc1 sin2 a2 sc212 sina2 cosa2 qlfl 2

rt rc2 sin2 a2 rc1 cos2 a2 sc212 sina2 cosa2 qtft 3

slt rc1 rc2 sina2 cosa2 sc21cos2 a2 sin2 a2 4where rl and rt are the average normal stresses in the l and t direc-tions, respectively, slt is the average shear stress in the lt coordi-nate, rc2 and rc1 are the average compressive and tensile stressesof concrete in the 2 and 1 directions, respectively, sc21 is the averageshear stress of concrete in the initial crack direction (i.e., 21 coor-dinate), a2 is the angle between the lt and 21 coordinate, fl and ftare the stress of reinforcement in the l and t directions, respectively.ql and qt are the reinforcement ratios in the l and t directions,respectively, which can be dened as follows:

ql Alp0td

5

L

dt

0

2dt

Shear flow path

Shear flow, qper unit length around perimeter P0

A

A(a) Torsional memberFig. 4. Torsional behavior of a SFRC memberas follows:

el e2 cos2 a2 e1 sin2 a2 c21 sina2 cosa2 8

et e2 sin2 a2 e1 cos2 a2 c21 sina2 cosa2 9

clt 2e1 e2 sina2 cosa2 c21cos2 a2 sin2 a2 10where el and et are the average strains in the l and t directions,respectively, clt is the average shear strain in the lt coordinate, e2and e1 are the average strain in the 2 and 1 directions, respectively,and c21 is the shear strain in the 21 coordinate. The relationshipbetween average shear strain (clt) and the twist angle per unitlength (h) can be dened as follows [30]:

h p02A0

clt 11

As indicated in Fig. 4a and b, a strain gradient at the concretestrut idealized with a thin tube occurs when members with non-circular sections are subjected to torsion [48,52,53,60]. According

T

L

dt

ds

stresses

strains

Effective outside surface

c ck f

,t cr fk frs

curvature

(b) Element Aand strain gradients in thin-walled tube.

where R is:

R 2e2sclt sin 2a2

4e2clt sin 2a2

19

5.3. Constitutive relationships

The constitutive relationships of the steel bars in tension andthe concrete in compression of FA-STM proposed by Hsu andZhang [56] shown in Fig. 6 are used in this study. Eq. (1) proposedin this study (Fig. 2) was applied as a tensile stressstrain relation-ship of SFRC. The stresses (rc2 and rc1) and strains (e2 and e1) in thecrack direction (21) can be determined by transforming the stres-ses and strains in the principal stress direction (dr) by the devia-tion angle of b as follows [70]:

rc2 rcd cos2 b rcr sin2 b 20

d

d

'cf

'cf

00

Eq. 26

nonsoftened d

d

d

d

rd

(a) Compressive stress-strain relationship of SFRC

Eq. 1-a

Eq. 1-b,cr sfrcf

fr

/c cr crE f

r

r

w

crf

Vecchio and Collins70

rr

,/cf cr sfrc crE f

0

=

=

Part B 45 (2013) 215231to Jeng and Hsu [53] and Jeng [54], the relationship between thecurvature of the strain gradient in a thin-walled tube and the twistangle per unit length (h) is expressed as follows:

wc h sin 2a2 12a

wt h sin 2a2 12bwhere wc and wt are the curvature of compression and the tensionstrut of concrete, respectively. As shown in Fig. 4b, the strain gradi-ent in the 2 and 1 directions is assumed to have a linear distributionwithin the effective thickness of the shear ow zone (td), which isexpressed as follows [53,54]:

td e2swc e1s

wt13

where e2s and e1s are the compressive and tensile strains at the sur-faces of the tube in the 2 and 1 directions, respectively. Under theassumption of a linear strain distribution, the relationship betweenthe average strains in element A within the effective depth (td) andthe maximum strain at the surface of the thin-walled tube can alsobe expressed, as follows [53]:

e2 e2s2 14a

e1 e1s2 14b

2

1 2

211, 2

222

21

2

rd

212 , 2

,

2lt

t

,

2lt

l

2

2

Fig. 5. Mohr circle for average strains.

222 H. Ju et al. / Composites:By substituting Eq. (11) into Eq. (12a) and then Eq. (13), theeffective thickness of the shear ow zone (td) is derived as follows[53]:

td 2A0e2sp0clt sin 2a215

where p0 is the perimeter of the centerline of shear ow, and A0 isthe cross-section area enclosed by the center line of shear ow,which can be expressed, respectively, as follows:

p0 pc 4td 16

A0 Ac 0:5pctd t2d 17where pc and Ac are the perimeter and the area of the gross concretesection. The effective thickness of the shear ow zone (td) also canbe modied by substituting Eqs. (16) and (17) into Eq. (15) as fol-lows [53]:

td 12R 4 pc 1R2

1 R

2

2p2c 4RR 4Ac

s24

35 180n sy

/s y yE f

=r0

cr

(b) Tensile stress-strain relationship of SFRC

sf

nfyf

Bare rebar

Eq. 36

sfsf

(c) Stress-strain relationship of steel rebarFig. 6. Stressstrain relationship of materials.

Gaussian quadrature [77] to determine an average tensile-stressfactor (kt) as follows:

kt ers2ecr forersecr6 1 32a

kt ers2ecr I

ersfcr;sfrcfor

ersecr

> 1 32b

where I is the area of the tensile stressstrain curve of SFRC aftercracking (er > ecr). Note that detailed calculation procedures are gi-ven in Appendix A.

As shown in Fig. 6c, the stressstrain relationship of reinforce-ment [48,50,53,67,69], which reects the tension-stiffening effectof the embedded bar in concrete, is adopted in this study asfollows:

fs Eses for es 6 en 33a

fs fy 0:91 2B 0:02 0:25B esey

for es > en 33b

where fs is the stress in the reinforcement, Es is the modulus of elas-ticity of the reinforcement, es is the strain in the reinforcement, fy isthe yield strength of the bare bar, fn is the smeared yield strength of

Parrc1 rcd sin2 b rcr cos2 b 21

e2 ed cos2 b er sin2 b 22

e1 ed sin2 b er cos2 b 23

where rcd and rcr are the average principal compressive and tensilestresses in concrete, respectively, and ed and er are the average prin-cipal compressive and tensile strains, respectively. The deviationangle (b = a2 a) between the initial crack angle (xed-crack angle)and the angle of the principal stress direction is calculated fromMohrs strain circle (Fig. 5) as follows [53,67,73]:

b 12tan1

c21e2 e1

24

Additionally, the shear stress of concrete in the crack direction(sc21) can be calculated by transforming the principal stresses bythe deviation angle (b) as follows [73]:

sc21 rd rr sinb cos b 25As shown in Fig. 6a, the compressive stressstrain relationship ofconcrete considering the softening effect of concrete is used in thisstudy as follows [56]:

rd ff 0c 2edfe0

ed

fe0

2" #for

edfe0

6 1 26a

rd ff 0c 1ed=fe0 14=f 1

2" #for

edfe0

> 1 26b

f 5:8f 0c

p 11 400erg

q 6 0:9 where g AtftyphAlflys

27

where f is the softening coefcient of concrete in compression, e0 isthe strain at the compressive strength of concrete, and fty and fly arethe yield strength of transverse and longitudinal reinforcement,respectively. It is reported in several studies on material character-istics of SFRC [24,7476] that the compressive strength of SFRC wasalmost same with that of conventional concrete, but the strain atthe compressive strength of SFRC (e0) appear to be somewhat great-er than that of conventional concrete. Based on study results men-tioned above, e0 of 0.003 was used in this study.

To transform a torsional member in the three-dimensionalspace into a two-dimensional plane element, the average compres-sive stress in the concrete strut with effective thickness (td) needsto be estimated considering the linear strain gradients induced bythe torsional moment. As shown in Figs. 4b and 7a, the averagecompressive stress of the concrete struts can be expressed usingthe average compressive stress factor (kc), as follows:

rcd kcff 0c 28

where kc is the ratio of the average compressive stress to the peakcompressive stress of the concrete struts. kc is obtained by integrat-ing Eq. (26) with respect to the compressive strain through theeffective depth (td), and then normalizing by the maximum com-pressive stress (ff 0c) and the maximum principal compressive strain(eds = 2ed) as follows:

kc edsfe0 eds23fe02

foredsfe0

6 1 29a

H. Ju et al. / Composites:kc 1 fe03eds eds fe03

3eds4e0 fe02for

edsfe0

> 1 29bAs shown in Figs. 4b and 7b, the tensile strain gradient withinthe effective thickness (td) in the perpendicular direction of thecompressive strut is also considered by using the average tensilestress factor (kt), based on which the average tensile stress (rcr)can be expressed, as follows [53]:

rcr ktfcr;sfrc 30where kt is the ratio of the average tensile stress to the tensile crack-ing stress of the SFRC strut. The average tensile stress factor (kt) isobtained in the same way as the average compressive stress factor[53].

kt 1ersfcr;sfrc

Z ers0

rfrerder 31

where ers is the maximum principal tensile strain (=2er), and rfreris calculated using Eq. (1). However, direct integration of Eq. (1b) isdifcult, thus Eq. (1b) is integrated numerically using the four-point

cdt

22 2

s =

2s'cf 'cd c ck f =

2ds

d

=

ds

(a) Strain and stress distribution in compressivestruts and average stress block

tdt

11 2

s =

1s,cr ff ,cr t cr fk f =

2rs

r

=

rs

(b) Strain and stress distribution in tensile strutsand average stress block

Fig. 7. Idealization of stress distributions in thin-walled tube to average stresses.

t B 45 (2013) 215231 223the reinforcement, and ey is the yield strain of the bare bar. Here, enis the smeared yield strain in the reinforcement that equals toey(0.932B), and B is dened as follows:

B 1q

fcrfy

1:534

where q is the reinforcement ratio, which should be greater than0.5%, and fcr is the shear cracking stress in the concrete.

5.4. Solution algorithm

The convergence conditions are imposed to the calculation pro-cedures at any loading states, combining the equilibrium equations(Eqs. (2) and (3)) as follows:

qlfl qt ft rl rt rc2 rc1 35

qlfl qt ft rl rt rc2 rc1 cos 2a2 2sc21 sin 2a2 36Fig. 8 shows a ow chart of nonlinear analysis procedures. Here, b = 0is used as an initial value under the assumption that the initial crackangle is identical to the angle of principal stress. The following is asummary of procedures for analyzing the torsional behavior of SFRC:

To verify the torsional behavior model presented in this study,38 SFRC specimens were collected from previous studies [20,24

START

21Assume

and d ds dSelect =

rAssume

Cal. by Eq. (28), (30)

, c c

d r

Cal. by Eq. (20), (21), (22), (23)

2 1 2 1, , , c c

Cal. by Eq. (16), (17), (18)

0 0, , dp A t

, , , f f

Cal. by Eq. (8), (9), (10), (25)

21, , , c

l t lt

224 H. Ju et al. / Composites: ParCheck if satisfy Eq. (36)l l t tf f

Check if satisfy Eq. (35)

Cal. by Eq. (4), (7), (11), (24)

Cal. by Eq. (5), (6), (33)

l l t tf f +

l t l t

, , , lt T

Is end point of strain?

ds

Yes

No

No

No

Yes

Yes

END

Fig. 8. Solution algorithm of the proposed model.29]. The material and dimensional properties of the specimensare shown in Table 3. All the collected specimens were reinforcedin the longitudinal and transverse directions, and more than theminimum amount of reinforcements were provided, as speciedin the structural concrete design codes [31,35]. Moreover, the berfactors (F) of the specimens ranged from 0.1 to 2.0, the volumefractions of steel ber (Vf) were between 0.3% and 3%, and the com-pressive strengths (f 0c) ranged from 17 MPa to 51 MPa.

Fig. 9 presents the comparison of test results on 38 SFRC tor-sional specimens and analysis results by the analytical model.The dotted lines (refer to Analysis-0) indicate the analysis resultsby the analytical torsional behavior model adopting the tensileconstitutive model presented in Eq. (1). The analysis model wellestimated the overall behavior of specimen including initial stiff-ness, but, torsional cracking moments of specimens were signi-cantly underestimated. This phenomena were also found by Jengand Hsu [53], who reported that torsional behavior model adoptingthe tensile constitutive model derived in shear underestimated thetorsional cracking moment. They also reported that this is becauseof the strain gradient effect which occurs within the effectivethickness in torsional members unlike the members subjected toshear, and for this reason, they increased the tensile crackingstrength for torsion by 2.1 times greater than that for shear. Thus,the tensile constitutive model derived from panel test in this studyshould also be modied to be applied to torsional behavior model.For this purpose, a tting coefcient for torsion (af) is introduced,which can be estimated by the ratio of the torsional cracking mo-ment obtained from test results to that of analysis results. The t-(1) Select a value for ed and eds = 2ed.(2) Assume a value of c21.(3) Assume a value of er.(4) Calculate rcd and rcr using Eqs. (28) and (30).(5) Calculate rc2, rc1, e2, and e1 using Eqs. (20)(23).(6) Calculate el, et, clt, and sc21 using Eqs. (8), (9), (10), and (25).(7) Calculate p0, A0, and td using Eqs. (16)(18).(8) Calculate ql, qt, fl, and ft using Eqs. (5), (6), and (33).(9) If qlfl + qtft does not satisfy Eq. (35), then repeat steps 36.

(10) If qlflqtft does not satisfy Eq. (36), then repeat steps 27.(11) Calculate T, h, and b using Eqs. (7), (11), and (24).(12) Select another value of ed in proper increments up to

eds = 0.0035, and repeat steps 211 to obtain the completeresponse.

Existing studies have reported that the SFRC member showedsufcient ductile behavior compared to the RC members[11,12,23,24,26]. Accordingly, the load after the maximum valuedecreases steadily as the angle of twist increases. But, when testsare conducted under load-control system, the post-peak can behardly captured by data acquisition system. Thus, determiningthe actual failure point of the specimens tested under load-controlis difcult. For this reason, the analysis of the specimens underload-control was terminated when both the transverse and longi-tudinal reinforcements yielded. For specimens under displace-ment-control, the analysis was terminated when the principalcompressive strain (eds) at the surface of thinwalled tube reachedat 0.0035 [20,26].

6. Verication

t B 45 (2013) 215231ting coefcient for torsion (af) of 1.7 was determined in this study,and the modied tensile cracking strength for torsional membercan be expressed as follows:

H. Ju et al. / Composites: Part B 45 (2013) 215231 2250

1

2

3

4

5

6

7

Torq

ue

(kN

Torq

ue

(kN

Torq

ue

(kN

Torq

ue

(kN

m

)

0 0.05 0.1 0.15 0.2Angle of Twist (rad/m)












R40C-F1Analysis-FAnalysis-0

(a) Vf = 0.3 %, F = 0.1

0

1

2

3

4

5

6

7


(b) Vf = 0.6 %, F = 0.2

0

1

2

3

4

5

6

7


(c) Vf = 0.9 % , F = 0.3

0

1

2

3

4

5

6

7


(d) Vf = 1.2 % , F = 0.4

0

1

2

3

4

5

6

7

R40L-F1Analysis-FAnalysis-0

(e) Vf = 0.3 %, F = 0.1

0

1

2

3

4

5

6

7

R40L-F2Analysis-F

Analysis-0

(f) Vf = 0.6 %, F = 0.2

0

1

2

3

4

5

6

7


(g) Vf = 0.9 %, F = 0.3

0

1

2

3

4

5

6

7


(h) Vf = 1.2%, F = 0.4

0

1

2

3

4

5

6

7

R40T-F1Analysis-FAnalysis-0

(i) Vf = 0.3 %, F = 0.1

0

1

2

3

4

5

6

7

R40T-F2

Analysis-F

Analysis-0

(j) Vf = 0.6 %, F = 0.2

0

1

2

3

4

5

6

7

R40T-F3Analysis-FAnalysis-0

(k) Vf = 0.9 %, F = 0.3

0

1

2

3

4

5

6

7

R40T-F4

Analysis-F

Analysis-0

(l) Vf = 1.2 % , F = 0.4

Fig. 9. Verications of proposed model by comparison with test results in literature.

, F

Par30(m) Vf = 1.5 %, F = 1.5

30 (n) Vf = 1.0 %

226 H. Ju et al. / Composites:fcr;sfrc af 0:33f 0c

p 3:5F1 500ecr0:51F

where af 1:7; ecr 0:33f 0c

q=Ec

37

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25

Torq

ue (k

N m

)

0

5

10

15

20

25

30

Torq

ue (k

N m

)To

rque

(kN

m)

Angle of Twist (rad/m)0 0.05 0.1

Angle of Twi

0 0.05 0.1 0.15 0.2 0.25Angle of Twist (rad/m)

0 0.05 0.1 0.Angle of Twi

T3Analysis-FAnalysis-0

0

5

10

15

20

25

T9Analysis-F

Analysis-0

(p) Vf = 1.0 %, F = 1.0

0

1

2

3

4

0

1

2

3

4

5

(q) Vf = 0.9 %, F

0

10

20

30

40

50

Torq

ue (k

N m

)

0

10

20

30

40

50

0 0.05 0.1Angle of Twist (rad/m)


0 0.05 0.1Angle of Tw

A-1.0Analysis-FAnalysis-0

C-1.0Analysis-FAnalysis-0

(s) Vf = 1.0 %, F = 0.3

(v) Vf = 1.0 %, F = 0.3

0

10

20

30

40

50

0 0.05Angle of Twi

(t) Vf = 1.5 %, F

(w) Vf = 1.0 %, F

Fig. 9. (con= 0.6 30 (o) Vf = 2.0 %, F = 2.0

t B 45 (2013) 215231The analysis results based on the modied tensile crackingstrength Eq. (37) are presented as a chain line in Fig. 9 (refer toAnalysis-F).

Fig. 9al shows comparisons of the Gunneswara Rao and RamaSeshus test results [20] with the analysis results of the analytical

0.15 0.2 0.25st (rad/m)

0 0.05 0.1 0.15 0.2 0.25Angle of Twist (rad/m)

15 0.2 0.25 0.3st (rad/m)


0

5

10

15

20

25


0

1

2

3

4

5

RF1

Analysis-F

Analysis-0

= 0.6

0

10

20

30

40

50


A-0.5Analysis-F

Analysis-0

(r) Vf = 0.5 %, F = 0.1

0.15 0.2ist (rad/m)


RR1Analysis-FAnalysis-0

RR3Analysis-FAnalysis-0

0.1st (rad/m)

A-1.5

Analysis-F

Analysis-0

= 0.5

= 0.3

0

10

20

30

40

50

0 0.05 0.1

Angle of Twist (rad/m)

B-1.0

Analysis-F

Analysis-0

(u) Vf = 1.0 %, F = 0.3

(x) Vf = 3.0 %, F = 1.1

tinued)

0.f Tw

0.f Tw

%, F

, %

(con

Par0

3

6

9

12

15

18


Angle of Twist (rad/m)0 0.02

Angle o

0 0.02Angle o

T1

Analysis-F

Analysis-0

(y) Vf = 0.5 %, F = 0.3

0

3

6

9

12

15

18 (z) Vf = 1.0

0123456789

10

T10

Analysis-F

Analysis-0

(ab) Vf = 1.0%, F = 0.4

0123456789

10f (ac) V = 1.5

Torq

ue (k

N m

)To

rque

(kN

m)

0.06 0.080.040.020

Fig. 9.

H. Ju et al. / Composites:model. These specimens were classied into three groups, (a)(d),(e)(h), and (i)(l), according to gt/gl (refer to the bottom of Table3). gt/gl of series (a)(d), (e)(h) and (i)(l) series was 1.36, 0.45and 2.17, respectively. Moreover, the volume fraction of steel berin each group varied from 0.3% to 1.2%. For instance, from (a) to (d),Vf increased from 0.3% to 1.2%. The analysis results of groups (a)(d) conrmed that the analytical model well captured the effectof the volume fraction of steel ber (Vf) on the torsional behaviorof SFRC. Such a tendency was also found in groups (e)(h) and(i)(l). Moreover, in comparing each group for instance, compar-ing (a), (e), and (i) the analytical model rationally evaluated theeffect of the reinforcement ratio (i.e., gt/gl) on the torsional behav-ior of SFRC. However, the initial torsional cracking strengths ofthese specimens were somewhat underestimated.

Fig. 9mp shows comparisons of the analysis results estimatedfrom the analytical model with the test results reported by Craiget al. [24]. gt/gl was 0.60 for all specimens, and the main test vari-ables were the volume fraction (Vf) and the aspect ratio (ld/df) ofsteel ber. The analysis results of the analytical model showedpretty close agreements with tests results of the SFRC specimensregardless of the volume fraction of steel ber (Vf) and the bertype. In particular, the behaviors of specimens T3, T8, and T9 withber factors (F) greater than 1.0 were estimated very accurately bythe analytical model. However, the failure points of these speci-mens showed some differences.

Fig. 9q shows the test results reported by Narayanan andKareem-Palanjian [25] and estimations by the analytical modelpresented in this study. Narayanan and Kareem-Palanjian carriedout experiments on a total of ten SFRC torsional specimens, asshown in Table 3. However, there is no information on the tor-que-twist relation of the specimens (i.e., torsional behavior), ex-cept for specimen RF1; thus, for those specimens, only thetorsional strengths were estimated by the analytical model as pro-vided in Table 3. For specimen RF1, the analytical model somewhat04 0.06 0.08ist (rad/m)

04 0.06 0.08ist (rad/m)


T2

Analysis-F

Analysis-0

= 0.6

0

1

2

3

4

5

6

7

8

9

T05

Analysis-F

Analysis-0

(aa) Vf = 0.5%, F = 0.2

T15

Analysis

Analysis-0

F = 0.6

tinued)

t B 45 (2013) 215231 227underestimated the torsional strength and overall behavior. How-ever, the torsional strengths of the rest of the specimens, were veryaccurately estimated, in which the average of test to analysis value(Tu,test/Tu,cal.) for the ten specimens was 1.09, the standard deviation(SD) was 0.141, and the coefcient of variation (COV) was 0.130.

Fig. 9rv presents comparisons of the experimental results re-ported by Mansur et al. [26] with the analysis results. SpecimensA-0.5, A-1.0, and A-1.5 shown in Fig. 9rt had an gl of about0.6%, and the volume fractions of steel ber (Vf) were 0.5%, 1.0%,and 1.5%, respectively. Moreover, for specimens B-1.0 and C-1.0shown in Fig. 9uv, the volume fraction of steel ber (Vf) was1.0%, and gl and gt were about 1.5 and 2.0 times greater than theA series, respectively. The analysis result of specimen A-0.5 witha small volume fraction of steel ber (Vf) was somewhat overesti-mated in terms of the strength and deformation capacity comparedto the test result, which is thought to be due to the balling of ber.In terms of the rest of the specimens, the analytical model appearsto accurately reect the effect of the volume fraction of steel ber(Vf) and the relative reinforcement ratio (gt/gl) on the torsionalbehavior of SFRC.

Fig. 9w and x shows comparisons of the test results by Chaliorisand Karayannis [27] with those of the presented analytical model.The volume fractions of steel ber (Vf) in the two specimens were1.0% and 3.0%, respectively, and the corresponding ber factors (F)were 0.3 and 1.1, respectively. The proposed model slightly overes-timated the torsional strength of the specimens, but accuratelypredicted the overall torsional behavior of the specimens.

Fig. 9yac shows comparisons of the experimental results re-ported by Al-Ausi et al. [28] and Kaushik and Sasturkar [29] withthe analysis results of the analytical model. As indicated by the tor-que-twist behavior of the specimens, it seems that the experimentswere performed under load-control; thus, the analysis was termi-nated when both the longitudinal and transverse reinforcementsyield. Although the analysis results showed a difference in the

Table 3Comparison of the proposed model with previous tests.

Ref. Specimen f 0c MPa H B (mm mm) x0 (mm) y0 (mm) gl (%) gt (%) fly (MPa) fly (MPa) s (mm) Vf (%) lfdf qf hu,test hu,cal.hu;testhu;cal:

Tu,test Tu,cal. Tu;testTu;cal:

rad/m 102 kN m[20] R40C-F1 40.05 200 100 72 172 1.00 1.36 90 0.30 90 0.30 41

0:540.5 12.63 13.22 0.955 5.56 5.60 0.992

R40C-F2 41.06 200 100 72 172 1.00 1.36 90 0.60 90 0.60 410:54

0.5 12.05 13.09 0.921 5.69 5.73 0.992

R40C-F3 41.98 200 100 72 172 1.00 1.36 90 0.90 90 0.90 410:54

0.5 12.32 12.94 0.952 5.73 5.87 0.976

R40C-F4 43.26 200 100 72 172 1.00 1.36 90 1.20 90 1.20 410:54

0.5 12.60 12.91 0.976 5.82 6.01 0.967

R40L-F1 41.28 200 100 74 174 1.57 0.70 100 0.30 100 0.30 410:54

0.5 15.77 14.59 1.081 4.11 4.49 0.914

R40L-F2 42.16 200 100 74 174 1.57 0.70 100 0.60 100 0.60 410:54

0.5 15.47 14.56 1.063 4.19 4.63 0.904

R40L-F3 43.37 200 100 74 174 1.57 0.70 100 0.90 100 0.90 410:54

0.5 16.21 14.67 1.105 4.23 4.79 0.884

R40L-F4 44.06 200 100 74 174 1.57 0.70 100 1.20 100 1.20 410:54

0.5 14.53 14.67 0.990 4.23 4.94 0.858

R40T-F1 41.47 200 100 72 172 0.57 1.23 100 0.30 100 0.30 410:54

0.5 15.77 18.26 0.864 3.85 4.40 0.874

R40T-F2 42.81 200 100 72 172 0.57 1.23 100 0.60 100 0.60 410:54

0.5 15.84 19.09 0.830 3.93 4.52 0.871

R40T-F3 43.06 200 100 72 172 0.57 1.23 100 0.90 100 0.90 410:54

0.5 15.23 19.14 0.796 3.98 4.63 0.859

R40T-F4 43.87 200 100 72 172 0.57 1.23 100 1.20 100 1.20 410:54

0.5 14.51 19.27 0.753 4.02 4.75 0.845

[24] T3 32.19 304.8 152.4 112.4 264.8 1.09 0.65 177.8 1.50 177.8 1.50 500:5

1.0 4.48 8.01 0.559 16.84 17.11 0.984

T4 28.95 304.8 152.4 112.4 264.8 1.09 0.65 177.8 1.00 177.8 1.00 300:5

1.0 7.06 9.15 0.771 14.13 14.61 0.967

T8 33.78 304.8 152.4 112.4 264.8 1.09 0.65 177.8 2.00 177.8 2.00 500:5

1.0 8.59 7.90 1.087 20.23 19.79 1.022

T9 29.64 304.8 152.4 112.4 264.8 1.09 1.30 88.89 1.00 88.89 1.00 500:5

1.0 6.14 8.67 0.709 16.50 20.23 0.816

[25] RF1 42.3 178 85 56.5 149.5 0.77 0.44 60 0.90 60 0.90 380:39

1.0 8.71 24.19 0.360 2.80 2.65 1.055

RF2 51.3 178 85 53.92 146.9 0.77 0.73 105 0.59 105 0.59 300:3

0.75 27.59 2.74 2.62 1.046

RF3 49.1 178 85 53.92 146.9 0.77 0.51 150 0.82 150 0.82 300:3

0.75 26.52 2.56 2.45 1.046

RF4 46.1 178 85 56.5 149.5 0.77 0.25 105 1.09 105 1.09 300:3

0.75 23.35 2.60 2.33 1.116

RF5 48.6 178 85 56.5 149.5 0.77 0.17 150 1.16 150 1.16 300:3

0.75 21.72 2.76 2.29 1.206

RF6 48.6 178 85 53.92 146.9 0.25 1.28 60 0.52 60 0.52 300:3

0.75 23.17 2.18 2.37 0.917

RF7 46.1 178 85 53.92 146.9 0.25 0.73 105 1.11 105 1.11 300:3

0.75 8.52 2.18 2.19 0.996

RF8 44.7 178 85 56.5 149.5 0.25 0.44 60 1.42 60 1.42 300:3

0.75 38.95 2.67 2.09 1.278

RF9 47.5 178 85 56.5 149.5 0.25 0.25 105 1.61 105 1.61 300:3

0.75 1.31 2.63 2.00 1.314

RF10 49.1 178 85 53.92 146.9 0.51 1.28 60 0.84 60 0.84 300:3

0.75 26.10 2.74 3.03 0.902

[26] A-0.5 25.8 300 300 260 260 0.63 0.69 120 0.50 120 0.50 300:8

1.0 4.57 3.83 1.194 27.34 33.79 0.809

A-1.0 21.4 300 300 260 260 0.63 0.69 120 1.00 120 1.00 300:8

1.0 5.28 4.05 1.304 29.01 35.18 0.825

A-1.5 28 300 300 260 260 0.63 0.69 120 1.50 120 1.50 300:8

1.0 5.26 3.77 1.398 34.67 37.87 0.916

B-1.0 21.4 300 300 260 260 0.95 1.03 80 1.00 80 1.00 300:8

1.0 5.96 5.13 1.161 36.46 46.71 0.781

C-1.0 21.4 300 300 260 260 1.27 1.37 60 1.00 60 1.00 300:8

1.0 5.90 5.43 1.087 40.86 51.79 0.789

[27] RR1 18.96 200 100 62 162 1.57 0.56 200 1.00 200 1.00 300:8

1.0 8.80 11.00 0.800 2.73 3.58 0.762

RR3 16.89 200 100 62 162 1.57 0.56 200 3.00 200 3.00 300:8

1.0 10.09 10.88 0.928 3.15 4.14 0.761

[28] T1 40.22 310 152 114.5 272.5 0.83 0.73 100 0.50 100 0.50 300:5

0.5 5.53 5.44 1.016 13.95 14.49 0.963

T2 40.15 310 152 114.5 272.5 0.83 0.73 100 1.00 100 1.00 300:5

0.5 6.09 5.55 1.097 15.67 15.55 1.007

[29] T05 24.22 300 125 69 244 0.49 1.05 45 0.50 45 0.50 38:60:46

0.5 5.11 5.94 0.860 7.50 8.05 0.932

T10 26.57 300 125 69 244 0.49 1.05 45 1.00 45 1.00 38:60:46

1.0 4.89 5.63 0.868 9.00 8.58 1.049

T15 25.51 300 125 69 244 0.49 1.05 45 1.50 45 1.50 38:60:46

1.0 5.33 5.93 0.899 8.50 9.14 0.930

Mean 0.944 Mean 0.951SD 0.213 SD 0.130COV 0.225 COV 0.136

C: compression failure, P: pull-out failure, gl: longitudinal reinforcement ratio to sectional area (=Al/Ac), gt: transverse reinforcement ratio to sectional area (=Atph/(Acs)) where, ph = 2(x0 + y0), Ac = BH.

228H.Ju

etal./Com

posites:Part

B45

(2013)215

231

kt 1ersfcr;f

Z ers0

rfrerder Ecf e2rs=2ersfcr;f

ers=2ecr

ers2ecr

forersecr6 1

A1As for Eq. (3b), which shows the tensile stressstrain relation aftercracking, direct integration with respect to tensile strain (er), is dif-cult; thus, Eq. (3b) can be integrated numerically using the Gauss-ian method as follows:Z 1 Xn

x = 0.000. . . 8/9 = 0.888. . .

Part B 45 (2013) 215231 229SFRC beams, based on which the following conclusions were made:

(1) The tensile behavior model for SFRC proposed in this studyconsidered the bond properties according to the volumefraction of steel ber (Vf), aspect ratio (ld/df), and ber typesusing the ber factor (F) in a simple manner, and the resultsobtained using the proposed model agreed with the resultsof the SFRC panel tests.

(2) Moreover, the proposed tensile behavior model rationallyconsidered the key inuential factors in behavior of SFRCin tension and accurately estimated the tension stiffeningbehavior and strain hardening behavior.

(3) The modied xed-angle softened-truss model, whichadopted the proposed tensile behavior model, reected thedeviation angle between the principal stress and xed-crackdirection, and, for this reason, the analytical model accu-rately estimated the torsional behavior of the SFRC memberwith various ber volume contents, ber types, membersizes, reinforcement ratios, and so on, compared to the exist-ing models.

(4) The analytical model provided excellent estimation of theoverall torsional behavior and the torsional strength, com-pared to the experimental results of the 38 SFRC torsionalmembers.

Acknowledgments

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Education, Science and Technology(2012R1A1A2002444).

Appendix A

The average tensile stress factor, kt, is calculated as follows:

kt 1ersfcr

Z ers0

rfrerder 28

where rfrer can be expressed as follows:rfr Ecf er for er 6 ecr 3a

rfr fcr 3:5F

1 500er0:51Ffor er > ecr 3binitial stiffness of the specimens, the overall torsional stiffness andbehavior after cracking were reasonably well estimated by the pro-posed model.

As shown in Table 3, the ratios of the experimental and analysisresults in terms of the torsional strength of all the specimens(Tu,test/Tu,cal.) showed a mean, a SD, and a COV of 0.951, 0.130 and0.136, respectively. Moreover, for the ratios of the twist angle perunit length at ultimate (hu,test/hu,cal.), the average, SD, and COV were0.944, 0.213 and 0.225, respectively, which are considered excel-lent results compared to existing prediction models.

7. Conclusions

In this study, based on the results of the SFRC panel tests, a sim-ple and rational constitutive model of SFRC in tension was pro-posed, and the proposed tensile behavior model was applied tothe modied xed-angle softened-truss model in order to estimatethe torsional behavior of SFRC members. Moreover, the proposedmodel was veried by comparing the experimental results of 38

H. Ju et al. / Composites:As inEq. (3a), the tensile stressstrain curveprior to cracking is linear,and the average tensile stress factor (kt) can be derived as follows:1yxdx

i1Wiyi A2

where Wi is the Gaussian weight factor, and yi is the function to beintegrated. To guarantee the accuracy of the numerical method inthis study, a four-point Gaussian quadrature [77] was used. Substi-tuting the range of 1 to 1 for tensile strain ranging from ecr to ers,the following equation can be obtained:Z ersecr

rfrerder X4i1

Wirfri A3

X1, X2, X3, and X4, which substituted the location xi (Table A1) withthe tensile strain (er) can be calculated as follows:

X1 ecr ers ecr2

ers ecrx12

A4a

X2 ecr ers ecr2

ers ecrx22

A4b

X3 ecr ers ecr2

ers ecrx32

A4c

X4 ecr ers ecr2

ers ecrx42

A4d

Moreover, as the Gaussian weight factorWi is a weight value corre-sponding to unity, it should be substituted with the value corre-sponding to the function intended to be applied. That is, thesubstitute value, Wsub,i, is:

Wsub;i Wi ers ecr2 A5

Therefore, the integrated value I of the stressstrain relationafter the cracking of SFRC against tensile strain (er) can be ex-pressed as follows:

I Z ersecr

rfre1der X4i1

Wirfri

rfrX1W1 rfrX2W2 rfrX3W3 rfrX4W4 A6The average tensile stress factor (kt) in the post-cracking region isnally derived as follows:

kt ers2ecr I

ersfcr;ffor

ersecr

> 1 A7

Table A1Gauss points for the integration from 1 to 1 [77].

No. of points Locations (xi) Associated weights (Wi)

1 x1 = 0.000... 2.0002 x1, x2 = 0.57735026918962 1.0003 x1, x3 = 0.77459666924148 5/9 = 0.555. . .2

4 x1, x4 = 0.8611363116 0.3478548451x2, x3 = 0.3399810436 0.6521451549

ParReferences

[1] ACI Committee 544. Design consideration for steel ber reinforced concrete(ACI 544.4R-88). ACI Struct J 1988; 85(5): 56380.

[2] Romualdi JP, Mandel JA. Tensile strength of concrete affected by uniformlydistributed and closely spaced short lengths of wire reinforcement. ACI J Proc1964;61(6):65771.

[3] Mitchell D, Collins MP. Diagonal compression eld theory a rational modelfor structural concrete in pure torsion. ACI Struct J 1974;71(8):396408.

[4] Romualdi JP, Batson GB. The mechanics of crack arrest in concrete. J Eng MechDiv, Proc ASCE 1963;89(EM3):14768.

[5] Dupont D, Vandewalle L. Calculation of crack widths with the s-e method, testand design methods for steel bre reinforced concrete: background andexperiences. In: Proceedings of the RILEM TC162-TDF workshop, RILEMtechnical committee 162-TDF, Bochum, Germany; 2003. p. 11944.

[6] Lee DH, Hwang JH, Ju H, Kim KS, Kuchma DA. Nonlinear nite element analysisof steel ber-reinforced concrete members using direct tension force transfermodel. Finite Elem Anal Des 2012;49(2):26686.

[7] Kim KS, Lee DH, Hwang JH, Kuchma DA. Shear behavior model for steel ber-reinforced concrete members without transverse reinforcements. Compos B:Eng 2012;43(2):232434.

[8] Narayanan R, Darwish IYS. Use of steel bers as shear reinforcement. ACI StructJ 1987;84(3):21627.

[9] Kwak YK, Eberhard MO, Kim WS, Kim JB. Shear strength of steel ber-reinforced concrete beams without stirrups. ACI Struct J 2002;99(4):5308.

[10] Ashour SA, Hassanain GS, Wafa FF. Shear behavior of high-strength berreinforced concrete beams. ACI Struct J 1992;89(2):17684.

[11] Lim TY, Paramsivam P, Lee SL. Shear and moment capacity of reinforced steel-ber concrete beams. Mag Concr Res 1987;39(140):14860.

[12] Lim TY, Paramsivam P, Lee SL. Analytical model for tensile behavior of steel-ber concrete. ACI Mater J 1987;84(4):28698.

[13] Tan KH, Mansur MA. Shear transfer in reinforced ber concrete. J Mater CivEng ASCE 1990;2(4):20214.

[14] Abrishami HH, Mitchell D. Inuence of steel bers on tension stiffnening. ACIStruct J 1997;94(6):76976.

[15] Bischoff PH. Tension stiffening and cracking of steel ber-reinforced concrete. JMater Civ Eng ASCE 2003;15(2):17482.

[16] Voo JYL, Foster SJ. Variable engagement model for bre reinforced concrete intension. UNICIV report no R-420 June 2003. The University of New SouthWales, Sydney, Australia; 2003. p. 186.

[17] Chalioris CE. Experimental study of the torsion of reinforced concretemembers. J Struct Eng Mech 2006;23(6):71337.

[18] Karayannis CG, Chalioris CE. Experimental validation of smeared analysis forplain concrete in torsion. J Struct Eng ASCE 2000;126(6):64653.

[19] Macgregor JG, Wight JK. Reinforced concrete: mechanics and design. Prentice-Hall; 2005. p. 1111.

[20] Gunneswara Rao TD, Rama Seshu D. Analytical model for the torsionalresponse of steel ber reinforced concrete members under pure torsion. CemConcr Compos 2005;27(4):493501.

[21] Susetyo J. Fibre reinforcement for shrikage crack control in prestressed, precastsegmental bridges. PhD dissertation, University of Toronto; 2009. 307pp.

[22] Susetyo J, Gauvreau P, Vecchio FJ. Effectiveness of steel ber as minimumshear reinforcement. ACI Struct J 2011;108(4):48896.

[23] Mansur MA, Paramasivam P. Steel bre reinforced concrete beams in puretorsion. Int J Cem Compos Lighw Concr 1982;4(4):3945.

[24] Craig RJ, Dunya S, Riaz J, Shirazi H. Torsional behavior of reinforced brousconcrete beams. In: Fiber reinforced concrete-international symposium, SP-81.American Concrete Institute, Detroit; 1984. p. 1749.

[25] Narayanan R, Kareem-Palanjian AS. Torsion in beams reinforced with bars andbers. J Struct Eng ASCE 1986;112(1):5366.

[26] Mansur MA, Nagataki S, Lee SH, Oosumimoto Y. Torsional response ofreinforced brous concrete beams. ACI Struct J 1989;86(1):3644.

[27] Chalioris CE, Karayannis CG. Effectiveness of the use of steel bres on thetorsional behavior of anged concrete beams. Cem Concr Compos2009;31(5):33141.

[28] Al-Ausi MA, Abdul-Whab HMS, Khidair RM. Effect of bres on the strength ofreinforced concrete beams under combined loading. In: The internationalconference on recent developments in bre reinforced cements andconcretes. UK: University of Wales College of Cardiff; 1989. p. 66475.

[29] Kaushik SK, Sasturkar PJ. Simply supported steel bre reinforced concretebeams under combined torsion, bending and shear. In: The internationalconference on recent developments in bre reinforced cements andconcretes. UK: University of Wales College of Cardiff; 1989. p. 68798.

[30] Bredt R. Kritische Bemerkungen zur Drehungselastizitat. Z Vereines DeutscherIngenieure, Band 1896;40(28):78590.

[31] ACI Committee 318. Building code requirements for structural concrete andcommentary (ACI 318M08). American Concrete Institute, Farmington Hills;2008. 473pp.

[32] American Association of State Highway and Transportation Ofcials. AASHTOLRFD bridge design specications. 3rd ed. AASHTO, Washington, DC; 2004.1450pp.

[33] Comite Euro-International du Beton. CEB-FIP model code 1990. Thomas

230 H. Ju et al. / Composites:Telford, London; 1990. 437pp.[34] CSA Committee A23.3-04. Design of concrete structures for buildings (CAV3-

A23.3-04). Canadian Standards Association, Canada; 2004. 232pp.[35] KCI-M-07. Design specications for concrete structures. Korea ConcreteInstitute; 2007. 523pp.

[36] Rausch E. Design of reinforced concrete in torsion (Berechnung desEisenbetons gegen Verdrehung). PhD thesis, Technische Hochschule, Berlin,Germany; 1929. 53pp.

[37] Ritter W. Die Bauweise Hennebique. Schweizerische Bauzeitung1899;33(7):5961.

[38] Mrsch E. Concrete-steel construction. New York: McGraw-Hill BookCompany; 1909. 368pp.

[39] Collins MP, Mitchell D. Prestressed concrete structure. Prentice Hill; 1991. 766pp.

[40] Anderson P. Rectangular concrete section under torsion. ACI J Proc1937;34(1):111.

[41] Lessig NN. Determination of the load carrying capacity of reinforced concreteelements with rectangular cross-section subjected to exure andtorsion. Moscow: Insitute Betonai Zhelezobetona; 1959. p. 428.

[42] Hsu TTC. Torsion of reinforced concrete. New York: Van Nostrand Reinhold,Inc.; 1984. 516pp.

[43] Hsu TTC. Torsion of structural concrete: behavior of reinforced concreterectangular members. Torsion of structural concrete, ACI SP-18. AmericanConcrete Institute; 1968. p. 261306.

[44] ACI Committee 318. Building code requirements for reinforced concrete (ACI31889). American Concrete Institute; 1989. 355pp.

[45] Lampert P. Torsion und Biegung von Stahlbetonbalken (Torsion and Bending ofReinforced Concrete Beams). Bericht 27, Institute f}ur Baustatik, Zurich; 1970.

[46] Lampert P, Th}urliman B. Ultimate strength and design of reinforced concretebeams in torsion and bending, vol. 31-I. Zurich: International Association ofBridge and Structural Engineering, Publication; 1971. p. 10731.

[47] Lampert P, Collins MP. Torsion, bending, and confusion an attempt toestablish the facts. ACI J Proc 1972;69(8):5004.

[48] Hsu TTC, Mo YL. Softening of concrete in torsional members theory and tests.ACI J Proc 1985;82(3):290303.

[49] Hsu TTC, Mo YL. Softening of concrete in torsional members-prestressedconcrete. ACI J Proc 1985;82(5):60315.

[50] Hsu TTC. Softened truss model theory for shear and torsion. ACI Struct J1988;85(5):62435.

[51] Vecchio FJ, Collins MP. Modied compression eld theory for reinforcedconcrete elements subjected to shear. ACI J Proc 1986;83(2):21931.

[52] Rahal KN, Collins MP. Analysis of sections subjected to combined shear andtorsion a theoretical model. ACI Struct J 1995;92(4):45969.

[53] Jeng CH, Hsu TTC. A softened membrane model for torsion in reinforcedconcrete members. Eng Struct 2009;32(9):194454.

[54] Jeng CH. Simple rational formulas for cracking torque and twist of reinforcedconcrete members. ACI Struct J 2010;107(2):18998.

[55] Vecchio FJ. Disturbed stress eld model for reinforced concrete: formulation. JStruct Eng ASCE 2000;126(9):10707.

[56] Hsu TTC, Zhang LX. Nonlinear analysis of membrane elements by xed-anglesoftened-truss model. ACI Struct J 1997;94(5):48392.

[57] Feo L, Mancusi G. Modeling shear deformability of thin-walled compositebeams with open cross-section. Mech Res Commun 2010;37(3):3205.

[58] Fraternali F, Feo L. On a moderate rotation theory of thin-walled compositebeams. Compos B: Eng 2000;31(2):14158.

[59] Ascione L, Feo L, Mancusi G. On the statical behaviour of bre-reinforcedpolymer thin-walled beams. Compos B: Eng 2000;31(8):64354.

[60] Rahal KN, Collins MP. Effect of thickness of concrete cover on shear-torsioninteraction an experimental investigation. ACI Struct J 1995;92(3):33442.

[61] Koutchoukali N, Belarbi A. Torsion of high-strength reinforced concrete beamsand minimum reinforcement requirement. ACI Struct J 2001;98(4):4629.

[62] You YM, Belarbi A. Thickness of shear ow zone in a circular RC column underpure torsion. Eng Struct 2011;33(9):243547.

[63] Rahal KN. Combined torsion and bending in reinforced and prestressedconcrete beams using simplied method for combined stress-resultants. ACIStruct J 2007;104(4):40211.

[64] Ashour SA, Samman TA, Radain TA. Torsional behavior of reinforced high-strength concrete deep beams. ACI Struct J 1999;96(6):104958.

[65] Chalioris CE. Analytical model for the torsional behaviour of reinforcedconcrete beams retrotted with FRP materials. Eng Struct2007;29(12):326376.

[66] El-Niema EI. Fiber reinforced concrete beams under torsion. ACI Struct J1993;90(5):48995.

[67] Hsu TTC, Mo YL. Unied theory of concrete structures. Wiley and Sons; 2010.p. 500.

[68] Karayannis CG. Nonlinear analysis and tests of steel-ber concrete beams intorsion. Struct Eng Mech 2000;9(4):32338.

[69] Pang XB, Hsu TTC. Fixed-angle softened-truss model for reinforced concrete.ACI Struct J 1996;93(2):197207.

[70] Chao S, Naaman AE, Parra-Montesinos GJ. Bond behavior of reinforcing bars intensile strain-hardening ber-reinforced cement composites. Struct J2009;106(6):897906.

[71] Harajli MH, Hout M, Jalkh W. Local bond stress-slip behavior of reinforcingbars embedded in plain and ber concrete. ACI Mater J 1995;92(4):34354.

[72] Lee JY, Mansur MY. New algorithm for xed-angle softened-truss model. ProcICE Struct Build 2006;159(6):34959.

t B 45 (2013) 215231[73] Lee JY, Kim SW, Mansour MY. Nonlinear analysis of shear-critical reinforcedconcrete beams using xed angle theory. J Struct Eng ASCE2011;137(10):101729.

[74] Choi KK, Park HG, Wight JK. Shear strength of steel ber-reinforced concretebeams without web reinforcement. ACI Struct J 2007;104(1):1221.

[75] Fanella DA, Naaman AE. Stressstrain properties of ber reinforced mortar incompression. ACI J 1985;82(4):47583.

[76] Nataraja MC, Dhang N, Gupta AP. Stressstrain curves for steel-ber reinforcedconcrete under compression. Cem Concr Compos 1999;21:38390.

[77] Smith IM, Grifths DV. Programming the nite element analysis. 4th ed. JohnWiley and Sons, Ltd.; 2004. p. 628.

H. Ju et al. / Composites: Part B 45 (2013) 215231 231

Torsional behavior model of steel-fiber-reinforced concrete members modifying fixed-angle softened-truss model1 Introduction2 Research significance3 Review of previous researches4 The constitutive model for SFRC in tension5 Modified fixed-angle softened truss model5.1 Equilibrium equations5.2 Compatibility equations5.3 Constitutive relationships5.4 Solution algorithm

6 Verification7 ConclusionsAcknowledgmentsAppendix AReferences

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