Flexural Behavior of Two-Span Continuous CFRP RC Beams

This paper investigates the feasibility of replacing steel bars with carbon-fiber-reinforced polymer (CFRP) bars in continuous reinforced concrete (RC) beams. A numerical model is introduced. Model predictions are compared with the experimental results that are available in the literature. A comprehensive numerical investigation is then performed on two-span CFRP/steel RC beams with ρb2 = 0.61–3.03% and ρb1/ρb2 = 1.5, where ρb1 and ρb2 are tensile bar ratios (ratios of tensile bar area to effective cross-sectional area of beams) over positive and negative moment regions, respectively. The study shows that replacing steel bars with CFRP bars greatly improves the crack mode at a low bar ratio. The ultimate load of CFRP RC beams is 89% higher at ρb2 = 0.61% but 7.2% lower at ρb2 = 3.03% than that of steel RC beams. In addition, CFRP RC beams exhibit around 13% greater ultimate deflection compared to steel RC beams. The difference of moment redistribution between CFRP and steel RC beams diminishes as ρb2 increases. ACI 318-19 appears to be conservative, and it leads to more accurate predictions of moment redistribution in CFRP RC beams than that in steel RC beams.


Introduction
The corrosion of steel bars is responsible for the structural deterioration of reinforced concrete (RC) members around the world, especially those in aggressive environment [1]. Replacing conventional steel bars with non-corrosive fiber-reinforced polymer (FRP) bars can effectively solve the corrosive issue. Apart from their anti-corrosiveness, FRPs have other attractive advantages over steels, such as higher strength, lighter weight and nonmagnetism [2]. As a result, FRP materials are widely employed to reinforce/strengthen various concrete members, such as sheets [3,4], bars [5,6] and tendons [7,8]. Nevertheless, FRPs are brittle in nature, and this may result in sudden structural collapse without sufficient warnings. Therefore, a careful assessment on the replacement of steel bars with FRPs in RC elements is essential. A great number of works have been conducted to evaluate the flexural ductility [9,10], deflection [11,12], cracking [13,14] and shear behavior [15] of simply supported FRP RC beams.
Continuous RC beams are preferred in engineering practice, as they have higher stiffness and load-carry capacity than their simply supported counterparts. In the flexural strength design of continuous FRP RC beams, the plastic hinge theory used for continuous steel RC beams is not valid, because of the linear-elastic property of FRP composites. Therefore, ACI 440.1R-06 [2] recommended that no moment redistribution in continuous FRP RC beams be considered. However, redistribution of moments does occur in continuous FRP RC beams during the inelastic range of loading [16][17][18].
Research on continuous FRP RC beams has been performed by different investigators [19][20][21][22]. Among various works, Lou et al. [23] evaluated the moment redistribution against neutral axis depth in carbon and glass FRP (CFRP and GFRP) RC continuous

Materials
The stress-strain relationship for concrete in compression recommended by Eurocode 2 [30] is expressed by the following: where σ c is the concrete stress; ε c is the concrete strain; f ck is the concrete cylinder compressive strength, taken equal to 30 MPa for RC beams used in this numerical investigation; ε 0 is the concrete strain at peak stress; and k is a coefficient depending on the concrete grade. The concrete in tension is assumed to be linearly elastic up to cracking, followed by linear tension-stiffening. CFRP bars are linearly elastic up to rupture. The rupture strength, rupture strain and elastic modulus of CFRP bars used in this numerical investigation are 1450 MPa, 1.09% and 133 GPa, respectively [11].
Steel bars are modeled by a bilinear elastic-hardening law, where the hardening modulus is equal to 1.5% of the elastic modulus. The yield strength and elastic modulus of steel bars used in this numerical investigation are 530 MPa and 200 GPa, respectively.

Finite Element Method
The Timoshenko beam theory has been used for modeling of RC beams [29]. In a two-node Timoshenko beam element, the transverse displacement w and rotation θ are expressed in terms of the nodal displacements by: (2) where N 1 = (l − x)/l, N 2 = x/l, in which l is the element length; the subscripts 1 and 2 represent end nodes. Hence, the curvature, κ, and shear strain, γ, are expressed by the following: where u e = w 1 θ 1 w 2 θ 2 T (4) Bending moment, M, and shear force, Q, are then expressed by the following: where EI is the flexural stiffness obtained from the pre-generated moment-curvature relation, GA is the shear stiffness and k s is the shear correction factor. The principle of virtual work leads to the following equilibrium equations: where Substituting Equation (6) into Equation (7) yields the following element stiffness equations: where The stiffness equations for the structure are assembled in the global coordinate system from the contribution of all the elements. After applying proper boundary conditions, the nonlinear equilibrium equations are solved by the incremental-iterative method. The iterative procedure for each increment involves four basic steps, namely formation of the current stiffness matrix, solution of the equilibrium equations, determination of the current state for each element and check of convergence. During the solution process, when any of the constituent materials reaches its ultimate strain capacity, the beam fails and the analysis is therefore terminated. A computer program implementing the aforementioned procedure was written. The program is capable of conducting the full-range nonlinear analysis of continuous steel/FRP RC beams up to failure.

Comparison of Numerical and Test Data
Two continuous RC beams tested by Ashour and Habeeb [24] were selected. One beam was reinforced with CFRP bars (designated as C-C-5), while the other beam was reinforced with steel bars (designated as S-C-6). Both beams were of a rectangular section (200 × 300 mm), continuous over two spans (2750 mm for each span) and subjected to center-point loading symmetrically applied on the spans, as shown in Figure 1. In Beam C-C-5, two CFRP bars with a diameter of 12 mm were provided in both the top and bottom layers of the beam. In Beam S-C-6, four steel bars with a diameter of 12 mm were provided Two continuous RC beams tested by Ashour and Habeeb [24] were selected beam was reinforced with CFRP bars (designated as C-C-5), while the other beam reinforced with steel bars (designated as S-C-6). Both beams were of a rectangular s (200 × 300 mm), continuous over two spans (2750 mm for each span) and subjec center-point loading symmetrically applied on the spans, as shown in Figure 1. In C-C-5, two CFRP bars with a diameter of 12 mm were provided in both the top and b layers of the beam. In Beam S-C-6, four steel bars with a diameter of 12 mm were pro in both the top and bottom layers of the beam. Both specimens were provided with reinforcement consisting of steel bars that are 8 mm diameter, with a spacing of 14 The rupture strength, rupture strain and elastic modulus of CFRP bars were 1061 0.53% and 200 GPa, respectively. The yield strength and elastic modulus of steel bars 510.8 MPa and 200 GPa, respectively. The value of fck was 28 MPa for Beam C-C-5 an MPa for Beam S-C-6. According to the numerical analysis, crushing failure happens in Beam S-Cconcrete at midspan reaches its ultimate compressive strain of 0.0035), while the ru failure of CFRP bars takes place in Beam C-C-5 (i.e., CFRP bars at center support their rupture strength of 1061 MPa). These failure modes of the specimens are cons with the experimental observations. The predicted load-deflection behavior of the mens is compared to the experimental results in Figure 2. It is observed that the cracking stiffness by the numerical analysis appears to be higher than the experim one, while the difference is more apparent for the CFRP RC beam (C-C-5) than for th RC beam (S-C-6). This can be explained by the fact that the bond-slip effect, which is important in CFRP bars than in steel bars, is neglected in the numerical modeling. dition, the numerical model does not account for the confinement effect in concrete ing to substantial underestimate in the ultimate deflection in steel RC beams faili concrete crushing. Despite some discrepancy, the numerical model can capture rea bly the key response characteristics of both specimens (i.e., cracking, yielding and ing in Beam S-C-6; and cracking and rupture in Beam C-C-5). According to the numerical analysis, crushing failure happens in Beam S-C-6 (i.e., concrete at midspan reaches its ultimate compressive strain of 0.0035), while the rupture failure of CFRP bars takes place in Beam C-C-5 (i.e., CFRP bars at center support reach their rupture strength of 1061 MPa). These failure modes of the specimens are consistent with the experimental observations. The predicted load-deflection behavior of the specimens is compared to the experimental results in Figure 2. It is observed that the post-cracking stiffness by the numerical analysis appears to be higher than the experimental one, while the difference is more apparent for the CFRP RC beam (C-C-5) than for the steel RC beam (S-C-6). This can be explained by the fact that the bond-slip effect, which is more important in CFRP bars than in steel bars, is neglected in the numerical modeling. In addition, the numerical model does not account for the confinement effect in concrete, leading to substantial underestimate in the ultimate deflection in steel RC beams failing by concrete crushing. Despite some discrepancy, the numerical model can capture reasonably the key response characteristics of both specimens (i.e., cracking, yielding and crushing in Beam S-C-6; and cracking and rupture in Beam C-C-5).

Results and Discussion
Two-span RC beams under center-point loading symmetrically applied on the spans,

Results and Discussion
Two-span RC beams under center-point loading symmetrically applied on the spans, as shown Figure 3, were used. The investigated variables are the type of bars (CFRP and steel) and the bar ratio. The arrangement of reinforcing bars is as follows: ρ b1 /ρ b2 = 1.5, ρ b2 = 0.61-3.03% and ρ b3 = 0.36%, where ρ b1 , ρ b2 and ρ b3 are ratios of tensile bars over positive, negative moment regions and compressive bars, respectively. The bar ratio is defined by where A b is the bar area, b is the cross-sectional width and d b is the distance from the centroid of tensile bars to the extreme compressive fiber of the cross-section.

Results and Discussion
Two-span RC beams under center-point loading symmetrically applied on the spans, as shown Figure 3, were used. The investigated variables are the type of bars (CFRP and steel) and the bar ratio. The arrangement of reinforcing bars is as follows: ρb1/ρb2 = 1.5, ρb2 = 0.61-3.03% and ρb3 = 0.36%, where ρb1, ρb2 and ρb3 are ratios of tensile bars over positive, negative moment regions and compressive bars, respectively. The bar ratio is defined by ρb = Ab/(bdb), where Ab is the bar area, b is the cross-sectional width and db is the distance from the centroid of tensile bars to the extreme compressive fiber of the cross-section.

Failure and Cracking Mode
Failure of CFRP and steel RC beams occurs when the critical section reaches the ultimate concrete compressive strain of 0.0035 (see Figure 4). For steel RC beams, crushing failure transits from the midspan to the center support when the bar ratio increases from a low to high level. For CFRP RC beams, the beam concrete is crushed at the center support, regardless of the bar ratio. At ρb2 = 0.61%, CFRP bars are close to their rupture strength when crushing failure happens, indicating an approximately balanced ratio of CFRP bars. Figure 4a shows that, for steel RC beams with a low ρb2 level of 0.61%, there appear large concrete tensile strains at the critical sections against relatively small ones over other regions, indicating marked cracking concentration. By using CFRP bars instead of steel bars, the tensile strain distribution or the crack mode is greatly improved, i.e., the crack

Failure and Cracking Mode
Failure of CFRP and steel RC beams occurs when the critical section reaches the ultimate concrete compressive strain of 0.0035 (see Figure 4). For steel RC beams, crushing failure transits from the midspan to the center support when the bar ratio increases from a low to high level. For CFRP RC beams, the beam concrete is crushed at the center support, regardless of the bar ratio. At ρ b2 = 0.61%, CFRP bars are close to their rupture strength when crushing failure happens, indicating an approximately balanced ratio of CFRP bars. Figure 4a shows that, for steel RC beams with a low ρ b2 level of 0.61%, there appear large concrete tensile strains at the critical sections against relatively small ones over other regions, indicating marked cracking concentration. By using CFRP bars instead of steel bars, the tensile strain distribution or the crack mode is greatly improved, i.e., the crack concentration is relieved and the crack region is more extended. Therefore, it can be concluded that, if concrete beams are lightly reinforced, replacing steel bars with CFRP bars can lead to significant improvement in the crack mode. Figure 4b demonstrates that, at a high ρ b2 level of 2.42%, the maximum tensile strains in the critical sections in both CFRP and steel RC beams are small and the strains over other regions exceed the cracking strain, indicating that small cracks appear all over the beam. CFRP RC beams show a slightly greater crack width at midspan but a slightly smaller crack width at the center support, as compared to steel RC beams. Figure 5 shows the moment-curvature curves of CFRP and steel RC beams with ρ b2 = 0.61 and 1.82%. It is clearly demonstrated that steel RC beams exhibit three-stage moment-curvature behavior. Each stage is represented by an approximately linear segment, while each stage transition is featured by a marked reduction in bending stiffness. The first stage, characterized by the elastic behavior, continues up to the occurrence of cracking, followed by the elastically cracked stage until yielding of steel and then by post-yielding stage up to the structural collapse. CFRP RC beams experience only the first and second stages, due to the lack of yielding of FRP bars. Steel and CFRP RC beams behave identically in the first stage, while, in the second stage, steel RC beams show a stiffer behavior than CFRP RC beams attributed to the modulus difference between steel and FRP bars.

Moment-Curvature Behavior
can lead to significant improvement in the crack mode. Figure 4b demonstrates that, high ρb2 level of 2.42%, the maximum tensile strains in the critical sections in both CF and steel RC beams are small and the strains over other regions exceed the cracking str indicating that small cracks appear all over the beam. CFRP RC beams show a sligh greater crack width at midspan but a slightly smaller crack width at the center supp as compared to steel RC beams.  Figure 5 shows the moment-curvature curves of CFRP and steel RC beams with = 0.61 and 1.82%. It is clearly demonstrated that steel RC beams exhibit three-stage m ment-curvature behavior. Each stage is represented by an approximately linear segm while each stage transition is featured by a marked reduction in bending stiffness. T first stage, characterized by the elastic behavior, continues up to the occurrence of cra ing, followed by the elastically cracked stage until yielding of steel and then by post-yie  ing stage up to the structural collapse. CFRP RC beams experience only the first and second stages, due to the lack of yielding of FRP bars. Steel and CFRP RC beams behave identically in the first stage, while, in the second stage, steel RC beams show a stiffer behavior than CFRP RC beams attributed to the modulus difference between steel and FRP bars. The variation in ultimate curvatures (κ) at midspan and center support with varying ρb2 is shown in Figure 6 and presented in Table 1. At a low ρb2 level of 0.61%, CFRP bars lead to substantially (i.e., 50.6% at midspan and 36.8% at center support) lower curvature than steel bars. As ρb2 increases, the decrease in curvatures of CFRP RC beams is slower  The variation in ultimate curvatures (κ) at midspan and center support with varying ρ b2 is shown in Figure 6 and presented in Table 1. At a low ρ b2 level of 0.61%, CFRP bars lead to substantially (i.e., 50.6% at midspan and 36.8% at center support) lower curvature than steel bars. As ρ b2 increases, the decrease in curvatures of CFRP RC beams is slower than that of steel RC beams. As a result, CFRP RC beams exhibit a curvature comparable to that of steel RC beams at high ρ b2 levels of 2.42% and 3.03%. The variation in ultimate curvatures (κ) at midspan and center support with varying ρb2 is shown in Figure 6 and presented in Table 1. At a low ρb2 level of 0.61%, CFRP bars lead to substantially (i.e., 50.6% at midspan and 36.8% at center support) lower curvature than steel bars. As ρb2 increases, the decrease in curvatures of CFRP RC beams is slower than that of steel RC beams. As a result, CFRP RC beams exhibit a curvature comparable to that of steel RC beams at high ρb2 levels of 2.42% and 3.03%.  The variation in ultimate moments (M) at midspan and center support with varying ρ b2 is shown in Figure 7 and presented in Table 1. In steel RC beams, the moment at center support is lower than that at midspan, attributed to the fact that a smaller bar area is provided in the hogging region compared to the sagging region. In CFRP RC beams, however, the moments at midspan and center support are almost identical. This can be explained by the fact that a higher stress in CFRP bars at the center support is achieved in failure, thus compensating the smaller bar area in the section. At a low ρ b2 level of 0.61%, CFRP RC beams show significantly (i.e., 71.5% at midspan and 125.5% at center support) higher moments than steel RC beams. With an increase in ρ b2 level, the increase in moments in CFRP RC beams is slower than that in steel RC beams, attributed to an vided in the hogging region compared to the sagging region. In CFRP RC beams, how ever, the moments at midspan and center support are almost identical. This can be ex plained by the fact that a higher stress in CFRP bars at the center support is achieved i failure, thus compensating the smaller bar area in the section. At a low ρb2 level of 0.61% CFRP RC beams show significantly (i.e., 71.5% at midspan and 125.5% at center support higher moments than steel RC beams. With an increase in ρb2 level, the increase in mo ments in CFRP RC beams is slower than that in steel RC beams, attributed to an apparen decrease in stress in CFRP bars. Consequently, CFRP bars may result in lower moment than steel bars at a high ρb2 level.    Figure 8 shows the load-deflection behavior of CFRP and steel RC beams with ρ b2 = 0.6 and 1.82%. Similar to the moment-curvature behavior, the load-deflection curve of CFRP RC beams consists of two stages, namely elastic stage controlled by concrete and postcracking stage dominated by tensile bars. Each stage is featured by a straight line. For steel RC beams, the first yielding at the center support has limited influences on the loaddeflection response, while the second yielding at midspan leads to a remarkable reduction in structural stiffness. For steel RC beams, the first yielding at the center support has limited influences on th load-deflection response, while the second yielding at midspan leads to a remarkable re duction in structural stiffness. The variation in the ultimate load (P) and deflection (δ) with varying ρb2 is shown i Figures 9 and 10, respectively. The results are also presented in Table 1. It is observed tha as ρb2 increases, the enhancement of the ultimate load of CFRP RC beams is not as appar ent as that of steel RC beams. At a low ρb2 level of 0.61%, the ultimate load of CFRP RC beams is significantly (89%) greater than that of steel RC beams, attributed to that CFR bars develop substantially higher stress than steel bars. However, at ρb2 = 3.03%, the ult mate load of CFRP RC beams turns to be 7.2% lower than that of steel RC beams. Th ultimate deflection of CFRP RC beams is around 13% greater than that of steel RC beam over the entire ρb2 levels investigated.  The variation in the ultimate load (P) and deflection (δ) with varying ρ b2 is shown in Figures 9 and 10, respectively. The results are also presented in Table 1. It is observed that, as ρ b2 increases, the enhancement of the ultimate load of CFRP RC beams is not as apparent as that of steel RC beams. At a low ρ b2 level of 0.61%, the ultimate load of CFRP RC beams is significantly (89%) greater than that of steel RC beams, attributed to that CFRP bars develop substantially higher stress than steel bars. However, at ρ b2 = 3.03%, the ultimate load of CFRP RC beams turns to be 7.2% lower than that of steel RC beams. The ultimate deflection of CFRP RC beams is around 13% greater than that of steel RC beams over the entire ρ b2 levels investigated.

Load-Deflection Behavior
The variation in the ultimate load (P) and deflection (δ) with varying ρb2 is shown in Figures 9 and 10, respectively. The results are also presented in Table 1. It is observed that, as ρb2 increases, the enhancement of the ultimate load of CFRP RC beams is not as apparent as that of steel RC beams. At a low ρb2 level of 0.61%, the ultimate load of CFRP RC beams is significantly (89%) greater than that of steel RC beams, attributed to that CFRP bars develop substantially higher stress than steel bars. However, at ρb2 = 3.03%, the ultimate load of CFRP RC beams turns to be 7.2% lower than that of steel RC beams. The ultimate deflection of CFRP RC beams is around 13% greater than that of steel RC beams over the entire ρb2 levels investigated.  Figure 11 shows the development of reactions and moments in CFRP and steel RC beams. The results are produced for ρb2 = 0.61%. The actual values in the graphs are obtained from the nonlinear computer analysis proposed in this study, while the elastic values are obtained based on linear-elastic theory. It can be seen from the figure that the actual values are identical to the elastic ones at the initial loading up to cracking. Thereafter, the actual values begin to deviate from the elastic ones due to redistribution of moments. Because the first cracking appears at the center support, the cracking moments are redistributed from the center support towards the midspan. As a consequence, the reactions grow faster at the end support, and they grow slower at the center support. Correspondingly, the moments increase quicker at midspan and slower at the center support.

Reaction and Moment Development
In CFRP RC beams, the post-cracking reaction or moment develops linearly with increasing load up to failure, as illustrated in Figure 11a. In addition, over the entire inelastic range of loading, the deviation between actual and elastic reactions or moments is not apparent, indicating a low redistribution of moments in these beams. In steel RC beams, yielding of steel bars may play a critical role in the development of reaction and moment, leading to substantially more deviation between actual and elastic values when compared to CFRP RC beams. Upon the first yielding of steel bars at the center support, further moments are redistributed away from the center support, leading to a further growth of the rate of increase in end support reaction and midspan moment, and correspondingly,  Figure 11 shows the development of reactions and moments in CFRP and steel RC beams. The results are produced for ρ b2 = 0.61%. The actual values in the graphs are obtained from the nonlinear computer analysis proposed in this study, while the elastic values are obtained based on linear-elastic theory. It can be seen from the figure that the actual values are identical to the elastic ones at the initial loading up to cracking. Thereafter, the actual values begin to deviate from the elastic ones due to redistribution of moments. Because the first cracking appears at the center support, the cracking moments are redistributed from the center support towards the midspan. As a consequence, the reactions grow faster at the end support, and they grow slower at the center support. Correspondingly, the moments increase quicker at midspan and slower at the center support.

Moment Redistribution against Bar Strain
Moment redistribution is quantified by the following: 1 / e M M β = − (11) where β is the degree of redistribution, M is the actual moment and Me is the elastic moment.
Moment redistribution relies strongly on the strain in tensile bars. Figure 12 shows the moment redistribution versus tensile bar strain curves for CFRP and steel RC beams with ρb2 = 0.61% and 1.82%. Prior to cracking, moment redistribution does not occur (β = 0), and the strain in tensile bars is negligible. Moment redistribution takes place after cracking. In general, the curves consist of two distinct stages for CFRP RC beams, while there are two additional stages for steel RC beams. In the first stage, moment redistribution develops linearly with increasing bar strain until the crack evolution stabilizes. In this stage, CFRP RC beams exhibit approximately the same behavior to that of steel RC beams. The second stage is characterized by stabilizing redistribution. For CFRP RC beams, this stage continues until the ultimate failure, accompanied by a substantial variation in bar strain. For steel RC beams, the third stage, triggered by the yielding of steel bars at the center support, is characterized by a quick development of moment redistribution with a In CFRP RC beams, the post-cracking reaction or moment develops linearly with increasing load up to failure, as illustrated in Figure 11a. In addition, over the entire inelastic range of loading, the deviation between actual and elastic reactions or moments is not apparent, indicating a low redistribution of moments in these beams. In steel RC beams, yielding of steel bars may play a critical role in the development of reaction and moment, leading to substantially more deviation between actual and elastic values when compared to CFRP RC beams. Upon the first yielding of steel bars at the center support, further moments are redistributed away from the center support, leading to a further growth of the rate of increase in end support reaction and midspan moment, and correspondingly, a further diminution of the rate of increase in center support reaction and moment, as illustrated in Figure 11b. When steel bars at midspan begin to yield, moments turn to be redistributed from the midspan towards the center support, thereby resulting in a diminution of the rate of increase in the end support reaction and midspan moment, while there is a growth of the increase rate of the center support reaction and moment.
Based on the above discussion, it can be concluded that the developments of support reactions and bending moments, and thereby moment redistribution, are influenced by concrete cracking and steel yielding for steel RC beams, while they are influenced by concrete cracking only for CFRP RC beams.

Moment Redistribution against Bar Strain
Moment redistribution is quantified by the following: where β is the degree of redistribution, M is the actual moment and M e is the elastic moment. Moment redistribution relies strongly on the strain in tensile bars. Figure 12 shows the moment redistribution versus tensile bar strain curves for CFRP and steel RC beams with ρ b2 = 0.61% and 1.82%. Prior to cracking, moment redistribution does not occur (β = 0), and the strain in tensile bars is negligible. Moment redistribution takes place after cracking. In general, the curves consist of two distinct stages for CFRP RC beams, while there are two additional stages for steel RC beams. In the first stage, moment redistribution develops linearly with increasing bar strain until the crack evolution stabilizes. In this stage, CFRP RC beams exhibit approximately the same behavior to that of steel RC beams. The second stage is characterized by stabilizing redistribution. For CFRP RC beams, this stage continues until the ultimate failure, accompanied by a substantial variation in bar strain. For steel RC beams, the third stage, triggered by the yielding of steel bars at the center support, is characterized by a quick development of moment redistribution with a limited increase in bar strain. The fourth stage, triggered by yielding of steel bars at midspan, is featured by stabilizing redistribution of moments with varying bar strain up to the ultimate failure.  The change in the strain in tensile bars (εt) and moment redistribution (β) in failure with the ρb2 level is displayed in Figures 13 and 14, respectively. The results are also presented in Table 1. It is observed that the bar strain at center support is higher than that at midspan. The strain in CFRP bars is significantly lower than that in steel bars at a low ρb2 level, but it is close or comparable to that in steel bars at a high ρb2 level. Moment redistribution at center support is substantially higher than that at midspan. The midspan-tosupport-redistribution ratio (absolute value) is around 0.6. Moment redistribution decreases as ρb2 increases, and this is attributed to the reduction in flexural ductility. The decrease rate of CFRP RC beams is less than that of steel RC beams. CFRP bars lead to significant lower moment redistribution than steel bars at a low ρb2 level, but the difference narrows as ρb2 increases. The ratio of moment redistribution in CFRP RC beams to that in steel RC beams varies from 0.42 (at ρb2 = 0.61%) to 0.85 (at ρb2 = 3.03%). The change in the strain in tensile bars (ε t ) and moment redistribution (β) in failure with the ρ b2 level is displayed in Figures 13 and 14, respectively. The results are also presented in Table 1. It is observed that the bar strain at center support is higher than that at midspan. The strain in CFRP bars is significantly lower than that in steel bars at a low ρ b2 level, but it is close or comparable to that in steel bars at a high ρ b2 level. Moment redistribution at center support is substantially higher than that at midspan. The midspanto-support-redistribution ratio (absolute value) is around 0.6. Moment redistribution decreases as ρ b2 increases, and this is attributed to the reduction in flexural ductility. The decrease rate of CFRP RC beams is less than that of steel RC beams. CFRP bars lead to significant lower moment redistribution than steel bars at a low ρ b2 level, but the difference narrows as ρ b2 increases. The ratio of moment redistribution in CFRP RC beams to that in steel RC beams varies from 0.42 (at ρ b2 = 0.61%) to 0.85 (at ρ b2 = 3.03%). support-redistribution ratio (absolute value) is around 0.6. Moment redistribution decreases as ρb2 increases, and this is attributed to the reduction in flexural ductility. The decrease rate of CFRP RC beams is less than that of steel RC beams. CFRP bars lead to significant lower moment redistribution than steel bars at a low ρb2 level, but the difference narrows as ρb2 increases. The ratio of moment redistribution in CFRP RC beams to that in steel RC beams varies from 0.42 (at ρb2 = 0.61%) to 0.85 (at ρb2 = 3.03%).

Redistribution Quantification Models based on Bar Strain
When calculating the design moments in continuous RC beams, the current design codes permit an elastic analysis with an adjustment of the elastic moment through the use of β. In ACI 318-19 [27], the parameter εt is adopted for calculating the permissible redistribution: The maximum redistribution is 20%, and εt is not less than 0.0075. Figure 15 shows the numerically obtained β-εt relationship in failure of the center support section of CFRP and steel RC beams, along with the ACI 318-19 curve. The numerical simulations show that, at a given εt level, CFRP bars lead to lower moment redistribution than steel bars, and the difference is increasingly notable with the increasing εt level. As far as the β-εt relationship is concerned, ACI 318-19 is consistent with the numerical results, indicating that this code can reflect the tendency of the redistribution variation with the εt level. In addition, all of the numerical data are outside of the code curve, suggesting conservative predictions by ACI 318-19.

Redistribution Quantification Models based on Bar Strain
When calculating the design moments in continuous RC beams, the current design codes permit an elastic analysis with an adjustment of the elastic moment through the use of β. In ACI 318-19 [27], the parameter ε t is adopted for calculating the permissible redistribution: The maximum redistribution is 20%, and ε t is not less than 0.0075. Figure 15 shows the numerically obtained β-ε t relationship in failure of the center support section of CFRP and steel RC beams, along with the ACI 318-19 curve. The numerical simulations show that, at a given ε t level, CFRP bars lead to lower moment redistribution than steel bars, and the difference is increasingly notable with the increasing ε t level. As far as the β-ε t relationship is concerned, ACI 318-19 is consistent with the numerical results, indicating that this code can reflect the tendency of the redistribution variation with the ε t level. In addition, all of the numerical data are outside of the code curve, suggesting conservative predictions by ACI 318-19. support section of CFRP and steel RC beams, along with the ACI 318-19 curve. The nu-merical simulations show that, at a given εt level, CFRP bars lead to lower moment redistribution than steel bars, and the difference is increasingly notable with the increasing εt level. As far as the β-εt relationship is concerned, ACI 318-19 is consistent with the numerical results, indicating that this code can reflect the tendency of the redistribution variation with the εt level. In addition, all of the numerical data are outside of the code curve, suggesting conservative predictions by ACI 318-19.  Figure 16 shows a comparison of code and numerical predictions regarding the β-ρ b2 relationship for CFRP and steel RC beams. It can be observed that ACI 318-19 can reflect the influence of bar ratio on moment redistribution in RC beams. In addition, according to ACI 318-19, CFRP RC beams generally exhibit lower moment redistribution than steel RC beams; this is especially notable at a low ρ b2 level. This observation is also consistent with the numerical results. Therefore, the effect of bar type on moment redistribution is also reflected in ACI 318-19. From Figures 15 and 16, it is seen that ACI 318-19, which is shown to be conservative, leads to more accurate predictions of moment redistribution in CFRP RC beams than that in steel RC beams.
Materials 2021, 14, x FOR PEER REVIEW 14 of 17 Figure 16 shows a comparison of code and numerical predictions regarding the β-ρb2 relationship for CFRP and steel RC beams. It can be observed that ACI 318-19 can reflect the influence of bar ratio on moment redistribution in RC beams. In addition, according to ACI 318-19, CFRP RC beams generally exhibit lower moment redistribution than steel RC beams; this is especially notable at a low ρb2 level. This observation is also consistent with the numerical results. Therefore, the effect of bar type on moment redistribution is also reflected in ACI 318-19. From Figures 15 and 16, it is seen that ACI 318-19, which is shown to be conservative, leads to more accurate predictions of moment redistribution in CFRP RC beams than that in steel RC beams. Lou et al. [28] proposed a modification of Equation (12) by introducing the parameter ρb2/ρb1 to predict the moment redistribution in steel RC beams:  Figure 17 shows a comparison of the predicted moment redistribution by ACI 318-19 and Lou et al. [28] equations with the numerical results. It is seen in Figure 17a that the equation by Lou et al. [28] shows much more accurate predictions in moment redistribution in steel RC beams than the ACI 318-19 equation. However, at a low ρb2 of 0.61%, the equation by Lou et al. [28] leads to an overestimate when calculating the moment redistribution in steel RC beams. In CFRP RC beams, the equation by Lou et al. [28] is generally Lou et al. [28] proposed a modification of Equation (12) by introducing the parameter ρ b2 /ρ b1 to predict the moment redistribution in steel RC beams: Figure 17 shows a comparison of the predicted moment redistribution by ACI 318-19 and Lou et al. [28] equations with the numerical results. It is seen in Figure 17a that the equation by Lou et al. [28] shows much more accurate predictions in moment redistri-bution in steel RC beams than the ACI 318-19 equation. However, at a low ρ b2 of 0.61%, the equation by Lou et al. [28] leads to an overestimate when calculating the moment redistribution in steel RC beams. In CFRP RC beams, the equation by Lou et al. [28] is generally unsafe, although it leads to accurate predictions for ρ b2 > 1.82%, as illustrated in Figure 17b. According to numerical simulations, the β-ρ b2 relationship for CFRP RC beams is approximately linear. However, according to simplified equations by ACI 318-19 and Lou et al. [28], the relationships exhibit an obviously nonlinear behavior with similar curve shape. This can be explained by the fact that both simplified equations are associated to the parameter ε t , which varies with varying ρ b2 in a nonlinear manner, as demonstrated in Figure 13.

Conclusions
A numerical study was conducted to examine the flexural behavior of two-span continuous CFRP RC beams, and the results were compared to those of steel RC beams. The main conclusions of the study are as follows:  Using CFRP bars instead of steel bars can greatly improve the crack mode of a RC beam with a low bar ratio. While yielding plays a critical role in steel RC beams, the global flexural response (e.g., development of deformations, reactions and moments) of CFRP RC beams is primarily influenced by cracking.  As ρb2 increases, the ultimate load of CFRP RC beams increases slower than that of steel RC beams. CFRP RC beams show 89% higher ultimate load at ρb2 = 0.61% but 7.2% lower ultimate load at ρb2 = 3.03% than steel RC beams. CFRP RC beams exhibit around 13% greater ultimate deflection than steel RC beams.  CFRP RC beams show lower moment redistribution due to the lacking of yielding,

Conclusions
A numerical study was conducted to examine the flexural behavior of two-span continuous CFRP RC beams, and the results were compared to those of steel RC beams. The main conclusions of the study are as follows:

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Using CFRP bars instead of steel bars can greatly improve the crack mode of a RC beam with a low bar ratio. While yielding plays a critical role in steel RC beams, the global flexural response (e.g., development of deformations, reactions and moments) of CFRP RC beams is primarily influenced by cracking.
• As ρ b2 increases, the ultimate load of CFRP RC beams increases slower than that of steel RC beams. CFRP RC beams show 89% higher ultimate load at ρ b2 = 0.61% but 7.2% lower ultimate load at ρ b2 = 3.03% than steel RC beams. CFRP RC beams exhibit around 13% greater ultimate deflection than steel RC beams. • CFRP RC beams show lower moment redistribution due to the lacking of yielding, when compared to steel RC beams. The redistribution difference between CFRP and steel RC beams is notable at a low ρ b2 level and becomes less important with increasing ρ b2 . • ACI 318-19 appears to be conservative and leads to more accurate predictions of moment redistribution in CFRP RC beams than that in steel RC beams. The equation by Lou et al. [28] shows much more accurate predictions of moment redistribution in steel RC beams than ACI 318-19 but is generally unsafe when predicting the moment redistribution in CFRP RC beams. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.