Accepted Manuscript. Factors affecting moment redistribution at ultimate in continuous beams prestressed with external CFRP tendons

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1 Aepted Manusript Fators affeting moment redistribution at ultimate in ontinuous beams prestressed with external CFRP tendons Tiejiong Lou, Sergio M.R. Lopes, Adelino V. Lopes PII: S (14) DOI: Referene: JCOMB 3026 To appear in: Composites: Part B Reeived Date: 3 February 2014 Revised Date: 9 April 2014 Aepted Date: 9 May 2014 Please ite this artile as: Lou, T., Lopes, S.M.R., Lopes, A.V., Fators affeting moment redistribution at ultimate in ontinuous beams prestressed with external CFRP tendons, Composites: Part B (2014), doi: /j.ompositesb This is a PDF file of an unedited manusript that has been aepted for publiation. As a servie to our ustomers we are providing this early version of the manusript. The manusript will undergo opyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the prodution proess errors may be disovered whih ould affet the ontent, and all legal dislaimers that apply to the journal pertain.

2 Fators affeting moment redistribution at ultimate in ontinuous beams prestressed with external CFRP tendons Tiejiong Lou 1, Sergio M. R. Lopes *1, Adelino V. Lopes 2 1. CEMUC, Department of Civil Engineering, University of Coimbra, Coimbra , Portugal 2. Department of Civil Engineering, University of Coimbra, Coimbra , Portugal (*) Corresponding author, sergio@de.u.pt; tel.: Abstrat: A numerial investigation of redistribution of moments in ontinuous onrete beams prestressed with external arbon fiber reinfored polymer (CFRP) tendons at failure loads is desribed. A finite element analysis (FEA) model is introdued, and an extensive parametri study is arried out on two-span ontinuous beams. The fators examined in this study inlude the ontent of non-prestressed steel, tendon eentriities, tendon area, effetive prestress, span-to-height ratio, onrete strength, CFRP modulus of elastiity and load type. The results obtained from FEA are ompared with those obtained from various odes. The study shows that the importane of some fators is not refleted in the odes. When used to alulate the degree of moment redistribution in these beams, the parameter ε t (net strain in extreme tension steel) seems to be more reasonable than the parameter /d (ratio of neutral axis depth to setion effetive depth). A simplified equation for alulating the degree of moment redistribution at ultimate is proposed. Keywords: A. Carbon fiber; B. Strength; C. Finite element analysis (FEA); C. Numerial analysis 1

3 1. Introdution In ivil engineering, the use of fiber reinfored polymer (FRP) materials is beoming more and more popular due to their noteworthy advantages of high orrosive resistane and high strength [1]. Extensive efforts have reently been made to examine the overall behavior of FRP-reinfored and strengthened onrete members [2-5]. In the field of external prestressing, FRP omposites are promising to be widely used as external tendons for the rehabilitation and onstrution of various engineering strutures. Among the FRP omposites, arbon FRP (CFRP) is reognized as an ideal material to replae the onventional prestressing steel. Previous theoretial [6] and experimental [7] studies indiated that external CFRP and steel tendon beams exhibit very similar strutural behavior. In engineering pratie, the appliation of external tendons in ontinuous onrete beams is rather ommon. It is well known that the redistribution of moments takes plae when a ontinuous onrete beam begins to assume inelasti behavior. A reasonable onsideration of the moment redistribution is important for the flexural strength analysis and design of the ontinuous beams. Over past years, a number of works have been performed to study the moment redistribution behavior and the fators affeting the redistribution of moments in reinfored [8-11] and bonded prestressed onrete beams [12-14]. However, the studies of the redistribution of moments in ontinuous onrete beams prestressed with unbounded or external tendons, partiularly FRP tendons, are very limited [15,16]. The redistribution of moments is linked to the dutile behavior of onrete beams. Beause of the 2

4 brittleness of the FRP materials and unbounded nature of external tendons, the onrete beams prestressed with external FRP tendons may exhibit different moment redistribution behavior ompared to the onventional onrete beams. As a onsequene, the urrent rules related to the moment redistribution in onventional onrete beams may not be appliable to the external FRP tendon beams. This paper presents a numerial investigation onduted to evaluate the redistribution of moments in two-span ontinuous prestressed onrete beams with external CFRP tendons at the ultimate limit state. The results obtained from the finite element analysis (FEA) are ompared with those obtained from various odes. A wide range of fators are examined, inluding the ontent of non-prestressed tension steel, eentriities of external tendons at midspan and enter support, amount of external tendons, effetive prestress, span-to-height ratio, onrete strength, CFRP tendons elasti modulus and type of loading. Based on the results of the parametri analysis, a reasonable simplified equation inluding the most important parameters for the alulation of the amount of moment redistribution at ultimate is proposed. 2. Measurement of moment redistribution and odes of pratie Several approahes have been used to measure quantitatively the amount of moment redistribution in a statially indeterminate struture. One of the approahes was based on a plasti adaption ratio (PAR) defined by [17] PAR = P / P (1) where P ol is the atual ultimate load; and P pl is the ultimate load alulated by a ol pl 3

5 plasti analysis. PAR = 1 indiates full redistribution of moments. Some investigators [18] defined the plasti adaption ratio using three ultimate loads as follows: PAR1 = ( P P )/( P P ) (2) ol el pl el where P el is the ultimate load alulated by an elasti analysis. PAR1 = 0 (P ol = P el ) orresponds to zero redistribution, while PAR1 = 1 (P ol = P pl ) orresponds to full redistribution. Cohn [19] defined the degree of moment redistribution by β = 1 M / M e (3) where M is the atual moment; M e is the elasti moment alulated based on the theory of elastiity. β = 0 indiates nil redistribution. This definition is adopted by various odes. In alulating the design moments in ontinuous flexural members, the odes allow designers to take advantage of a linear analysis with an adjustment of the elasti moments through the use of the degree of moment redistribution β. However, the empirial equations for alulation of β in various odes are quite different. In the ACI ode [20], the degree of moment redistribution for prestressed onrete beams with suffiient bonded reinforement is alulated using the net strain in extreme tension steel ε t by β (%) 1000ε t (4) with a maximum of 20%. Also, the moment redistribution an be done only when ε t is not less than at the setion where the moment is redued. The CSA ode [21] indiates that the negative moment alulated by an elasti 4

6 analysis an be inreased or dereased by β (%) / d (5) with a maximum of 20%. In Eq. (5), /d is the ratio of the neutral axis depth to the effetive depth of a ross setion at the ultimate limit state. In Europe, EC2 [22] and MC10 [23] also uses the parameter /d to alulate the degree of moment redistribution: for f 50 MPa, ' β ( / ε u ) / d (6a) for f > 50 MPa, ' β ( / ε u ) / d (6b) with a maximum of 30% for high- and normal-dutility steel and of 20% for low-dutility steel. In Eq. (6), ' f is the onrete ylinder ompressive strength; and ε u is the ultimate onrete ompressive strain. 3. Nonlinear model A previously developed numerial model [24] is used here to ondut the parametri evaluation of the redistribution of moments in ontinuous onrete beams prestressed with external CFRP tendons. The time-dependent effets are negleted in the present study, but the modeling of these effets an be found elsewhere [25]. The proposed model is based on the finite element method, and aounts for both geometri and material nonlinearities. Apart from the variation in the external tendon depth, the oupling between axial and flexural deformations is also inluded within 5

7 the geometri nonlinearity. The following basi assumptions are adopted in the analysis: (1) A plane setion remains plane after deformations; that is, the strain distribution aross the depth of a onrete setion is linear. (2) Non-prestressed steel ompletely bonds with the surrounding onrete; that is, the strains between bonded reinforement and onrete are perfetly ompatible. (3) The fritions between deviators and external tendons are negligible. This simplifiation may lead to a higher predited stress inrease in external tendons at the initial loading stage. After raking, the tendon stress inreases quikly, so the effet of frition loss on the response of load versus tendon stress inrease tends to diminish with inreasing load up to the ultimate. (4) The shear deformation is negligible. This simplifiation is reasonable for slender beams suh the ones used for prestressed onrete beams. The onstitutive laws for materials used in the urrent analysis are as follows: The stress-strain relationship for onrete in ompression suggested by Hognestad [26] is adopted. It is omposed of a paraboli asending branh and a linear desending branh as shown in Fig. 1(a), and is expressed as follows: for asending branh, σ 2 ' 2ε ε = f ε0 ε0 (7a) for desending branh, ' ε ε 0 σ = f εu ε0 (7b) 6

8 where σ and ε = onrete stress and strain, respetively; f ' = onrete ylinder ompressive strength; 0 ε = 0.002; and u ε = The onrete in tension is assumed to be linear elasti up to raking, followed by linear desending stress-strain behavior up to zero stress, as shown in Fig. 1(b) where f t = onrete tensile strength and r ε = raking strain. The CFRP prestressing tendon is linear elasti up to rupture, as shown in Fig. 1() where f σand f ε = CFRP tendon stress and strain, respetively; f f = CFRP tensile strength; and E f = CFRP modulus of elastiity. The non-prestressed steel is assumed to be elasti-perfetly plasti in both tension and ompression, as shown in Fig. 1(d) where s σ and s ε = steel stress and strain, respetively; f y = steel yield strength; and E y = steel modulus of elastiity. The onrete beam is divided into a number of beam elements, and the ross setion of eah element is subdivided into disrete layers to inlude different material properties. The ontribution of external tendons to the onrete beam is made by transforming the urrent prestressing fore into equivalent nodal loads applied on the finite element model. A load ontrol or displaement ontrol inremental method, together with the Newton-Raphson iterative algorithm, is used to solve the nonlinear equilibrium equations of the struture. The iterative proedure for eah inrement involves four basi steps: (1) form the urrent tangent stiffness matrix; (2) solve the equilibrium equations; (3) determine the urrent state for eah element; and (4) hek onvergene. During the solution proess, when the onrete strain at the extreme ompressive fiber of the ritial setion reahes the allowed maximum strain, the beam is assumed to be rushed. 7

9 The proposed method of analysis is apable of prediting the strutural behavior of externally prestressed onrete beams, both simply supported and ontinuous, over the entire loading range up to the ultimate. The model has been alibrated using a large number of experimental beams available in literature. The omparison between numerial preditions and experimental results for ontinuous external tendon speimens was reported in Lou et al. [27], where the predited load-defletion response and stress inrease in external tendons were shown to be in favorable agreement with the experimental ones. 4. Parametri study A two-span ontinuous prestressed onrete retangular beam with external CFRP tendons, as shown in Fig. 2(a), is used as a referene beam for the parametri analysis. The material parameters are as follows: unless otherwise stated, the areas of non-prestressed tension steel reinforement over positive moment region A s1 and negative moment region A s2 are 720 and 360 mm 2, respetively; the area of non-prestressed ompression steel reinforement A s3 is 360 mm 2 ; the yield strength f y of non-prestressed steel is taken as 450 MPa; the area of external tendons A p is 450 mm 2, and the modulus of elastiity E f and tensile strength f f of CFRP tendons are 147 GPa and 1840 MPa, respetively; the effetive prestress f pe for the CFRP tendons is onsidered to be 930 MPa; the onrete ompressive strength ' f and tensile strength f t are 40 and 3 MPa, respetively. In the finite element idealization, the onrete beam is divided into 36 beam 8

10 elements as shown in Fig. 2(b), and the ross setion of a beam element is subdivided into 10 onrete layers and two steel layers eah of whih represents the bottom or top non-prestressed steel reinforement. The external tendon is also divided into 36 tendon segments orresponding to the beam elements. Using this finite element model, the influene of various fators on the redistribution of moments at ultimate is evaluated. These fators inlude the non-prestressed steel area, A s2 /A s1 ratio, tendon eentriities, tendon area, effetive prestress, span-to-height ratio, onrete strength, CFRP elasti modulus and load type. Unless otherwise stated, the results (β, /d, ε t ) presented in the following setions of this paper are for the ritial negative moment (enter support) setion of the beams at the ultimate limit state Effet of non-prestressed steel area The effet of non-prestressed steel area is examined by varying A s2 from 360 to 2280 mm 2 and maintaining the A s2 /A s1 ratio at 0.8. Figure 3 shows the variation of β with the amount of non-prestressed steel. Both the FEA results and ode preditions are presented. The FEA results are obtained using Eq. (3) where the atual moment apaity M and elasti moment M e are omputed by FEA. In the alulation of M, both geometri and material nonlinearities are onsidered. On the other hand, in the alulation of M e, all the materials are assumed to be linear elasti while the geometri nonlinearity is taken into aount. In order to obtain the elasti moment M e, the ultimate load orresponding to the atual moment apaity M is applied and the inremental load method is employed to solve the equilibrium equations. A summary of results in relation to moment redistribution for different amounts of non-prestressed 9

11 steel is given in Table 1. From Fig. 3 and Table 1, it is seen that, aording to the FEA preditions, the β value inreases with inreasing A s2 up to 1800 mm 2 and then gradually dereases with ontinuing inrease of the steel area. This observation an be attributed to the ombined effets of dutility and stiffness differene between ritial setions. When A s2 (A s2 /A s1 = 0.8) inreases, the flexural dutility tends to dereases (the less the dutility, the lower the moment redistribution) while the stiffness differene between ritial setions enlarges (the larger the stiffness differene, the higher the moment redistribution). Therefore, if the effet of stiffness differene transends the effet of dutility (for A s2 inreased up to 1800 mm 2 ), the moment redistribution inreases; on the other hand, if the effet of dutility prevails against the effet of stiffness differene (for A s2 inreased beyond 1800 mm 2 ), the moment redistribution dereases. It is also observed that, aording to the preditions by various odes, the moment redistribution onsistently dereases as the amount of non-prestressed steel inreases. This implies that the odes aounts for the setion dutility only, negleting the stiffness differene between ritial setions. As a onsequene, the ode preditions fail to reflet aurately the atual trend of the variation of β with the amount of non-prestressed steel. In this analysis, it is seen that EC2 and the CSA ode are non-onservative partiularly at a low amount of non-prestressed steel, while the ACI ode is generally onservative Effet of A s2 /A s1 The effet of A s2 /A s1 is examined assuming a minimum non-prestressed steel (A s1 10

12 = A s2 = 360 mm 2 ) and varying A s1 or A s2 from 360 to 1800 mm 2. Figure 4(a) shows the variation of β with the A s2 /A s1 or A s1 /A s2 ratio aording to FEA preditions. A omparison between the β values predited by FEA and various ode equations is illustrated in Fig. 4(b) and Table 1. It is observed from Fig. 4(a) that the A s2 /A s1 (or A s1 /A s2 ) ratio strongly affets the degree of moment redistribution, attributed primarily to the hange in the stiffness differene between the ritial midspan and enter support setions. The β value inreases signifiantly with the inrease of A s1 /A s2 or dereases signifiantly with the inrease of A s2 /A s1. When A s2 /A s1 inreases to a level of about 1.6, the positive redistribution at the enter support disappears and the negative redistribution begins to appear. The negative redistribution, whih indiates that the atual moment is greater than the elasti value, beomes more and more signifiant with ontinuing inrease of A s2 /A s1. From Fig. 4(b) and Table 1, it is observed that for a fixed value of A s2, the effet of the A s2 /A s1 ratio is slightly refleted in the ACI ode but negleted in other odes where the parameter /d is used. On the other hand, for a fixed value of A s1, the β values predited by various odes gradually dereases as A s2 /A s1 inreases, but the importane of the parameter A s2 /A s1 is signifiantly underestimated. It should be noted that this observation is attributed to the hange in the dutility of the enter support setion rather than the hange in the stiffness differene between ritial setions. For a minimum amount of non-prestressed steel over the enter support (A s2 = 360 mm 2 ), the ACI ode is generally onservative exept when the non-prestressed steel over 11

13 midspan is lose to the minimum amount. The CSA ode is non-onservative for A s1 /A s2 less than about 2, while EC2 is non-onservative for A s1 /A s2 less than about 3. On the other hand, for a minimum non-prestressed steel over midspan (A s1 = 360 mm 2 ), all the odes are non-onservative, partiularly at high values of A s2 /A s Effet of midspan and enter support tendon eentriities To study the influene of tendon eentriities on the degree of moment redistribution, four levels of the midspan eentriity e 1 or enter support eentriity e 2 are seleted: 0, 100, 200 and 300 mm. The variation of β with the midspan or enter support tendon eentriity is shown in Fig. 5(a). A omparison between the β values predited by FEA and various ode equations is illustrated in Fig. 5(b) and Table 2. It is observed from Fig. 5(a) that the β value inreases with the inrease of e 1 but dereases with inreasing e 2. The dereasing rate is muh more signifiant than the inreasing rate. The β value inreases by 18.5% as e 1 inreases from 0 to 300 mm, while dereases by 44.72% as e 2 inreases from 0 to 300 mm. The important influene of the tendon eentriity is partly attributed to the hange in the stiffness differene between ritial midspan and enter support setions, and partly attributed to the hange in seondary moments, whih is mainly ontrolled by the profile of the prestressing tendons. From Table 2, it an be observed that the hange in the values of ε t and /d with varying e 1 is negligible, indiating that the effet of the variable e 1 is not inluded in all ode equations, as an be seen in Fig. 5(b). On the other hand, as e 2 inreases, the value of ε t remains almost unhanged while the value of /d quikly dereases. In fat, 12

14 the variable e 2 (e 1 as well) does not affet the neutral axis depth. The signifiant variation in the value of /d with e 2 is due to the hange of the effetive depth, d, of the enter support setion. From Fig. 5(b) and Table 2, it an also be observed that the effet of the variable e 2 is negleted in the ACI ode, while it is inorretly inluded in the CSA ode and EC2 beause the trend predited by these ode equations is opposite to the atual trend by FEA. In addition, the ACI ode is onservative while EC2 is non-onservative. The CSA ode may be non-onservative for low levels of e 1 or high levels of e Effet of tendon area and effetive prestress The tendon area A p and effetive prestress f pe are two variables that determine the effetive prestressing fore N pe (= A p f pe ) that is a fundamental parameter in the design of prestressing. To study the effet of N pe on the moment redistribution, either A p varies from 0 to 600 mm 2 (f pe = 930 MPa) or f pe varies from 0 to 1240 MPa (A p = 450 mm 2 ) so as to produe N pe from 0 to 558 kn. Figure 6(a) shows the variation of β with the effetive prestressing fore. It is observed that the β value quikly dereases as the effetive prestressing fore inreases. The phenomenon is partiularly obvious when the amount of external tendons varies. When A p = 0, namely, in the ase of a reinfored onrete ontinuous beam, the β value is as high as 41.55%. The value is signifiantly redued to 28.95% when the RC beam is slightly prestressed with external tendons of 150 mm 2. On the other hand, when f pe = 0 (A p = 450mm 2 ), the β value is 31.43%, whih is muh lower than that for A p = 0. 13

15 For different levels of A p and f pe, a omparison between the β values by FEA and various ode equations is illustrated in Fig. 6(b) and Table 3. It is observed in Table 3 that as A p or f pe inrease, the value of ε t gradually dereases while the value of /d inreases gradually. As a onsequene, all the odes take into aount the effet of these variables, as shown in Fig. 6(b). It an also be seen that the ACI ode is onservative, while EC2 is non-onservative exept at a very low level of N pe. The CSA ode may be non-onservative at high levels of N pe Effet of span-to-height ratio and onrete strength Figure 7(a) shows the variation of β with the span-to-height ratio L/h (ratio of span to overall height of a ross setion). The results are produed using onrete strengths ' f of 30 and 50 MPa. For onrete strength of 50 MPa, the maximum redistribution of moments in the beams appears at the ultimate limit state. For onrete strength of 30 MPa, on the other hand, the maximum redistribution of a very slender beam may not take plae at ultimate due to softening load-deformation behavior during the loading proess. For example, for L/h of 33.33, the maximum redistribution of moments, ourred at the maximum load, is 17% higher than the redistribution at ultimate, as shown in Fig. 7(a). Provided that there is no softening behavior, a higher span-to-height ratio produes an obviously higher redistribution at ultimate, while a lower onrete strength leads to a slightly higher redistribution. However, a long beam with lower onrete strength may exhibit softening load-deformation behavior, hereby ausing lower redistribution at ultimate ompared to the one with higher onrete strength, as shown in Fig. 7(a). 14

16 For different levels of L/h and ' f, a omparison between the β values by FEA and various ode equations is illustrated in Fig. 7(b) and Table 4. It is observed from Table 4 that as L/h inreases, the value of ε t gradually inreases while the derease in /d is negligible, provided that there is no softening load-deformation behavior. Therefore, the effet of L/h is refleted in the ACI ode but negleted in the CSA ode and EC2, as shown in Fig. 7(b). It is also observed that a higher onrete strength leads to a higher value of ε t and a lower value of /d, hereby ausing higher redistribution aording to the ode equations. However, this is opposite to the fat that a higher onrete strength produes a lower redistribution as disussed previously. In EC2, the effet of onrete strength is onsidered using Eq. (6a) for normal-strength onrete and (6b) for high-strength onrete. It is also seen that the ACI ode is onservative while EC2 is non-onservative. The CSA ode may be non-onservative for a low level of L/h Effet of CFRP elasti modulus and load type The CFRP omposites over a wide range of modulus of elastiity whih may vary from 80 to 500 GPa [23]. In this study, four levels of the CFRP tendon elasti modulus E f are seleted, namely, 80, 147, 270 and 500 GPa. The orresponding tensile strengths f f are 1440, 1840, 2160 and 2500 MPa, respetively. Figure 8(a) shows the variation of β with the CFRP modulus of elastiity for enter-point loading and uniform loading. It is observed that the β value dereases slightly as E f inreases. In addition, uniform loading mobilizes an obviously higher redistribution ompared to enter-point loading. In this analysis, the β value for uniform loading is about

17 times that for enter-point loading. For different levels of E f and different types of loading, a omparison between the β values predited by FEA and various ode equations is illustrated in Fig. 8(b) and Table 5. It is observed in Table 5 that the variable E f affets the values of ε t and /d. The load type also influenes the value of ε t but has null effet on the value of /d. As a onsequene, the effet of the variable E f is onsidered in all the ode equations, while effet of the load type is onsidered in the ACI ode but negleted in the CSA ode and EC2, as illustrated in Fig. 8(b). In this analysis, the ACI ode is onservative, but may be over-onservative when uniform loading is used. EC2 is non-onservative, partiularly for enter-point loading. The CSA ode may be non-onservative in the ase of low CFRP modulus of elastiity and enter-point loading. 5. Proposed equation for alulating the degree of redistribution Among various fators examined in the parametri study, the A s2 /A s1 ratio is found to be a leading parameter affeting the moment redistribution. The results presented in Setion 4.2 (Effet of A s2 /A s1 ) show that the degree of moment redistribution dereases remarkably from 41.97% to % when A s2 /A s1 inreases from 0.2 to 5 (see Table 1). This indiates that the moment redistribution depends on not only the dutility of one ritial setion as refleted in the ode equations, but also on the strutural harateristis of the whole beam. In addition, the parameter ε t (adopted by the ACI ode) seems to be better than the parameter /d (adopted by the CSA ode and EC2) when used to alulate the degree of moment redistribution in ontinuous 16

18 external tendon beams, beause ε t an reflet more important fators affeting the moment redistribution. Therefore, a simplified equation inluding the two parameters, A s2 /A s1 and ε t, may be reasonable to alulate the degree of moment redistribution, sine this equation an take into aount both the strutural harateristis and the setion dutility. Based on the above disussion, the ACI ode equation indiated by Eq. (4) an be modified as follows: β (%) = λ(1000 ε t ) (8) in whih λ is a oeffiient related to the parameter A s2 /A s1. To get the form of λ, the relationship between β / (1000 ε t ) and ln(a s2 /A s1 ) for the beams analyzed in Setion 4.2 is plotted in Fig. 9. Aording to the fit urves, λ is related to A s2 /A s1 by λ = ln( A / A ) for A 2 / A 1 1 (9a) s2 s1 s s 2 λ = ln( As2 / As 1) 2.76ln ( As2 / As1) for s2 s1 A / A > 1 (9b) Figure 10 illustrates the orrelation of simplified equations with the atual β values. In addition to the beams of the present numerial test, 16 two-span unbonded prestressed onrete beam speimens tested by Zhou and Zheng [15] are also used for the orrelation. The atual values of β are obtained from FEA (for numerial test speimens) or experiment (for laboratory test speimens). It an be seen from Fig. 10(a) that the data that the ACI ode equation is fitted to the atual values are rather sattered. By introduing the oeffiient λ, the modified equation proposed in this study orrelates well with the atual values, as shown in Fig. 10(b). In addition, most of the data shown in Fig. 10(b) are in the safe side, indiating that the proposed equation is generally onservative in prediting the degree of moment redistribution at 17

19 ultimate in suh beams. 6. Conlusions Based on the parametri study onduted on two-span ontinuous onrete beams prestressed with external CFRP tendons, the following onlusions regarding the redistribution of moments at ultimate an be drawn: (1) The A s2 /A s1 ratio is one of the most important fators affeting the moment redistribution. The variation of moment redistribution with the amount of non-prestressed steel depends on the ombined effets of dutility and stiffness differene between ritial setions. (2) The redistribution of moments is signifiantly redued when a reinfored onrete beam is strengthened by external prestressing. The moment redistribution dereases quikly as the effetive prestressing fore inreases. The eentriities of external tendons have important influene on the moment redistribution. (3) A higher span-to-height ratio generally leads to obviously higher moment redistribution. Uniform loading produes muh higher moment redistribution than enter-point loading. The moment redistribution slightly dereases with the inrease of the CFRP tendon modulus of elastiity. (4) The parameter ε t (adopted by the ACI ode) is superior to the parameter /d (adopted by the CSA ode and EC2) when used to alulate the degree of moment redistribution in ontinuous external tendon beams, beause ε t an reflet more important fators affeting the moment redistribution. 18

20 (5) A simplified equation inluding two important parameters, A s2 /A s1 and ε t, is proposed to alulate the degree of moment redistribution. The proposed equation exhibits a quite good fit to the atual values obtained from FEA and experiment. Aknowledgments This researh is sponsored by FEDER funds through the program COMPETE Programa Operaional Fatores de Competitividade and by national funds through FCT Fundação para a Ciênia e a Tenologia under the projet PEst-C/EME/UI0285/2013. The work presented in this paper has also been supported by FCT under Grant No. SFRH/BPD/66453/2009. Referenes [1] Shmidt JW, Bennitz A, Taljsten B, Goltermann P, Pedersen H. Mehanial anhorage of FRP tendons A literature review. Constrution and Building Materials 2012; 32: [2] Seo SY, Feo L, Hui D. Bond strength of near surfae-mounted FRP plate for retrofit of onrete strutures. Composite Strutures 2013; 95: [3] Nigro E, Cefarelli G, Bilotta A, Manfredi G, Cosenza E. Guidelines for flexural resistane of FRP reinfored onrete slabs and beams in fire. Composites Part B: Engineering 2014; 58: [4] Neto P, Alfaiate J, Vinagre J. A three-dimensional analysis of CFRP onrete bond behavior. Composites Part B: Engineering 2014; 59:

21 [5] D Antino T, Pellegrino C. Bond between FRP omposites and onrete: Assessment of design proedures and analytial models. Composites Part B: Engineering 2014; 60: [6] Lou T, Lopes SMR, Lopes AV. Numerial analysis of behaviour of onrete beams with external FRP tendons. Constrution and Building Materials 2012; 35: [7] Bennitz A, Shmidt JW, Nilimaa J, Taljsten B, Goltermann P, Ravn DL. Reinfored onrete T-beams externally prestressed with unbonded arbon fiber-reinfored polymer tendons. ACI Strutural Journal 2012; 109(4): [8] Carmo RNF, Lopes SMR. Dutility and linear analysis with moment redistribution in reinfored high-strength onrete beams. Canadian Journal of Civil Engineering 2005; 32(1): [9] Carmo RNF, Lopes SMR. Available plasti rotation in ontinuous high-strength onrete beams. Canadian Journal of Civil Engineering 2008; 35(10): [10] Oehlers DJ, Haskett M, Mohamed Ali MS, Griffith MC. Moment redistribution in reinfored onrete beams. Proeedings of the Institute of Civil Engineers Strutures and Buildings 2010; 163 (SB3): [11] Lou T, Lopes SMR, Lopes AV. Evaluation of moment redistribution in normal-strength and high-strength reinfored onrete beams. ASCE Journal of Strutural Engineering 2013; doi: /(ASCE)ST X [12] Lopes SMR, Harrop J, Gamble AE. Study of moment redistribution in prestressed onrete beams. ASCE Journal of Strutural Engineering 1997; 123(5):

22 [13] Kodur VKR, Campbell TI. Evaluation of moment redistribution in a two-span ontinuous prestressed onrete beam. ACI Strutural Journal 1996; 93(6): [14] Kodur VKR, Campbell TI. Fators governing redistribution of moment in ontinuous prestressed onrete beams. Strutural Engineering and Mehanis 1999; 8(2): [15] Zhou W, Zheng WZ. Experimental researh on plasti design method and moment redistribution in ontinuous onrete beams prestressed with unbonded tendons. Magazine of Conrete Researh 2010; 62(1): [16] Lou T, Lopes SMR, Lopes AV. External CFRP tendon members: Seondary reations and moment redistribution. Composites Part B: Engineering 2014; 57: [17] Tihy M, Rakosnik J. Plasti analysis of onrete frames (with partiular referene to limit states design). Collet (Publishers) Ltd., London, England; [18] Mouessian A, Campbell TI. Predition of the load apaity of two-span ontinuous prestressed onrete beams. PCI Journal 1988; 33(2): [19] Cohn MZ. Continuity in prestressed onrete partial prestressing. Partial prestressing, from theory to pratie, Vol. I, Survey Reports, NATO ASI Series, Martinus Nijhoff Publishers, Boston, Mass., 1986; [20] ACI Committee 318. Building ode requirements for strutural onrete (ACI ) and ommentary. Amerian Conrete Institute, Farmington Hills, MI;

23 [21] CSA. Design of onrete strutures. A , Canadian Standards Assoiation, Mississauga, Ontario, Canada; [22] CEN. Euroode 2 (EC2): Design of onrete strutures Part 1-1: General rules and rules for buildings. EN , European Committee for Standardization, Brussels, Belgium; [23] FIB. Model Code 2010 (MC10). Bulletins 55 and 56, International Federation for Strutural Conrete, Lausanne, Switzerland; [24] Lou T, Xiang Y. Finite element modeling of onrete beams prestressed with external tendons. Engineering Strutures 2006; 28(14): [25] Lou T, Lopes SMR, Lopes AV. Nonlinear and time-dependent analysis of ontinuous unbonded prestressed onrete beams. Computers & Strutures 2013; 119: [26] Hognestad E. A study of ombined bending and axial load in reinfored onrete members. Bulletin No. 399, University of Illinois Engineering Experiment Station, Urbana, IL; [27] Lou T, Lopes SMR, Lopes AV. Flexural response of ontinuous onrete beams prestressed with external tendons. ASCE Journal of Bridge Engineering 2013; 18(6):

24 σ f ' σ f t σ f ff 1 σ s f y E s 1 E f ε s ε 0 ε u ε ε r 10ε r ε ε f (a) (b) () (d) Fig. 1 Stress-strain diagrams for materials. (a) onrete in ompression; (b) onrete in tension; () CFRP prestressing tendons; (d) non-prestressed steel 23

25 As3 P 6667 mm As2 P Centroidal axis e0=0 mm As1 e1=150 mm e2=150 mm As3 e1=150 mm e0=0 mm 2222 mm 2222 mm 5000 mm 5000 mm 5000 mm 5000 mm L=10000 mm L=10000 mm 600 mm 50 mm Ap 550 mm 300 mm (a) P Beam element node P Centroidal axis mm (b) Fig. 2 Referene beam used for parametri evaluation and its finite element model. (a) beam details; (b) finite element model 24

26 EC2 limit β ACI & CSA limit 0.10 Eq. (4) - ACI 0.05 Eq. (5) - CSA Eq. (6) - EC2 A s2 /A s1 = 0.8 FEA A s2 (mm 2 ) Fig. 3 Effet of non-prestressed steel area on the degree of moment redistribution aording to FEA and ode preditions 25

27 β A s1 /A s2 (A s2 =360 mm 2 ) A s2 /A s1 (A s1 =360 mm 2 ) A s2 /A s1 or A s1 /A s2 (a) β A s2 =360 mm 2 EC2 limit ACI and CSA limit Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA β EC2 limit ACI and CSA limit Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA A s1 =360 mm A s1 /A s2 A s2 /A s1 (b) Fig. 4 Effet of A s2 /A s1 or A s1 /A s2 on the degree of moment redistribution. (a) FEA results; (b) omparison between FEA and ode preditions 26

28 e 1 e β e 1 or e 2 (mm) (a) β EC2 limit ACI and CSA limit β EC2 limit ACI and CSA limit Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA e 1 (mm) Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA e 2 (mm) (b) Fig. 5 Effet of midspan and enter support tendon eentriities on the degree of moment redistribution. (a) FEA results; (b) omparison between FEA and ode preditions 27

29 A p =variable, f pe =930 MPa A p =450 mm 2, f pe =variable β A p f pe (kn) (a) Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA EC2 limit β ACI and CSA limit EC2 limit A p (mm 2 ) β ACI and CSA limit Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA f pe (MPa) (b) Fig. 6 Effet of tendon area and effetive prestress on the degree of moment redistribution. (a) FEA results; (b) omparison between FEA and ode preditions 28

30 f ' =30 MPa f ' =50 MPa Maximum 0.24 β Ultimate L/h (a) EC2 limit 0.30 EC2 limit β 0.20 ACI and CSA limit β 0.20 ACI and CSA limit Eq. (4) - ACI f ' =30 MPa Eq. (5) - CSA 0.05 Eq. (6) - EC2 FEA L/h 0.10 Eq. (4) - ACI f ' =50 MPa Eq. (5) - CSA 0.05 Eq. (6) - EC2 FEA L/h (b) Fig. 7 Effet of span-to-height ratio and onrete strength on the degree of moment redistribution. (a) FEA results; (b) omparison between FEA and ode preditions 29

31 Center-point loading Uniform loading β E f (GPa) (a) EC2 limit EC2 limit β ACI and CSA limit β ACI and CSA limit Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA E f (GPa) Center-point loading Eq. (4) - ACI Eq. (5) - CSA Eq. (6) - EC2 FEA E f (GPa) Uniform loading (b) Fig. 8 Effet of CFRP elasti modulus and load type on the degree of moment redistribution. (a) FEA results; (b) omparison between FEA and ode preditions 30

32 4 2 Y = X [β/(1000ε t )] (%) Y = X X ln(a s2 /A s1 ) Fig. 9 Relationship between β / (1000 ε t ) and ln(a s2 /A s1 ) 31

33 β by simplified equation (%) ACI equation Present numerial test -20 Zhou and Zheng [15] Atual value of β (%) β by simplified equation (%) Proposed equation Atual value of β (%) Present numerial test Zhou and Zheng [15] (a) (b) Fig. 10 Correlation of simplified equations with atual β values. (a) ACI ode equation; (b) proposed equation 32

34 Table 1 Results in relation to moment redistribution for different ontents of non-prestressed steel A s1 A s2 Beam (mm 2 ) (mm 2 ) A s2/a s1 ε t (%) /d (%) M (kn m) M e (kn m) Eq. (4) (ACI) β (%) Eq. (5) Eq. (6) (CSA) (EC2) FEA B B B B B B B B B B B B B B B

35 Table 2 Results in relation to moment redistribution for different levels of e 1 and e 2 Beam e 1 (mm) e 2 (mm) ε t (%) /d (%) M (kn m) M e (kn m) Eq. (4) (ACI) β (%) Eq. (5) (CSA) Eq. (6) (EC2) FEA B B B B B B B B

36 Table 3 Results in relation to moment redistribution for different levels of A p and f pe Beam A p (mm 2 ) f pe (MPa) ε t (%) /d (%) M (kn m) M e (kn m) Eq. (4) (ACI) β (%) Eq. (5) (CSA) Eq. (6) (EC2) FEA B B B B B B B B B B

37 Table 4 Results in relation to moment redistribution for different levels of L/h and ' f Beam L/h ' f (MPa) ε t (%) /d (%) M (kn m) M e (kn m) Eq. (4) (ACI) β (%) Eq. (5) (CSA) Eq. (6) (EC2) FEA B B B B B B B B

38 Table 5 Results in relation to moment redistribution for different levels of E f and different types of loading Beam E f (GPa) Load type ε t (%) /d (%) M (kn m) M e (kn m) Eq. (4) (ACI) β (%) Eq. (5) (CSA) Eq. (6) (EC2) FEA B B CPL B B B B UL B B Note: CPL = enter-point loading; UL = uniform loading 37

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