Balanced Ratio of Concrete Beams Internally Prestressed with Unbonded CFRP Tendons

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1 International Journal of Conrete Strutures and Materials Vol.11, No.1, pp.1 16, Marh 2017 DOI /s ISSN / eissn Balaned Ratio of Conrete Beams Internally Prestressed with Unbonded CFRP Tendons C. Lee 1), *, S. Shin 2), and H. Lee 2) (Reeived Otober 9, 2015, Aepted September 28, 2016, Published online Deember 27, 2016) Abstrat: The ompression or tension-ontrolled failure mode of onrete beams prestressed with unbonded FRP tendons is governed by the relative amount of prestressing tendon to the balaned one. Expliit assessment to determine the balaned reinforement ratio of a beam with unbonded tendons (q U pfb ) is diffiult beause it requires a priori knowledge of the deformed beam geometry in order to evaluate the unbonded tendon strain. In this study, a theoretial evaluation of q U pfb is presented based on a onept of three equivalent retangular urvature bloks for simply supported onrete beams internally prestressed with unbonded arbon-fiber-reinfored polymer (CFRP) tendons. The equivalent urvature bloks were iteratively refined to losely simulate beam rotations at the supports, mid-span beam defletion, and member-dependent strain of the unbonded tendon at the ultimate state. The model was verified by omparing its preditions with the test results. Parametri studies were performed to examine the effets of various parameters on q U pfb. Keywords: balaned ratio, arbon-fiber-reinfored polymer, modeling, prestressed onrete, unbonded. 1. Introdution Carbon Fiber Reinfored Polymers (CFRPs) have a number of valuable advantages: orrosion-free; high strength in tension; lower unit weight than steel; and low linear expansion oeffiient. As a substitute material for steels, different types of CFRP have been suggested in various appliations of onrete strutures, mainly for flexure as internal or external CFRP tendons and for CFRP stirrups (Elrefai et al. 2012; Lee et al. 2013; Han et al. 2015; Lee et al. 2015a, b; Girgle and Petr 2016). Conrete beams prestressed with an insuffiient amount of FRP tendons are subjet to a atastrophi brittle failure of the beam resulting from a sudden release of elasti energy at the moment of tensile rupture of the FRP tendons. A setion is regarded as the tension-ontrolled setion if the FRP tendon rupture governs the beam failure with a prestressing ratio (q pf ) less than the balaned ratio of the prestressing tendon (q pfb ). On the other hand, if onrete rushing governs beam failure where q pf is greater than q pfb, the setion is regarded as the ompression-ontrolled setion (ACI 440.1R ; ACI 440.4R ). 1) Shool of Arhiteture and Building Siene, Chung- Ang University, Seoul 06974, Korea. *Corresponding Author; dlee@au.a.kr 2) Graduate Shool, Chung-Ang University, Seoul 06974, Korea. Copyright The Author(s) This artile is published with open aess at Springerlink.om The balaned reinforement ratio of bonded FRP tendons (q B pfb ) is presented in ACI 440.4R-04 (2011), and was developed based on the ompatibility at a setion. Linearly varying strength redution fators were suggested from 0.65 for ompression-ontrolled setions to 0.85 for tension-ontrolled setions with CFRP (aramid fiber-reinfored polymer) tendons. Determining q U pfb beomes more hallenging for prestressed beams with unbonded FRP tendons, as the tensile stress of the unbonded tendons depends on the averaged elongation of onrete at the level of unbonded tendons along the beam span (Naaman and Alkhairi 1991a, b; Kato and Hayashida 1993; Maissen and De Semet 1995; Grae et al. 2006, 2008; Du et al. 2008; Heo et al. 2013). Consequently, it is expeted that q U pfb is always smaller than qb, and prestressing the beam with a reinforement ratio of unbodned pfb CFRP tendon (q U pf ) greater than qb pfb would preserve the ompression-ontrolled setion with a higher degree of plastiity, avoiding an abrupt brittle failure. However, this may result in overdesign and underutilize maximum tensile apaity of unbonded tendon. In order to realize a more struturally reliable and yet ost-effetive design of prestressed onrete beams with unbonded CFRP tendons, it seems neessary to have a rational evaluation of q U pfb available. An estimation of q U pfb is not provided in the ACI 440.4R- 04 (2011) or any other literature to authors best knowledge. In this study, modeling of tensile strain of unbonded FRP tendon at ultimate state is presented first and ompared with test results of the beam prestressed with unbonded CFRP tendons. Based on the developed iterative algorithms, a pratial formula of Df pfu is presented. The model is then modified and the iterative algorithm determining q U pfb is presented. 1

2 2. Equivalent Curvature Bloks at Ultimate State 2.1 Assumptions Figure 1 shows the load defletion urves of two typial unbonded-cfrp-prestressed onrete beams tested by Heo et al. (2013). Both beams were post-tensioned with straight CFRP tendons, but reinfored with different types of auxiliary bonded bars: CFRP bars for the TB45 beam and steel bars for the RU50 beam. The experimentally-observed load defletion urves are haraterized in approximation with the linearly inreasing segments between the harateristi points at different loading states: the initial raking state ( C in Fig. 1a, b), the yielding state with RU50 beam reinfored with auxiliary bonded steel bars ( Y in Fig. 1b); and the ultimate state ( U in Fig. 1(a)). Based on these and other experimental observations (Kato and Hayashida 1993; Maissen and De Semet 1995; Grae et al. 2006, 2008; Heo et al. 2013), the following assumptions were made in the modeling of three equivalent urvature bloks: (1) The urvature distributions an be losely approximated with linear segments between harateristi setions along the span at the ultimate state first raked setion ( s ), first yielded setion ( ys ) if steel bars with a yielding property were used as auxiliary bonded reinforement and ultimate setion ( us ), as shown in Fig. 1(); (2) The plane setion before deformation remains plane after deformation; (3) The shear deformation is relatively smaller than the flexural deformation and an thus be ignored; and (4) A positive bending moment induing a positive urvature generates ompression and tension, respetively, in the top and bottom fibers of the onrete in a beam setion. 2.2 Equivalent Retangular Curvature Bloks and Unbonded Tendon Strain For the beam subjeted to 4-point loading (Fig. 2a), the onept of one retangular equivalent retangular urvature blok (Figs. 3b and 3) has been used to predit the tensile strain of unbonded steel tendons (e pfu ) at the ultimate state (Harajli 1990; Naaman and Alkhairi 1991a, b; Lee et al. 1999). This onept impliitly assumes a linearly-hanging defletion of the beam between the support and the loading point, as shown in Fig. 2d. Due to a relatively lower modulus of elastiity than that of steel, relatively larger urvatures and defletions are indued under the same level of flexural load for beams prestressed with CFRP tendons than their ounterparts prestressed with steel tendons (ACI 440.4R ). Should the onept of a single equivalent urvature blok be used, a rational assessment of the beam defletion and the member-dependent tensile strain of unbonded CFRP tendons ould be hindered due to ignorane of the urvature distributions between the supports and the loading points (Figs. 2b and 2). In order to reflet the ontribution of urvature distributions between the support and loading point, three equivalent urvature bloks were assumed in this study. For ultimate loading states ( U in Fig. 1), the orresponding distributions of moments, linearly-approximated urvatures, and three equivalent urvature bloks along the span are illustrated in Fig. 3a, respetively. In Figs. 3a and 3b, M se (/ se ) denotes the moment (urvature) at a partiular setion se for the ultimate state, as shown in Fig. 1, where se = es for the setion at the beam end; s and ys for the first-raked and first-yielded setions, Fig. 1 The experimentally observed and theoretially modeled load defletion urves of the internally unbonded CFRP prestressed onrete beams. a Beam speimen TB45. b Beam speimen RU50. Charateristi points in theoretial load defletion modeling (not to sale). 2 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

3 Fig. 2 Approximation by one equivalent urvature blok. a Beam under 4-point loading. b Atual and linearly approximated urvature distributions. One equivalent urvature blok. d Defletion of the beam under the assumption of one equivalent urvature blok. Fig. 3 Moment, urvature distributions and defletions at ultimate state. a Moment distributions. b Linearly approximated urvature distributions. Three equivalent retangular urvature bloks. d Close approximations of beam defletion and rotation at support by three equivalent retangular urvature bloks to those by linearly varying urvatures. respetively, whih were first observed from the beam end; and us for the ultimate setion. At the ultimate loading state, one enter retangular urvature blok between the loading points with / p and L p and two symmetrially-loated adjaent retangular bloks, eah with / eq and L eq, were assumed. The use of three equivalent retangular urvature bloks enables the approximation of beam defletion between the loading points and rotations at the beam end supports, as a result of whih the overall beam defletion ould losely approximate that of linearly-varying urvature distributions /(x) in Fig. 3b (Fig. 3d). In addition, adopting three retangular urvature bloks failitates the integration of urvatures, as this onept assumes the distribution of onstant strain at beam setions within a blok along the beam span. For the 4-point loading ase, the magnitude (/ eq ) and length (L eq ) of the adjaent equivalent retangular urvature bloks between the loading points and supports in Fig. 3 were determined aording to Eq. (1). Defletion based on the linearly varying urvatures is at most ubi polynomial as it an be obtained by integrating linearly varying urvatures twie with respet to beam axis. International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 3

4 On the other hand, defletion based on three equivalent urvature blok is at most quadrati equation. Therefore, Eq. (1) whih provides the same beam end rotations and beam defletion between loading points an losely approximate ubially varying defletion of the beam resulting from the linearly varying urvatures. / eq L eq ¼ Z Lp =2þL sp L p =2 / ðþdx x for the same beam rotation at the supports / eq L eq L sp L Z Lp =2þL sp eq ¼ / ðþx x dx 2 L p =2 ð1aþ ð1bþ for the same beamdefletion in L p where L sp is the distane between the support and loading point (mm). For the 3-point loading ase, L p = L/10 ould be used (Lee et al. 1999). It is worth mentioning that the model adjusts the size of the equivalent bloks for a given magnitude of L p so that the beam end rotation and mid-span defletion beome the same as those obtained from the linearly varying urvature distributions between harateristi setions. An evaluation of Eq. (1) results in the following general expressions for / eq and L eq : / eq ¼ 3 ðkþ2 4 X and L eq ¼ 2 3 X K ð2þ where K ¼ x es / es þ x s / s þ x ys / ys þ x us / us, and X ¼ j es / es þ j s / s þ j ys / ys þ j us / us. The expressions for x se and j se are given in Table 1. In Table 1, values of L s were obtained based on moment distribution in Fig. 3a. Note that x se and j se in Eq. (2) are funtions of the unknown strain values of the unbonded CFRP tendon. The amount of additional stress of unbonded tendon depends on the elongations of onrete at the level of the tendon in the L eq region (D eq ) and the L p region (D p ) (Fig. 3): Z Lp =2þL eq d pf D eq ¼ 2 1 e eq dx L p =2 eq ¼ 2 / eq ðd pf eq ÞL eq ð3aþ D p ¼ 2 Z Lp =2 0 d pf 1 e u dx ¼ / u ðd pf u ÞL p u ð3bþ where eq and e eq are the depth of the neutral axis from the top ompressive onrete fiber (mm) and the ompressive strain of the topmost onrete fiber in a setion at ultimate state in region L eq, respetively. In Eq. (3a), eq was obtained from the ondition of setional equilibrium, where the stress strain relationships of onrete suggested by Kent and Park (1971) was used for ompression. Z eq 0 bðzþr H ðeðzþþ dz ¼ A pf E pf e pfu þ A bf E bf / eq ðd bf eq Þ ð4þ where A pf and E pf are the ross-setional area (mm 2 ) and the modulus of elastiity of the unbonded CFRP tendon (MPa), respetively, A bf, d bf, and E bf are the total setional area (mm 2 ), effetive depth (mm), and modulus of elastiity of non-prestressed auxiliary bonded reinforement (MPa), respetively, b(z) is the width of the beam at z (mm), z is the vertial distane measured from the neutral axis to the top of the setion (mm), e(z), e o and e r are the onrete ompressive strains at z (mm/mm), at topmost onrete fiber in a setion at maximum stress (= 0.002), and at residual stress (=0.2f 0 ompressive onrete strength; and r H ðþ¼ e 8 >< >: f 0 f 0 ) (mm/mm), respetively, f e eo e e o ½1 0:5 e re o ðee o Þ is the speified for 0 e e o 0:2 f 0 for e r \e : ðmpaþ: for e o \e e r The u in Eq. (3b) was obtained from setional equilibrium at ultimate state with equivalent retangular ompressive stress blok provided by ACI Committee 318 (2014). The additional strain of the unbonded FRP tendon (De pfu ) an be assessed by dividing the onrete elongation by the beam span (L): De pfu ¼ D eq þ D p L ð5þ Table 1 Expressions of multipliers for K and X in / eq ¼ 3 ðkþ2 4 X and L eq ¼ 2 3 X K (Eq. (2)). Setions L s x s j s Loations Symbols (s) Beam end es Mes M us L sp L es x es (3L sp - L es ) Craked setion s M esþm s M us L sp - (L s? L us ) x r (L s? 2L ys? 3L us ) M Yield setion ys ysm s M us L sp - (L ys? L us ) x ys (L ys? 2L us ) Ultimate setion us 1 Mys M us L sp - L us x us L us 4 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

5 2.3 Moment and Curvature at Charateristi Setions The moment and urvature at first raking setion from the beam ends (M es and / es in Figs. 3a and 3b, respetively) an be expressed as Eq. (6): M es ¼A pf E pf e pfu e and / es ¼ M s E I where E,I and e are the modulus of elastiity of onrete (MPa), moment of inertia of the setion (mm 4 ), and the eentriity of the unbonded tendon (mm), respetively. pffiffiffiffi Using the modulus of rupture of onrete (f r ¼ 0:63 f 0 in MPa), the magnitude of the internal moment for the raked setion within the region of the maximum moment and the orresponding urvature an be expressed as r 2 M s ¼ f r S 2 þ e pfu A pf E pf þ e ð7aþ h 2 / s ¼ A pf E pf e pfu e þ M s E I E I ð6þ ð7bþ where h 2 is the height of the ross setion (mm), and S 2 is the setion modulus with respet to the bottom surfae of the beam (mm 3 ). The moment and urvature at first yielding setion from the beam ends (M s and / s in Figs. 3a and 3b, respetively) an be found with the yielding of the auxiliary bonded steel bars. The neutral axis depth, ys, was then determined from the setional equilibrium. Z ys 0 bðzþr H ðeðzþþ dz ¼ A pf E pf e pfu þ A bf f bf ð8þ where f bf and e y are the tensile stress of the bonded bar (MPa)and the ompressive strains of the topmost onrete fiber in a setion at the first yielded setion (mm/mm), respetively. The moment and urvature at a beam setion at whih the bonded auxiliary steel bars begin to yield an be obtained one the neutral axis depth, ys, was obtained from Eq. (9): M ys ¼ Z ys 0 / ys ¼ e y ys bðzþr H ðeðzþþ z dz þ A pf E pf e pfu ðd pf ys Þ þa bf f bf ðd bf ys Þ ð9aþ ð9bþ At ultimate setion, the neutral axis depth, us, was obtained via an equivalent retangular ompressive stress blok (ACI Committee 318, 2014) in setional equilibrium with the top most onrete ompressive strain at its ultimate strain (e u = in this study): Z us us ð1b 1 Þ 0:85 f bðzþdz ¼ A pf E pf e pfu þ A bf f bf ð10þ 8 < 0:85 0 f 0 28 MPa where b 1 ¼ 0:85 0:05 f =7 28 f 56 MPa : 0:65 56 MPa f 0 Using us, the moment and urvature orresponding to the region between the loading points with maximum moment were determined by Eq. (11): M us ¼ Z us usð1b 1 Þ þa bf f pf ðd bf us Þ / us ¼ e u us 0:85 f bðzþz dz þ A pf E pf e pfu ðd pf us Þ ð11aþ ð11bþ 3. Iterative Algorithm for the Predition of e pfu at Ultimate State 3.1 Modified Algorithm to Inlude Fritional Losses In order to find a fixed point of unbonded tendon strain at the ultimate state, equivalent urvature bloks are iteratively refined as given in the following algorithms. The wobble and urvature frition losses assoiated with urrent equivalent bloks at ultimate state are inluded (Nilson 1987; Jeon et al. 2015). In the following, the subsript s represents ritial setions ( s = r, ys, or us ). A numerial example is given in Appendix 1. (1) At the beginning of iteration, assume De pfu as a fration of the differene between the strain of the maximum stress of the CFRP tendon (e pfm ) and the sum of the initial prestressing strain (e pfi ) and deompression strain (e d ): De pfu;s ¼ a e pfm ðe pfi þ e d Þ ; 0:0 a 1:0 ð12aþ e pfu;s;o ¼ e pfi þ e d þ De pfu;s ð12bþ (2) Using e pfu;s;o, evaluate (M es, / es ), (M s, / s ), (M ys, / ys ), and (M us, / us ) based on Eqs. (6), (7), (9), and (11), respetively. Then obtain the harateristi lengths from Table 1. (3) From Eq. (2), obtain / eq and L eq. From Eq. (5), find De pfu ¼ D eqþd p L. (4) Obtain the revised De pfu,s resulting from the losses by wobble frition (j) and urvature frition (g). 8 < 1 2 / u L p ð¼ h us Þ forultimatesetion h s ¼ / u þ R D E L s L / L : p=2 eq sl p=2 L eq 1 dx forothersetions De pfu;s ¼ e ðjlþgh sþ De pfu where, \E [ ¼ L eq if E 0 L s if E\0 ð13aþ ð13bþ International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 5

6 (5) Find the updated value of e pfu;s;n for eah ritial setion by: e pfu;s;n ¼ e pfi þ e d þ De pfu;s (6) If max e pfu;s;ne pfu;s;o e pfu;s;o tolerane, then the strain of the unbonded tendon is onverged with e pfu;s ¼ e pfu;s;n. Otherwise, let e pfu;s;o ¼ e pfu;s;n and repeat from step 2). For a wide range of the initially assumed values of the additional strain for the unbonded tendon (a in Eq. (12) between 0 and 1.0), the model showed stability by onverging to the idential values of e pfu within two to four iterations as an be seen in Fig. 4 for all ases with or without fritional loss. In order to investigate the effet of frition on strain inrement, a relatively onservative value of j = /m for wobble frition and g = 0.264/rad. provided by Yu and Zhang (2011) were used. It was found that only a marginal differene of 1 % (5 %) for total strain (strain inrement) was observed due to fritional loss. Therefore, no further fritional effet is inluded in subsequent analysis. Fig. 4 Examples of onvergene trend of the model predition with two different ways: a in Eq. (12a) given in the initial raking state only; and the same a assigned at the beginning stage of eah state. a RU50. b TO45. Table 2 Mehanial properties of tendons, auxiliary bonded reinforement and stirrup. Types Diameter (mm) Effetive area (mm 2 ) f pfm or f bfm (MPa) CFRP E pf or E bf (MPa) e pfm or e bfm (%) Steel f y (MPa) Tendons CFCC , , , DWC , Auxiliary bars DWC , Steel Stirrups DWC , Steel f pfm : tensile strength of CFRP tendons (MPa); f bfm : tensile strength of non-prestressed auxiliary bonded reinforements (MPa). 6 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

7 Table 3 Details of tested beam speimens. Speimens Prestressing reinfoement ratio q U pf =qb pfb Initial prestrssing fpfi/fpfm Loading type, Lsp (mm) Conrete ompressive strength, f 0 (MPa) Reinforement Unbonded FRP tendons Bonded reinforing bar Types Diameter (mm) Apf (mm 2 ) Types Diameter (mm) Abf (mm 2 ) RU point (900) 35 CFCC Steel RU point (900) 35 CFCC Steel RO point (900) 35 CFCC Steel CFCC RO point (900) 40 DWC CFRP TB point (1200) 40 DWC CFRP TO point (1200) 40 DWC CFRP DWC CFCC: CFRP tendons manufatured by Tokyo Rope Ltd.; DWC: CFRP tendons manufatured by DongWon Constrution Ltd. International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 7

8 3.2 Validity of the Model The validity of the model was examined by omparing its preditions with those from the tested beams reported in the authors previous study (Lee et al. 2015a, b). As the details of tested prestressed beams with unbonded CFRP tendons have been reported elsewhere (Heo et al. 2013), the disussion herein is foused primarily on omparisons between model preditions and test results on flexural apaity and at the ultimate state of the tested beams. The mehanial properties of the materials and the details of the tested beams are presented in Tables 2, 3, and Fig. 5. In Table 4 and Fig. 6, omparisons between model preditions and test results for flexural load apaity and unbonded tendon strain are presented. Gage readings from beams RU70 and TB45 are not inluded in the omparison, as their values, presumably hampered by the frition between the gages and the surrounding onrete, were erroneous. The model was able to predit an ultimate load apaity for the tested speimens with reasonable auray regardless of the setional shape, prestressing reinforement ratio, Fig. 5 Beam speimens ( open irle for onrete strain gages and open square for unbonded CFRP tendon strain gages). a RU50, RU70, or RO55. b RO50. TB45. d TO45. 8 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

9 Table 4 Comparisons of model preditions with test results. Speimens Prestressing reinforement ratios Max. load Tendon strains q U pf q B pfb q U pfb q U pf =qu pfb Test (kn) Thy/Test epfu Depfu Test (mm/mm) Model/Test Test (mm/mm) Model/Test Equation (14)/ Equation (15)/ Test Test RU RU N.A N.A N.A N.A N.A N.A RO RO TB N.A N.A N.A N.A N.A N.A TO Average Standard deviation N.A. not available. Fig. 6 Comparisons between test results and model preditions on load versus strain of unbonded CFRP tendon. amount of initial prestressing, and type of auxiliary bar. The average (l) and standard deviation (r) of the ratios of beam strength at ultimate state predited by the model to that from the test were 0.96 and 0.09, respetively. Those for the ratios of the predited values of e pfu (De pfu ) to the orresponding experimental values at ultimate state are 1.07 and 0.06 (1.27, 0.24), respetively. 3.3 Pratial Equation of Df ps Using the developed model, a pratial formula Df ps is presented. The onept of bond redution fator (X u ), suggested by Alkhairi (1991), was adopted. The least square method was used to find the best fitting oeffiients in X u given as a linear funtion of d p /L and L p /L (Sivaleepunth et al. 2006). The developed model was used to generate a number of Df ps values for the beam prestressed with unbonded CFRP tendons (Fig. 7) 324 values of Df ps with auxiliary bonded steel bars and 324 with auxiliary bonded CFRP bars. In Fig. 8, the ranges of different parameters in generating the values of Df ps are presented. All the preditions made by the model were divided by 1.27, whih is the average of the ratios of model predition to test result. From regression analysis performed on these generated values, Eq. (14) was suggested for the evaluation of Df ps. d pf D f ps ¼ X u E pf e u 1 ð14aþ u Fig. 7 Standard beam for parametri studies. International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 9

10 Fig. 8 Comparisons of Df ps : test results and preditions by three equivalent urvature blok model versus preditions by Eqs. (14) or (15) (30 MPa B f 0 B 50 MPa, 0.4 B f pfi /f pfm B 0.6, 525 mm B d pf B 575 mm 1.0 B A bf /A bf,min B 2.0, 0 B L p /L B 0.5, 1.0 B q U pf /q B pfb B 1.2). a Equation (14) for beams with auxiliary bonded steel bars. b Equation (14) for beams with auxiliary CFRP bars. Equation (15) for beams with auxiliary bonded steel bars. d Equation (15) for beams with auxiliary CFRP bars. ( dp Lp 1:80 X u ¼ L þ 0:47 L þ 0:14 2:15 dp Lp L þ 0:64 L þ 0:21 for auxiliary bonded steel bars for auxiliary bonded CFRP bars ð14bþ Equation (15) is suggested in ACI 440.4R-04 (2011) for the predition of Df ps of the unbonded FRP tendon at ultimate state. 8 < 1:5 ðl=d Df ps ¼ pf Þ E d pf e u pf u 1 for1point mid span loading : 3:0 ðl=d pf Þ E d pf e u pf u 1 for4point or uniform loading ð15þ As an be seen in Table 4, Eq. (14) was able to predit test results with a reasonable auray. The l and r for the ratios of predition made by Eq. (14) to test result are 1.04 and 0.20, respetively. However, rather sattered preditions were made by Eq. (15) with 0.83 and 0.63 for the l and r, respetively. Figure 8 exhibits that better preditions were made by Eq. (14) than Eq. (15) for both values of Df ps from test result and model predition. Similar trends in prediting Df ps are shown between Eqs. (14) and (15): overestimation for RU50 and underestimation of RO55. It is worth mentioning that Eq. (15) was empirially suggested by Alkhairi (1991) for onrete beams with unbonded steel tendons in suh a way that most of the predited values are smaller than the experimental results. As a result, Eq. (15) estimated rather onservative values of Df ps for the beams with auxiliary bonded steel bars, ompared with those from the test results and three equivalent urvature blok model (Fig. 8). In addition, Eq. (15) for 4-point loading ase was empirially developed mostly based on the third point test results 10 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

11 for the beams with unbonded steel tendons and auxiliary bonded steel bars. Consequently, Eq. (15) resulted in a signifiant deviation from model preditions and test results when it predited the values of Df ps for the beams with auxiliary bonded CFRP bars (Fig. 8d). Equation (15) was also shown to disregard the effet of L p /L on Df ps. 4. Iterative Algorithm for the Determination U of q pfb 4.1 Development of a Balaned Ratio Based on the ompatibility between onrete and CFRP tendon strains, q B pfb an be given by Eq. (16): q B pfb ¼ 0:85 a 1 b 1 q ð1 þ a 2 Þ bf a 3 ð16þ where a 1 ¼ f 0 f pfm, a 2 ¼ e pfme pfi e d e u, a 3 ¼ f bf f pfm, q bf ¼ A bf bd pf, and f pfm is the tensile strength of CFRP tendons (MPa). For the balaned onrete beam prestressed with unbonded tendon, tendon strain reahes its maximum value, e pfm, at ultimate state: e pfi þ e d þðd eq þ D p Þ=L ¼ e pfm ð17þ A substitution of D eq and D p in Eq. (3) into Eq. (17) results in a q U pfb as given in Eq. (18): q U pfb ¼ 0:85 a 1 b h 1 1 þ L L p a 2 2ðd pf eq Þ a 3 / eq e u L eq L i q bf ð18þ In order to reflet their member dependeny of / eq, L eq U and eq in Eq. (18), q pfb was obtained by repeatedly renewing the values of / eq, L eq and eq in two loops. The following iterative sheme is employed: (1) Assume q U pfb ¼ 0:5 qb pfb (2) Let De pfu;o ¼ e pfm e pfi e d and e pfu;o ¼ e pfm (3) Obtain A pf ¼ q U pfb b d pf (4) Estimate the tensile fore of the tendon by T pf ¼ A pf E pf e pfu;o. Obtain the L eq and / eq from Eq. (2). (5) Obtain the updated value of De pfu,n from Eq. (5). Find the renewed unbonded tendon strain by (6) e pfu;n ¼ e pfi þ e d þ De pfu;n. If e pfu;ne pfu;o e pfu;o tolerane, then let e pfu ¼ e pfu;n and go to step 7). Otherwise, let e pfu;o ¼ e pfu;n and repeat from (7) step 4). If e pfue pfm e pfm tolerane, then the q U pfb is onverged and stop. Otherwise, update q U pfb in Eq. (18) using the revised values of / eq, L eq and eq. Repeat from step 3). 4.2 Estimation of q U pfb and Failure Modes for Tested Beams For eah tested beam, evaluated values of q B pfb using Eq. (16) and q U pfb via the above algorithms are presented in Table 4. All tested beams presented in Table 4 exept for beam RU70 are shown to be ompression-ontrolled setions with a q U pf greater than the orresponding q U pfb. Note that the q U pf s of beams RU50, RU70 and TB45 are less than the orresponding q B pfb s. The beam RO55 was found to reah its ultimate state without tensile frature regardless of the setional area of unbonded tendon due to a relatively large setional area of the auxiliary bars. From Table 4, it an be seen that the average of the ratios of theoretially estimated e pfu values to experimentally obtained ones is A more onservative q U pfb is, therefore, implied by the model with its greater predition on the tensile strain of unbonded tendon. From experiments, it was observed that all of the tested beams, inluding the underreinfored RU70 beam with a ratio of q U pf =qu pfb equal to 0.76, failed in ompression-ontrolled mode by the onrete rushing without tensile frature of the unbonded tendons (Heo et al. 2013). For the under-reinfored beam RU70, if the strain of the unbonded tendon inreases over its e pfm without frature at the ultimate state, the developed model predits the strain value at the ultimate state orresponding to the 1.05 times e pfm. This inrease in tendon strain seems to be relatively marginal ompared with the provided setional area of the tendon, whih was 24 % less than the setional area for the balaned ondition. The ompression-ontrolled failure in beam RU70 an be explained by the marginally inreasing tendon strain over e pfm even for relatively small q U pf =qu pfb in onjuntion with the onservative overestimation of the model for the tendon strain. 5. Parametri Studies for q U pfb Using the developed model, the effets of f 0, f pfi, d pfu, A bf and L p /L on q U pfb were studied. Figure 7 and Table 5 present the standard beam geometry and the ranges of different parameters, respetively. Beams having auxiliary bonded CFRP bars only were onsidered. 5.1 Effet of f 0 Figure 9(a) illustrates that an inrease in f 0 results in an almost proportional inrease in the values of q B pfb and qu pfb. For the setion with bonded tendons, the strain onfiguration of onrete through the depth of the beam setion is predetermined by the differene in e pfm and e pfi of the tendon. As a result, an inrease (derease) in the setional area of tendon is needed when onrete ompressive strength inreases (dereases). The tensile strain of the auxiliary bonded re-bars remains the same irrespetive of onrete ompressive strengths due to a predetermined strain onfiguration. A similar explanation an also be appliable to a setion prestressed with unbonded tendons. Compared with a ounterpart setion with bonded tendon, a setion with unbonded tendon would have redued neutral axis depth, whih results in relatively larger strains of onrete at the level of tendon and auxiliary bonded bars. International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 11

12 Table 5 Balaned ratio with eah parameter. No. Effets Initial prestress Compressive onrete strength fpfi (MPa) fpfi=fpfm f 0 (MPa) f 0 =f 0 ;std Abf (mm 2 ) Bonded FRP tendon Effetive depth Distane between loading points Abf/Abf,min dpf (mm) L/dpf Lp (mm) Balaned ratio Lp/L q U pfb q B pfb q U pfb =qb pfb 1 Standard fpfi f d pf Abf Lp International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

13 Fig. 9 Effets of main parameter on q pfb and e bf. a Conrete ompressive strength (f 0 ). b Initial prestress (f pfi=f pfm ). Effetive depth (d pf ). d Amount of bonded rebar (A bf ). e Relative distane between loading points, L p /L. 5.2 Effet of f pfi As shown in Fig. 9b, the inrease in f pfi resulted in an inrease in both q B pfb and qu pfb. With a larger value of f pfi, less additional tensile strain on the tendon is available at the ultimate state for the balaned beam. This inreases the depth of the neutral axis, inreasing the area of onrete in ompression; hene, the setional area of tendon inreases with the inrease in f pfi for the setional equilibrium. 5.3 Effet of d pf In Fig. 9, inreasing tendenies for both q B pfb and qu pfb and deeasing tendenies for the strains of auxiliary bonded rebars were observed with an inrease in d pf. Sine the amount of available tensile strain in the tendon in addition to its initial strain is the same regardless of the d pf, an inrease in d pf inreases the setional area of onrete in ompression due to the inreased depth of neutral axis and dereases the strain of the auxiliary bonded re-bars. Equilibrium at a setion, therefore, requires a larger setional area of tendon with a larger d pf. 5.4 Effet of A bf Figure 9d illustrates that with the inrease in A bf, both q B pfb and q U pfb derease while the strain of auxiliary bars remains almost onstant. As A bf inreases, the tensile resistane of bonded auxiliary bars inreases. For the beam prestressed International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 13

14 Fig. 9 ontinued Fig. 10 Relative effets of parameters on q U pfb /q B pfb (Values in parenthesis represent standard ones in Table 5). with bonded tendon, as the area of A bf inreases, the area of bonded tendon redues to satisfy setional equilibrium without hange in strain onfiguration orresponding to the balaned ondition. A similar explanation an also be appliable to an unbonded ase even though there would be some hanges in strain onfiguration at balaned ondition with the inrease of A bf due to the member-dependeny of e pfu. 5.5 Effet of L p /L The effet of L p /L is presented in Fig. 9e. When the ratio of L p /L inreases, the value of q U pfb (eu bf ) tends to inrease (derease) in an almost linear fashion. Derease in the value of e U bf with the inrease in L p /L implies less urvature at ritial setions, beam defletion, and e pfu at ultimate state with the inrease in L p /L. Consequently, a greater area of A pfu is needed with a smaller value of e pfu to balane a ompressive resultant in a setion resulting from a smaller urvature as L p /L inreases. In Fig. 10, the relative effets of eah parameter on q U pfb =qb pfb is illustrated based on the previous parametri studies. As shown in Table 5, the values of q U pfb =qb pfb were shown to be between 0.43 to A relatively small value of q U pfb =qb pfb suggests that a beam prestressed with an unbonded tendon and its reinforement ratio greater than the balaned ratio (q U pf qb pfb ) would fail in a more desirable ompression-ontrolled mode with suffiient safety. It an be observed in Fig. 10 that inrease in q U pfb was more heavily influened by inreases in L p, d pf, f pfi, and f 0 in this order. However, the inrease in A bf resulted in the derease in q U. pfb 6. Conlusions The following onlusions are drawn from this study. (1) The model based on three equivalent urvature bloks was validated by omparing its preditions with test results: the average and standard deviation of the ratios of the predited to the measured beam strengths (total strain of unbonded tendon) at the ultimate state were 0.96 and 0.09 (1.07 and 0.06), respetively. 14 International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)

15 (2) Performing regression analysis on the values of Df ps generated by the model, a pratial formula based on the onept of bond redution fator was suggested for the estimation of Df ps. The formula predited test results and model simulated values with reasonable auray, regardless of type of auxiliary bonded bars and different values of L p /L. (3) Preditions made by the equation provided by ACI 440.4R-04 showed signifiant deviations from the test results and model preditions, partiularly for the beams reinfored with unbonded CFRP tendons and auxiliary bonded CFRP bars. (4) By extending the onept of three equivalent urvature bloks, a stable iterative algorithm in two-loop was developed to find fixed points for the member-dependent quantities of the q U pfb. remained in a (5) It was found that the values of q U pfb =qb pfb range between 0.43 and 0.83 for the beams onsidered in this parametri study. Relatively small values of q U pfb =qb pfb imply that a beam prestressed with unbonded tendons would fail in a ompression-ontrolled mode with suffiient margin of safety if prestressed with q U pf qb pfb. (6) From the parametri studies, an inrease in q U was pfb more heavily influened by inreases in L p, d pf, f pfi, and f 0 in order. However, the inrease in A bf resulted in the derease in q U. pfb (7) The suggested model for q U pfb an be of useful guide for a ost-effetive design with minimum amount of unbonded CFRP tendons for ompression-ontrolled beam setion, avoiding a atastrophi failure resulting from undesired tensile frature of tendons. It is assumed that g = 0.264/rad. for urvature fritional loss. No wobble fritional loss is assumed as the span of the beam is relatively short. In the following, values in parenthesis represent the De pfu values obtained without fritional loss. Tolerane is given by Units of length, moment, rotational angle and urvatures are mm, kn m, radian and /mm, respetively. Iteration 1 (1) Let a = 0.5. Obtain the assumed De pfu;s ¼ a e pfm ðe pfi þ e d Þ ¼ 0:0048 (Eq. (12a)) and e pfu;s;o ¼ e pfi þ e d þ De pfu;s ¼ 0:0129 (Eq. (12b)). (2) (M es ; / es ), (M s ; / s ) and (M us ; / us ) = (15.5, 1.9), (42.6, 21) and (68.6, 720) by Eqs. (6), (7) and (11), respetively. (3) Obtain / eq = 550 and L eq = 535 (Eq. 2). D eq = 9.88, D p = 5.96 (Eq. (3)) and De pfu ¼ D eqþd p L = (Eq. 5). (4) h R us ¼ 1 2 / Du L p ¼ 0:018 E and h s ¼ h us þ Ls L p =2 / L eq s L p =2 L eq 1 dx ¼ 0:066 and for s = es and s, respetively (Eq. (13a)). De pfu;s ¼ e ðjlþghsþ De pfu ¼ 0:00536 ð0:00545þ, ( ) and ( ) for s = es, s and us, respetively (Eq. (13b)). (5) e pfu;s;n ¼ e pfi þ e d þ De pfu;s ¼ 0:01376 ð0:01385þ, ( ) and ( ) for s = es, s and us, respetively. e (6) pfu;s;n e pfu;s;o e pfu;s;o ¼ 0:12, 0.12 and 0.13 for s = es, s and us, respetively. max e pfu;s;n e pfu;s;o e ¼ e pfu;s;n e pfu;s;o pfu;s;o e pfu;s;o ¼ 0:13 [ 0:01 ð¼ toleraneþ: Aknowledgments This researh was supported by the Chung-Ang University Researh Sholarship Grants in Open Aess This artile is distributed under the terms of the Creative Commons Attribution 4.0 International Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the original author(s) and the soure, provide a link to the Creative Commons liense, and indiate if hanges were made. Appendix Example of Finding De pfu with and without Fritional Loss The speimen TB45 is onsidered. For this beam, e pfm, e pfi and e d are given as 0.018, and , respetively. Let e pfu;s;o ¼ e pfu;s;n ¼ 0:01376 ð0:01385þ, ( ) and ( ) for s = es, s and us, respetively. Repeat from step 2) in iteration 2. Iteration 2 (2) (M es, / es ), (M s, / s ) and (M us, / us ) = (16.2, 1.8), (43.2, 20) and (69.8, 710) by Eqs. (6), (7) and (11), respetively. (3) Obtain / eq = 540 andl eq = 541 (Eq. (2)).D eq = 9.70, D p = 5.85 (Eq. (3)) and De pfu = = (Eq. (5)). (4) h us = and h s = and for s = es and s, respetively (Eq. (13a)). De pfu,s = ( ), ( ) and ( ) for s = es, s and us, respetively (Eq. (13b)). (5) e pfu,s,n = ( ), ( ) and ( ) for s = es, s and us, respetively. e (6) pfu;s;n e pfu;s;o e pfu;s;o ¼ 0.007, and for s = es, s and us, respetively. max e pfu;s;ne pfu;s;o ¼ e pfu;s;ne pfu;s;o ¼ \ 0.01 e pfu;s;o e pfu;s;o International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017) 15

16 (= tolerane) then the strain of unbonded tendon is onverged withe pfu,us = e pfu,us,n = ( ), ( ) and ( ) for s = es, s and us, respetively. Stop iteration. Referenes ACI Committee 318. (2014). Building ode requirements for strutural onrete and ommentary (ACI ). Farmington Hills, MI: Amerian Conrete Institute. Alkhairi, F. M. (1991). On the behavior of onrete beams prestressed with unbonded internal and external tendon, Ph.D. thesis, University of Mihigan, Ann Arbor, MI. Amerian Conrete Institute (ACI). (2003). Guide for the design and onstrution of onrete reinfored with FRP bars (ACI 440.1R-03). Amerian Conrete Institute, Farmington Hills, MI. Amerian Conrete Institute (ACI). (2011). Prestressing onrete strutures with FRP tendons (ACI 440.4R-04), Amerian Conrete Institute, Farmington Hills, MI. Du, J. S., Au, F. T. K., Cheung, Y. K., & Kwan, A. K. H. (2008). Dutility analysis of prestressed onrete beams with unbonded tendons. Engineering Strutures, 30(1), Elrefai, A., West, J., & Soudki, K. (2012). Fatigue of reinfored onrete beams strengthened with externally post-tensioned CFRP tendons. Constrution and Building Materials, 29, Girgle, F., & Petr, Š. (2016). An anhoring element for prestressed FRP reinforement: simplified design of the anhoring area. Materials and Strutures, 49(4), Grae, N. F., Enomoto, T., Sahidanandan, S., & Puravankara, S. (2006). Use of CFRP/CFCC reinforement in prestressed onrete box-beam bridges. ACI Strutural Journal, 103(1), Grae, N. F., Singh, S. B., Puravankara, S., Mathew, S. S., Enomoto, T., Mohti, A. A., et al. (2008). Flexural behavior of preast onrete post-tensioned with unbonded, arbonfiber-omposite ables. Preast/Prestressed Conrete Institute Journal, 92(6), Han, Q., Wang, L., & Xu, J. (2015). Experimental researh on mehanial properties of transverse enhaned and hightemperature-resistant CFRP tendons for prestressed struture. Constrution and Building Materials, 98, Harajli, M. H. (1990). Effet of span-depth ratio on the ultimate steel stress in unbondedprestressed onrete members. ACI Strutural Journal, 87(3), Heo, S., Shin, S., & Lee, C. (2013). Flexural behavior of onrete beams internally prestressed with unbonded arbonfiber-reinfored polymer tendons. Journal of Composites for Constrution, 17(2), Jeon, S. J., Park, S. Y., Kim, S. H., Kim, S. T., & Park, Y. H. (2015). Estimation of frition oeffiient using smart strand. International Journal of Conrete Strutures and Materials, 9(3), Kato, T., & Hayashida, N. (1993). Flexural harateristis of prestressed onrete beams with CFRP tendons. International Symposium Fiber Reinfored Plasti Reinforement for Conrete Strutures, ACI Speial Publiation, , Kent, D. C., & Park, R. (1971). Flexural members with onfined onrete. Journal of the Strutural Division ASCE, 97(7), Lee, L. M., Moon, J. H., & Lim, J. M. (1999). Proposed methodology for omputing of unbonded tendon stress at flexural failure. ACI Strutural Journal, 96(6), Lee, C., Ko, M., & Lee, Y. (2013) Bend strength of omplete losed-type arbon fiber-reinfored polymer stirrups with retangular setion. Journal of Composites for Constrution. 18(1). Lee, C., Lee, S., & Shin, S. (2015a) Shear apaity of RC beams with arbon fiber-reinfored polymer stirrups with retangular setion, Journal of Composites for Constrution, Lee, C., Shin, S., & Lee, S. (2015b). Modelling of load-defletion of onrete beams internally prestressed with unbonded CFRP tendon. Magazine of Conrete Researh, 67(13), Maissen, A., & De Smet, C. A. M. (1995). Comparison of onrete beams prestressed with arbon fibre reinfored plasti and steel strands, non-metalli (FRP) reinforement for onrete strutures (pp ). Bagneux, Frane: RILEM. Naaman, A. E., & Alkhairi, F. M. (1991a). Stress at ulitmate in unbonded post-tensioning tendons : Part 1 Evaluation of the state-of-the-art. ACI Strutural Journal, 88(5), Naaman, A. E., & Alkhairi, F. M. (1991b). Stress at ultimate in unbonded post-tensioning tendons: Part 2 Proposed methodology. ACI Strutural Journal, 88(6), Nilson, A. H. (1987). Design of prestressed onrete (2nd ed.). Hoboken, NJ: Willey. Sivaleepunth, C., Niwa, J., Bui, D. K., Tamura, S., & Hamada, Y. (2006). Predition of tendon stress and flexural strength of externally prestressed onrete beams. Doboku Gakkai Ronbunshu E, 62(1), Yu, T. L., & Zhang, L. Y. (2011). Frition loss of externally prestressed onrete beams with arbon fiber-reinfored polymer tendons. Advaned Materials Researh, 163, International Journal of Conrete Strutures and Materials (Vol.11, No.1, Marh 2017)