Flexural Behavior of Two-Span Continuous CFRP RC Beams

Abstract

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 ρ_b2 = 0.61–3.03% and ρ_b1/ρ_b2 = 1.5, where ρ_b1 and ρ_b2 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 ρ_b2 = 0.61% but 7.2% lower at ρ_b2 = 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 ρ_b2 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.

1. 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 non-magnetism [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 beams, and they also examined various design codes that adopted the neutral axis depth for quantifying the allowable moment redistribution. They concluded that these codes were unsafe in the calculation of moment redistribution in FRP RC beams but the neglect of moment redistribution was over-conservative. In an experimental study, Ashour and Habeeb [24] tested three two-span and two simply supported CFRP RC beams as well as 1 two-span steel RC control beam. The main test variable was the area of CFRP bars. They found that an increase in bottom CFRP bar area resulted in an enhancement in the ultimate load of the beams while the top CFRP bars had negligible influence on the load-carrying capacity of continuous beams. Abushanab et al. [25] tested a total of 10 two-span continuous RC beams with either basalt FRP (BFRP) or steel bars. Their tests showed that BFRP RC beams exhibited larger cracking width, deflection and strain than steel RC beams. In addition, moment redistribution in the beams was considerably improved by increasing the sagging-to-hogging bar ratio or decreasing the stirrup spacing, while the influence of bar ratio was more effective than the influence of stirrup spacing. Basa et al. [26] presented the test results of six GFRP RC continuous beams with the investigated parameters including the sagging-to-hogging bar ratio and the type of GFRP bars. They concluded that in spite of the brittleness of GFRP bars, continuous beams reinforced with GFRP bars exhibited a certain level of ductility, offering sufficient warning prior to collapse.

Although extensive studies have been conducted, behavior of continuous FRP RC beams has not yet been fully addressed. For example, moment redistribution is closely related to the strain in tensile bars, but their relationship in continuous FRP RC beams is yet to be revealed. ACI 318-19 [27] adopted the strain in tensile bars as a single parameter for quantifying the moment redistribution in continuous steel RC beams. Lou et al. [28] proposed a modified ACI 318-19 equation by introducing a new parameter so as to account for the effect of relative stiffness between critical positive and negative moment sections on moment redistribution in continuous steel RC beams. However, the applicability of these approaches to continuous FRP RC beams is yet to be evaluated.

This paper describes an investigation into the feasibility of replacing steel bars with CFRP bars in continuous RC beams. A numerical model developed by Lou et al. [29] is applied. Model predictions are compared with experimental results of CFRP/steel RC continuous beam specimens available in the technical literature. Numerical tests are then performed on two-span continuous CFRP/steel RC beams. Particular attention is placed on the moment redistribution against the strain in tensile bars. Available models for quantifying the moment redistribution based on the bar strain in continuous RC beams are also evaluated.

2. Materials and Methods

2.1. Materials

The stress–strain relationship for concrete in compression recommended by Eurocode 2 [30] is expressed by the following:

$$\frac{{\sigma }_c}{f_{ck}+8}=\frac{k({\epsilon }_c/{\epsilon }_0)-{({\epsilon }_c/{\epsilon }_0)}^2}{1+(k-2)({\epsilon }_c/{\epsilon }_0)}$$

(1)

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; ε₀ 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.

2.2. 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:

$$w=N_1{w}_1+N_2{w}_2,\theta =N_1{\theta }_1+N_2{\theta }_2$$

(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:

$$\kappa =-\frac{d\theta }{dx}={\mathit{B}}_b{\mathit{u}}^e,\gamma =\frac{dw}{dx}-\theta ={\mathit{B}}_s{\mathit{u}}^e$$

(3)

where

$${\mathit{u}}^e={\left\{\begin{array}{cccc}{w}_1& {\theta }_1& w_2& {\theta }_2\end{array}\right\}}^T$$

(4)

$${\mathit{B}}_b=\left[\begin{array}{cccc}0& -\frac{d{N}_1}{dx}& 0& -\frac{d{N}_2}{dx}\end{array}\right],{\mathit{B}}_s=\left[\begin{array}{cccc}\frac{d{N}_1}{dx}& -N_1& \frac{d{N}_2}{dx}& -N_2\end{array}\right]$$

(5)

Bending moment, M, and shear force, Q, are then expressed by the following:

$$M=(EI){\mathit{B}}_b{\mathit{u}}^e,Q=(GA/k_s){\mathit{B}}_s{\mathit{u}}^e$$

(6)

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:

$${\mathit{P}}^e={\displaystyle {\int }_l{\mathit{B}}_b^{T}Mdx}+{\displaystyle {\int }_l{\mathit{B}}_s^{T}Qdx}$$

(7)

where

$${\mathit{P}}^e={\left\{\begin{array}{cccc}{Q}_1& M_1& Q_2& M_2\end{array}\right\}}^T$$

(8)

Substituting Equation (6) into Equation (7) yields the following element stiffness equations:

$${\mathit{P}}^e={\mathit{K}}^e{\mathit{u}}^e=({\mathit{K}}_b^e+{\mathit{K}}_s^e){\mathit{u}}^e$$

(9)

where

$${\mathit{K}}_b^e={\displaystyle {\int }_l{\mathit{B}}_b^T(EI){\mathit{B}}_{b}dx},{\mathit{K}}_s^e={\displaystyle {\int }_l{\mathit{B}}_s^T(GA/k_s){\mathit{B}}_{s}dx}$$

(10)

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.

3. 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 in both the top and bottom layers of the beam. Both specimens were provided with shear reinforcement consisting of steel bars that are 8 mm diameter, with a spacing of 140 mm. The rupture strength, rupture strain and elastic modulus of CFRP bars were 1061 MPa, 0.53% and 200 GPa, respectively. The yield strength and elastic modulus of steel bars were 510.8 MPa and 200 GPa, respectively. The value of f_ck was 28 MPa for Beam C-C-5 and 26.3 MPa for Beam S-C-6.

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).

4. 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 ρ_b = A_b/(bd_b), 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.

4.1. 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.

4.2. Moment–Curvature Behavior

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.

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. Values of typical response characteristics in failure.

Bar	ρb2 (%)	P (kN)	δ (mm)	κ (10−6 rad/mm)		M (kN·m)		εt (‰)		β (%)
				Mid	Sup	Mid	Sup	Mid	Sup	Mid	Sup
Steel	0.61	408.64	39.94	39.65	−40.19	460.69	−345.05	18.33	18.92	−15.85	26.76
	1.21	775.98	28.95	23.25	−27.08	862.60	−643.24	9.33	12.11	−15.05	25.97
	1.82	1066.75	24.59	12.68	−20.84	1161.10	−918.56	3.82	8.14	−12.63	22.09
	2.42	1229.71	21.54	9.39	−15.49	1289.30	−1151.03	2.31	5.02	−8.30	14.65
	3.03	1351.61	20.14	9.05	−12.51	1376.39	−1342.54	2.01	3.50	−5.01	8.92
CFRP	0.61	772.69	44.42	19.60	−25.39	790.22	−778.15	7.68	10.54	−6.62	11.20
	1.21	973.99	33.01	14.99	−18.82	988.29	−985.88	4.94	6.89	−5.67	9.71
	1.82	1060.95	27.03	12.16	−15.54	1074.22	−1074.91	3.71	5.07	−5.19	8.98
	2.42	1171.36	24.91	11.30	−14.21	1182.73	−1189.13	3.14	4.36	−4.71	8.21
	3.03	1254.32	23.43	10.75	−13.33	1262.98	−1277.50	2.73	3.83	−4.24	7.44

Note: Mid = midspan; Sup = center support.

. 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 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 apparent decrease in stress in CFRP bars. Consequently, CFRP bars may result in lower moments than steel bars at a high ρ_b2 level.

4.3. Load–Deflection Behavior

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 post-cracking 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 load–deflection response, while the second yielding at midspan leads to a remarkable reduction in structural stiffness.

The variation in the ultimate load (P) and deflection (δ) with varying ρ_b2 is shown in Figure 9 and Figure 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.

4.4. Reaction and Moment Development

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.

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.

4.5. Moment Redistribution against Bar Strain

Moment redistribution is quantified by the following:

$$\beta =1-M/M_e$$

(11)

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 (ε_t) and moment redistribution (β) in failure with the ρ_b2 level is displayed in Figure 13 and Figure 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 midspan-to-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%).

5. 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:

$$\beta (\%)=1000{\epsilon }_t$$

(12)

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.

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 Figure 15 and Figure 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 ρ_b2/ρ_b1 to predict the moment redistribution in steel RC beams:

$$\beta (\%)=\lambda (1000{\epsilon }_t)$$

(13)

$$\lambda =0.68-4.21\ln ({\rho }_{b2}/{\rho }_{b1})-2.05{\ln }^2({\rho }_{b2}/{\rho }_{b1})$$

(14)

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 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.

6. 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 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.