# Moment-Curvature Behaviors of Concrete Beams Singly Reinforced by Steel-FRP Composite Bars

Authors: Zeyang Sun; Yang Yang; Wenlong Yan; Gang Wu; Xiaoyuan He
Journal: Advances in Civil Engineering
 (2017)

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## Abstract

A steel-fiber-reinforced polymer (FRP) composite bar (SFCB) is a kind of rebar with inner steel bar wrapped by FRP, which can achieve a better anticorrosion performance than that of ordinary steel bar. The high ultimate strength of FRP can also provide a significant increase in load bearing capacity. Based on the adequate simulation of the load-displacement behaviors of concrete beams reinforced by SFCBs, a parametric analysis of the moment-curvature behaviors of concrete beams that are singly reinforced by SFCB was conducted. The critical reinforcement ratio for differentiating the beam’s failure mode was presented, and the concept of the maximum possible peak curvature (MPPC) was proposed. After the ultimate curvature reached MPPC, it decreased with an increase in the postyield stiffness ratio (rsf), and the theoretical calculation method about the curvatures before and after the MPPC was derived. The influence of the reinforcement ratio, effective depth, and FRP ultimate strain on the ultimate point was studied by the dimensionless moment and curvature. By calculating the envelope area under the moment-curvature curve, the energy ductility index can obtain a balance between the bearing capacity and the deformation ability. This paper can provide a reference for the design of concrete beams that are reinforced by SFCB or hybrid steel bar/FRP bar.


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## Body

## 1. Introduction

The corrosion of steel bars in concrete structures will demolish the bonding performance and lead to the cracks of concrete due to the volume expansion of corrosion products [1]. Fiber-reinforced polymer (FRP) is a type of composite with superior anticorrosion performance in concrete structures [2, 3]; the successive research and application of glass FRP (GFRP) for the repair of bridges and parking lots prompted the Canadian government to reexamine this type of material to extend the service life of concrete structures. Experimental and theoretical studies on concrete beams that are reinforced by GFRP were conducted by Benmokrane et al. (1996) [4], and the results of these studies indicated that the calculation method of the bearing capacity and the displacement of the American Concrete Institute (ACI) code [5] can be employed for a GFRP beam with minor modifications. For concrete bridges or marine structures in Canada, a GFRP-reinforced structure can maintain superior performance (Mufti et al., 2007) [6]; the adhesions between GFRP and concrete were undamaged in 5~8 years, and the alkaline substance did not penetrate the GFRP reinforcement. The Canadian Highway Bridge Design Code [7] has enabled designers to use GFRP bars as a primary means of reinforcement in concrete structures, but for a prestressed concrete structure, the GFRP bar’s service strength is limited to 25% of its ultimate strength due to its elastic property (brittle failure), and, according to ACI 440.4R-04 [8], GFRPs are not allowed for prestressing application because of their weak creep-rupture characteristics.To improve the performance of an FRP-reinforced concrete structure, a hybrid FRP bar with different ultimate strains was proposed [9] to achieve a certain degree of ductility, but this type of ductility was realized by the partial rupture of FRP with a low elongation rate, which will exhibit significant strength degradation during cyclic loading. Because a steel bar has a large elongation rate (approximately 15%), a better performance and a more acceptable cost can be achieved by combining steel and FRP. Two combinations exist: (1) a concrete beam hybrid reinforced by steel bars and FRP bars; (2) beams reinforced by steel-FRP composite bars. Twelve concrete beams that were reinforced by a steel bar, a GFRP bar, or a steel/GFRP bar were tested by Lau and Pam (2010) [10]; the results indicated that the ductility of hybrid reinforced beams was better than the ductility of a pure FRP-reinforced beam; the minimum reinforcement ratio for an FRP-reinforced beam could be reduced by 25% according to the ACI 440.1R-06 [11]. A study of twelve concrete beams, which were reinforced by steel bar and GFRP bar, was conducted by Safan (2013) [12], and the results showed that the failure modes of concrete beams were presented as concrete crushing after the yielding of the steel bar on the tensile side and a GFRP bar can maintain the flexural bearing capacity of a concrete beam with a relatively small reinforcement ratio. A ductility index was proposed by Pang et al. (2015) [13] to evaluate the load-displacement relationship of a hybrid reinforced concrete beam and the corresponding method of differentiation of the failure mode was suggested, but this method of differentiation was slightly complicated. An experimental study on six hybrid reinforced concrete beams was conducted by El Refai et al. (2015) [14], and the results indicated that the deflection of the hybrid reinforcement beam with the higher reinforcement ratio can be well predicted by current design codes; the crack width was calculated by ACI-440.1R-06 [11] by modifying the bonding coefficient between the FRP bar and concrete.Deterioration failure can occur in concrete beams that are reinforced by hybrid steel bars/FRP bars when a steel bar is subjected to corrosion. A better anticorrosion performance can be achieved by a steel-FRP composite bar (SFCB), which is a type of rebar with an inner steel bar and an outer FRP. Sixteen concrete beams reinforced by hybrid rebar were tested by Nanni et al. (1994) [15], and the theoretical analysis of the load-displacement curves revealed that the calculated displacement was slightly smaller than that of the test curves when the slip between the FRP and concrete was disregarded, and the load-carrying capacity of the hybrid reinforced beams can be calculated using traditional RC theory. Concrete beams that are reinforced by longitudinal steel-GFRP composite bars and GFRP stirrups were tested by Saikia et al. (2005) [16]; the composite bar was composed of an inner steel bar with a diameter of 6 mm that is helically wounded by GFRP with a thickness of 2 mm. Due to an underdeveloped manufacturing technology, a slip between the inner steel bar and the outer GFRP occurred. A factory-produced SFCB was achieved by Wu et al. by modifying the pultrusion process of FRP bars [17], and its characteristics include the following: (1) the initial elastic modulus and the postyield modulus can be designed by adjusting the steel/FRP ratio; (2) a durability of SFCB that is equivalent to the durability of FRP bar can be achieved (Figure 1(a) [18]); (3) the strength of the inner steel bar can be effectively employed, and the high ultimate strength of FRP can be employed as a reservation; (4) the performance/cost ratio of a concrete structure reinforced by steel-FRP composite bars can be optimized with considering the long-term performance and bearing capacity. The typical load-strain curves of steel-basalt FRP composite bars and steel-carbon FRP composite bars are illustrated in Figure 1(b), and the stress-strain model of SFCB has been comprehensively investigated [19]. It was demonstrated that there was no slip between the inner steel and outer FRP during the loading process. An experimental study of a concrete beam that is reinforced with SFCBs, steel bars, and pure FRP bars under static loading was conducted by the author’s group [20], and research on the effect of blast load [21] is to be conducted. In this paper, a parametric analysis of the moment-curvature behavior is conducted based on the simulation of the load-displacement curve of an SFCB-reinforced concrete beam; the corresponding curvature ductility index is also discussed. The research results of this paper can provide a reference for the design of hybrid reinforced concrete beams.Figure 1
Factory production of SFCB and its mechanical properties.
(a)
Steel-FRP composite bars (b)
Load-strain curves


## 2. Simulation of a Concrete Beam Reinforced by SFCB

### 2.1. Material Parameters

Experimental studies of six concrete beams were conducted by Sun et al. (2012) [20], and the SFCB beams exhibit a stable postyield stiffness after the yielding of SFCB’s inner steel bar, and concrete crushed after the rupture of SFCB’s outer FRP. The conventional RC beam had the largest ductility, whereas the ultimate load of the RC beam was approximately 31% of the ultimate load of the SFCB beam. To calculate the moment-curvature behavior of the concrete beam that is reinforced by SFCB, the comparison analysis between the experimental results of the SFCB beam and the calculated curves are conducted.Concrete02 in OpenSees (OS) [22] is adopted for concrete stress-strain behavior (Figure 2). The peak stress and strain of concrete can be expressed by (1) and (2) [23], and the relationship between the crush strain and the volumetric percentage of stirrups is shown in (3):(1)fc0=Kfc′=1+ρsvfyhfc′fc′,(2)εc0=0.002K=0.0021+ρsvfyhfc′,(3)εcu=0.004+ρsvfyhMPa300,where fc′ and εc0 are the peak stress and peak strain, respectively; εcu is the ultimate compressive strain of concrete; ρsv is the volumetric percentage of the stirrups; and fyh is the yield strength of the stirrups.Figure 2
Stress-strain behavior of the concrete model.The load-strain relationship of a SFCB is a trilinear model (Figure1(b)), where the rupture of SFCB’s outer FRP is defined as the failure point. As a result, the trilinear model can be simplified in two stages (see (4)). The first stage is before yielding (EI), and the second stage is between yielding and FRP rupture (EII). (4)fsf=EIεsf0≤εsf≤εsfyfsfy+EIIεsf-εsfyεsfy≤εsf≤εsfu,where fsf and εsf are the stress of SFCB and the strain of SFCB, respectively; EI is the elastic modulus before yielding; EII is the postyield modulus of SFCB; fsfy and εsfy are the yield stress of SFCB and the yield strain of SFCB, respectively; and fsfu and εsfu are the ultimate stress and the ultimate strain of SFCB at the point of FRP rupture, respectively.The mechanical behavior of SFCBs in OS is realized by established separate fibers of steel and FRP. For steel fiber, the key points are the yield point, the hardening point, the hardening slope, and the ultimate point, which can be described by the Chang and Mander (1994) model [24]. For FRP, a linear elastic element is defined to represent the elastic behavior. The postyield stiffness ratio rsf of an SFCB can be defined by (5), and the corresponding equivalent longitudinal reinforcement ratio (ρsfe) with regard to conventional steel reinforced concrete beam is defined by (6):(5)rsf=EfAfEsAs+EfAf=EfAfEsfAsf,(6)ρsfe=EfAfrsfEsAg,where Ef and Af are the elastic modulus of outer FRP and the cross-sectional area of the outer FRP, respectively; Es and As are the elastic modulus of the inner steel bar and the cross-sectional area of the inner steel bar, respectively; Esf and Asf are the elastic modulus of SFCB and the cross-sectional area of SFCB, respectively; and Ag is the total cross-sectional area of the concrete beam.


### 2.2. The Comparison between the Tested and the Calculated Results for Concrete Beams

The specimen details are presented in Figure3: the total length of the beam is 2000 mm, the cross section was 220 mm × 300 mm, and the shear-span ratio was 3. The diameter of the top bars was 12 mm diameter, the diameter of the stirrups was 8 mm diameter, and the average tested compressive strength of six concrete cubes (150 × 150 × 150 mm) was 48.8 MPa [20].Figure 3
Specimen design.The longitudinal reinforcements of the selected concrete beams were steel bar and S10B51, and the corresponding mechanical properties are listed in Table1. The notation “S10B51” indicates that the SFCB is composed of an inner steel bar with a diameter of 10 mm that is longitudinally wrapped by 51 bundles of 4000-tex basalt fibers, where “tex” is the weight (g) of one fiber bundle per kilometer. The elastic modulus of a steel bar is approximately 200 GPa, and that of a basalt fiber is nearly 90 GPa. As a result, the elastic modulus of an SFCB is smaller than that of a steel bar [19].Table 1
Mechanical properties of the selected reinforcements.
Reinforcement type d (mm) E I (GPa) E I I (GPa) f y (MPa) f u (MPa) Elongation rate (%) Steel bar 12 204 / 415 580 14.5 S10B51 18 96.13 29.80 192.27 548.4 2.6Five groups of dial gauges were evenly arranged on the front side of the mid-span, and five strain gauges were also installed on the opposite side with the same height (Figure4(a)) to verify the plane section assumption. The average strain of B-S10B51 that was measured by the dial gauges is presented in Figure 4(b): the horizontal axis represents microstrain and the vertical axis represents the distance from the beam’s bottom surface. The average strain of B-S10B51 along the section height can satisfy the plane section assumption, and the neutral axis was substantially unchanged before cracking and increased rapidly after cracking. Although the composite bar S10B51 had stable postyield stiffness, the neutral axis rapidly increases after the yielding of the inner steel bar to resist the increased moment.Figure 4
Verification of the plane section assumption of B-S10B51.
(a)
Measurement of average strain at mid-span (b)
The strain distribution of B-S10B51With verification of the plain section assumption, the average curvatures of the SFCB-reinforced concrete beam can be calculated by(7)ϕ=εc+εsfh0,where εc (absolute value) is the concrete compressive strain and h0 is the effective height of the beam’s cross section. As a result, MPPC ϕpeak_max of a hybrid reinforced beam can be determined by the FRP rupture strain εFRP, the section effective height h0, and the ultimate compressive strain of concrete εcu.The tested stress-strain curve of the composite bar S10B51 and the corresponding calculated curve by OS are shown in Figure5(a); the calculated curve was consistent with the tested value. As shown in Figure 5(b), the comparison between the calculated results and the tested results for a concrete beam reinforced by SFCB is presented, in which the yield strength and the ultimate strength of S10B51 were taken as 80% of the tested values. The calculated bearing capacity corresponded with the test results, whereas the corresponding ultimate mid-span displacement is approximately 81% of the tested values, which is caused by the large slip between the SFCB and the concrete at mid-span (Figure 6).Figure 5
Comparison between tested results and calculated results of SFCB-reinforced concrete beam.
(a)
Mechanical properties (b)
SFCB-reinforced concrete beamFigure 6
Failure mode of B-S10B51.


### 2.3. Critical Reinforcement Ratio

The calculation method of the RC beam can also be employed for beams that are reinforced by SFCB or hybrid FRP bar/steel bar. The failure mode of SFCB-reinforced concrete beam can be influenced by the yield strain of SFCB, FRP rupture strain, and concrete crush strain. The different ultimate state (strain distribution) of hybrid reinforced concrete beam is shown in Figure7(a), where the tensile capacity of cracked concrete is assumed to be zero. The failure mode can be divided into three cases according to the dominated parameters, εsfu or εcu. When the equivalent reinforcement ratio ρsfe exceeds ρb_Ie, the failure mode (Mode I) is concrete crushed before the tensile reinforcement reached yield strain. When the ultimate state is concrete crushed after SFCB’s inner steel bar yielded (without FRP rupture), the reinforcement ratio ranges between ρb_Ie and ρb_IIe (Mode II). The third failure mode is the SFCB’s outer FRP rupture after the inner steel yielded without concrete crushing (Mode III), and the corresponding ρsfe is smaller than ρb_IIe. When the failure mode includes concrete crushing, the compression stress of concrete can be simplified as a rectangular block (Figure 7(b)), the average stress intensity is expressed as α1fc′, and the compression height of concrete block can be represented as β1xc.Figure 7
Schematic strain distribution and equivalent concrete stress block.
(a)
Strain distribution of different failure modes (b)
Equivalent stress block of internal relationships atϕpeak_maxAccording to the static equilibrium of axial force and the plain section assumption, the critical reinforcement ratios (ρb_Ie and ρb_IIe) of SFCB-reinforced concrete beam can be calculated by equations (8) and (9) [25]:(8)ρb_Ie=α1β1fc′Es1εsfy1+εsfy/εcu,(9)ρb_IIe=α1β1fc′Es1εsfu-εsfyrsf+εsfy1+εsfu/εcu,where α1 and β1 are coefficients for the equivalent stress block of concrete in compression.


## 2.1. Material Parameters

Experimental studies of six concrete beams were conducted by Sun et al. (2012) [20], and the SFCB beams exhibit a stable postyield stiffness after the yielding of SFCB’s inner steel bar, and concrete crushed after the rupture of SFCB’s outer FRP. The conventional RC beam had the largest ductility, whereas the ultimate load of the RC beam was approximately 31% of the ultimate load of the SFCB beam. To calculate the moment-curvature behavior of the concrete beam that is reinforced by SFCB, the comparison analysis between the experimental results of the SFCB beam and the calculated curves are conducted.Concrete02 in OpenSees (OS) [22] is adopted for concrete stress-strain behavior (Figure 2). The peak stress and strain of concrete can be expressed by (1) and (2) [23], and the relationship between the crush strain and the volumetric percentage of stirrups is shown in (3):(1)fc0=Kfc′=1+ρsvfyhfc′fc′,(2)εc0=0.002K=0.0021+ρsvfyhfc′,(3)εcu=0.004+ρsvfyhMPa300,where fc′ and εc0 are the peak stress and peak strain, respectively; εcu is the ultimate compressive strain of concrete; ρsv is the volumetric percentage of the stirrups; and fyh is the yield strength of the stirrups.Figure 2
Stress-strain behavior of the concrete model.The load-strain relationship of a SFCB is a trilinear model (Figure1(b)), where the rupture of SFCB’s outer FRP is defined as the failure point. As a result, the trilinear model can be simplified in two stages (see (4)). The first stage is before yielding (EI), and the second stage is between yielding and FRP rupture (EII). (4)fsf=EIεsf0≤εsf≤εsfyfsfy+EIIεsf-εsfyεsfy≤εsf≤εsfu,where fsf and εsf are the stress of SFCB and the strain of SFCB, respectively; EI is the elastic modulus before yielding; EII is the postyield modulus of SFCB; fsfy and εsfy are the yield stress of SFCB and the yield strain of SFCB, respectively; and fsfu and εsfu are the ultimate stress and the ultimate strain of SFCB at the point of FRP rupture, respectively.The mechanical behavior of SFCBs in OS is realized by established separate fibers of steel and FRP. For steel fiber, the key points are the yield point, the hardening point, the hardening slope, and the ultimate point, which can be described by the Chang and Mander (1994) model [24]. For FRP, a linear elastic element is defined to represent the elastic behavior. The postyield stiffness ratio rsf of an SFCB can be defined by (5), and the corresponding equivalent longitudinal reinforcement ratio (ρsfe) with regard to conventional steel reinforced concrete beam is defined by (6):(5)rsf=EfAfEsAs+EfAf=EfAfEsfAsf,(6)ρsfe=EfAfrsfEsAg,where Ef and Af are the elastic modulus of outer FRP and the cross-sectional area of the outer FRP, respectively; Es and As are the elastic modulus of the inner steel bar and the cross-sectional area of the inner steel bar, respectively; Esf and Asf are the elastic modulus of SFCB and the cross-sectional area of SFCB, respectively; and Ag is the total cross-sectional area of the concrete beam.


## 2.2. The Comparison between the Tested and the Calculated Results for Concrete Beams

The specimen details are presented in Figure3: the total length of the beam is 2000 mm, the cross section was 220 mm × 300 mm, and the shear-span ratio was 3. The diameter of the top bars was 12 mm diameter, the diameter of the stirrups was 8 mm diameter, and the average tested compressive strength of six concrete cubes (150 × 150 × 150 mm) was 48.8 MPa [20].Figure 3
Specimen design.The longitudinal reinforcements of the selected concrete beams were steel bar and S10B51, and the corresponding mechanical properties are listed in Table1. The notation “S10B51” indicates that the SFCB is composed of an inner steel bar with a diameter of 10 mm that is longitudinally wrapped by 51 bundles of 4000-tex basalt fibers, where “tex” is the weight (g) of one fiber bundle per kilometer. The elastic modulus of a steel bar is approximately 200 GPa, and that of a basalt fiber is nearly 90 GPa. As a result, the elastic modulus of an SFCB is smaller than that of a steel bar [19].Table 1
Mechanical properties of the selected reinforcements.
Reinforcement type d (mm) E I (GPa) E I I (GPa) f y (MPa) f u (MPa) Elongation rate (%) Steel bar 12 204 / 415 580 14.5 S10B51 18 96.13 29.80 192.27 548.4 2.6Five groups of dial gauges were evenly arranged on the front side of the mid-span, and five strain gauges were also installed on the opposite side with the same height (Figure4(a)) to verify the plane section assumption. The average strain of B-S10B51 that was measured by the dial gauges is presented in Figure 4(b): the horizontal axis represents microstrain and the vertical axis represents the distance from the beam’s bottom surface. The average strain of B-S10B51 along the section height can satisfy the plane section assumption, and the neutral axis was substantially unchanged before cracking and increased rapidly after cracking. Although the composite bar S10B51 had stable postyield stiffness, the neutral axis rapidly increases after the yielding of the inner steel bar to resist the increased moment.Figure 4
Verification of the plane section assumption of B-S10B51.
(a)
Measurement of average strain at mid-span (b)
The strain distribution of B-S10B51With verification of the plain section assumption, the average curvatures of the SFCB-reinforced concrete beam can be calculated by(7)ϕ=εc+εsfh0,where εc (absolute value) is the concrete compressive strain and h0 is the effective height of the beam’s cross section. As a result, MPPC ϕpeak_max of a hybrid reinforced beam can be determined by the FRP rupture strain εFRP, the section effective height h0, and the ultimate compressive strain of concrete εcu.The tested stress-strain curve of the composite bar S10B51 and the corresponding calculated curve by OS are shown in Figure5(a); the calculated curve was consistent with the tested value. As shown in Figure 5(b), the comparison between the calculated results and the tested results for a concrete beam reinforced by SFCB is presented, in which the yield strength and the ultimate strength of S10B51 were taken as 80% of the tested values. The calculated bearing capacity corresponded with the test results, whereas the corresponding ultimate mid-span displacement is approximately 81% of the tested values, which is caused by the large slip between the SFCB and the concrete at mid-span (Figure 6).Figure 5
Comparison between tested results and calculated results of SFCB-reinforced concrete beam.
(a)
Mechanical properties (b)
SFCB-reinforced concrete beamFigure 6
Failure mode of B-S10B51.


## 2.3. Critical Reinforcement Ratio

The calculation method of the RC beam can also be employed for beams that are reinforced by SFCB or hybrid FRP bar/steel bar. The failure mode of SFCB-reinforced concrete beam can be influenced by the yield strain of SFCB, FRP rupture strain, and concrete crush strain. The different ultimate state (strain distribution) of hybrid reinforced concrete beam is shown in Figure7(a), where the tensile capacity of cracked concrete is assumed to be zero. The failure mode can be divided into three cases according to the dominated parameters, εsfu or εcu. When the equivalent reinforcement ratio ρsfe exceeds ρb_Ie, the failure mode (Mode I) is concrete crushed before the tensile reinforcement reached yield strain. When the ultimate state is concrete crushed after SFCB’s inner steel bar yielded (without FRP rupture), the reinforcement ratio ranges between ρb_Ie and ρb_IIe (Mode II). The third failure mode is the SFCB’s outer FRP rupture after the inner steel yielded without concrete crushing (Mode III), and the corresponding ρsfe is smaller than ρb_IIe. When the failure mode includes concrete crushing, the compression stress of concrete can be simplified as a rectangular block (Figure 7(b)), the average stress intensity is expressed as α1fc′, and the compression height of concrete block can be represented as β1xc.Figure 7
Schematic strain distribution and equivalent concrete stress block.
(a)
Strain distribution of different failure modes (b)
Equivalent stress block of internal relationships atϕpeak_maxAccording to the static equilibrium of axial force and the plain section assumption, the critical reinforcement ratios (ρb_Ie and ρb_IIe) of SFCB-reinforced concrete beam can be calculated by equations (8) and (9) [25]:(8)ρb_Ie=α1β1fc′Es1εsfy1+εsfy/εcu,(9)ρb_IIe=α1β1fc′Es1εsfu-εsfyrsf+εsfy1+εsfu/εcu,where α1 and β1 are coefficients for the equivalent stress block of concrete in compression.


## 3. Parametric Study

### 3.1. Typical Moment-Curvature Curves of the SFCB Beam

According to the design code of AASHTO [26], the selected parameters are as follows: the width of the beam section b = 200 mm; the aspect ratio h/b = 2~4; ρsfe = 0.3%~1.2%; fc′ = 30~90 MPa; the rupture strain of SFCB’s outer FRP εsf = 0.015~0.025, which represent carbon fiber (0.015) to basalt fiber (0.025); and the postyield stiffness ratio of SFCB rsf = 0.001~0.95. A total of 2880 moment-curvature analyses were performed in this paper; typical moment-curvature curves are illustrated in Figure 8 by varying rsf. With an increase in rsf, the yield point remained stable, the postyield stiffness increased, and the peak curvature (ϕpeak) increased and then decreased. Before the ultimate curvature reached ϕpeak_max, the increase of ϕpeak is caused by a decrease in the concrete compression depth, and the failure mode is concrete crushing. As a result, the ultimate curvature is dominated by the tensile strain of SFCB in this failure mode. After the rupture of SFCB’s outer FRP, the moment decreased to the level in which only the inner steel bar worked, and the inner steel bar decreased with an increase in the rsf.Figure 8
Moment-curvature analyses of SFCB beams.


### 3.2. Yield Point

The yield moment increases with an increase inρsfe, whereas the yield curvature slightly decreased. With an increase in the concrete compressive strength, both the yield moment and the yield curvature increased. A semiempirical equation was proposed by Aycardi et al. [27] with consideration of the effect of yield strain and effective depth, as shown in(10)ϕy=1.7fyEsh0=1.7εsfyh0.Because the yield strain and the column depth are fixed parameters, the key parameter is the concrete strain when tensile reinforcement yielded. The fitted yield curvature is presented in (11) with consideration of the reinforcement ratio and concrete strength:(11)ϕy_regressh0=εsfy+0.0396ρsfe+0.00035.The comparison between the fitted results and the yield curvature by OS is shown in Figure 9(a). Aycardi’s empirical equation overestimated the yield curvature, the proposed (11) is consistent with the results of OS, and the coefficient of determination (R2) of the fitted equation is 0.997.Figure 9
Fitted results of yield curvature and moment.
(a)
Curvature (b)
MomentThe fitted yield moment can be obtained based on the fitted yield curvature and the sectional force balance, as shown in(12)My_regressEsbh02=0.00157ρsfe+6.233E-07,where My_regress is the fitted moment. Figure 9(b) presents the comparison between the fitted results and the OS values; R2 is 0.998.


### 3.3. Ultimate Point

The ultimate point of a conventional RC structure was defined by 80% or 85% of the peak load capacity. For SFCB-reinforced concrete structures, the rupture of SFCB’s outer FRP will cause a significant decrease in the load capacity. As a result, the rupture of SFCB’s outer FRP was defined as the ultimate point of SFCB-reinforced concrete beam in this paper.


#### 3.3.1. The Influence of the Reinforcement Ratio

Before the section curvature reachedϕpeak_max, both the ultimate moment and curvature increase with an increase in ρsfe (Figure 10(a)). With an increase in rsf, the section ultimate curvature with a larger ρsfe increases faster than the section ultimate curvature with a smaller ρsfe. When the section ultimate curvature reached ϕpeak_max, the larger ρsfe is, the smaller the corresponding rsf is. As illustrated in Figure 10(b), when the ρsfe increased from 0.3% to 0.9%, ϕpeak_max decreased from 0.65 rad/m to 0.15 rad/m (reduced by 77%). After the curvature reached ϕpeak_max, with an increase in rsf, the section moment continued to increase while the curvature significantly decreased, which was caused by a decrease in the tensile strain of SFCB at the ultimate point.Figure 10
Influence of the reinforcement ratio.
(a)
r s f– moment (b)
r s f– curvature (c)
r s f– dimensionless moment (d)
r s f– dimensionless curvatureThe dimensionless ultimate moment and curvature are shown in Figures10(c) and 10(d). Before reaching the critical point (ϕpeak_max), the slopes of the dimensionless moments almost overlap. When rsf was fixed as 0.1, the ultimate curvatures were approximately ten times the yield curvature when other parameters changed. Before the ultimate curvature reached ϕpeak_max, with an increase in rsf, both the slope and the dimensionless value of the section moment with larger ρsfe decreased. The trend of the dimensionless curvature (Figure 10(d)) was similar to the trend of the original curvature (Figure 10(b)). As a result, only the dimensionless curvature was discussed when changing other parameters.


#### 3.3.2. The Influence of SFCB’s Rupture Strain

The rupture strain of SFCB’s outer FRP has no effect on the yield curvature or moment. The dimensionless moments and curvatures of the SFCB beam section by changing the rupture strain of SFCB were presented in Figure11; the failure modes are Mode II and Mode III. When failure Mode II occurred, the ultimate point was determined by the concrete crush strain. As a result, the ultimate moment and curvature with different SFCB rupture strains were equivalent. In Mode III, the larger the SFCB’s rupture strain was, the larger the ultimate moment and curvature were. With an increase in rsf, the ultimate curvature increased at a faster rate than the rate of increase in the ultimate moment, which was caused by the nonlinear property of concrete stress-strain in the compression zone.Figure 11
Influence of SFCB’s rupture strain.
(a)
r s f– dimensionless moment (b)
r s f– dimensionless curvature


#### 3.3.3. The Influence of Effective Depth

When the effective depth (h0) increases and other parameters remain unchanged, both the ultimate moment and the ultimate curvature will increase with an increase in h0. However, rsf (approximately to 0.25) does not change when the ultimate point reached critical point ϕpeak_max. The dimensionless moment and curvature are also listed in Figure 12, and the dimensionless curves almost overlap.Figure 12
Influence of effective depth.
(a)
r s f–moment (b)
r s f–curvature


### 3.4. Curvature before MPPC

The failure is determined by the rupture strain of SFCB’s outer FRP in failure Mode III, and the concrete in the compression zone kept undamaged. Based on the force equilibrium and elastic assumption, the concrete strain can be obtained as shown in(13)εc=ϕuh0-εsfu=γ1+γ12+4γ1εsfu2,where γ1 can be expressed as(14)γ1=2EsfeEcρsfeεy+rsfεsfu-εy.The comparison between the calculated ultimate curvature and the corresponding OS curvature is shown in Figure13.Figure 13
Comparison between the calculated curvatures and the corresponding OS results in failure Mode III.
(a)
All data (b)
Different reinforcement ratio (c)
Different SFCB’s rupture strain (d)
Differentfc′The calculated curvatures correspond with the corresponding OS results whenrsf is relatively small. The calculated curvatures became smaller than the calculated curvatures in OS when rsf is large (Figure 13(a)). The error was caused by the elastic assumption of compressive concrete, which enlarges the contribution of compressive concrete, and, therefore, the calculated ultimate curvature was underestimated when rsf is relatively large. The compressive strain of concrete will increase with an increase in the reinforcement ratio (Figure 13(b)), an increase in SFCB’s rupture strain (Figure 13(c)), or a decrease in fc′ (Figure 13(d)), which will produce a larger error between the calculated values and OS values; the maximum error is approximately 30%.


### 3.5. The Maximum Possible Curvature (MPPC)

A total of 98 sets of ultimate points were observed around the maximum possible peak curvature. The comparison between the calculatedϕpeak_max using (7) and the corresponding curvature in OS is shown in Figure 14(a), where the former was generally larger than the latter. The maximum error was approximately 20%, which was primarily caused by the error in concrete compressive strain. The fitted ϕpeak_max is listed in (15), and the corresponding comparison between the fitted value and the OS results is presented in Figure 14(a), which indicates that (15) can yield a better prediction of ϕpeak_max:(15)ϕpeak_max=εsfu+0.985εcu-0.0045h0.Figure 14
Maximum possible peak curvature and critical reinforcement ratio.
(a)
Comparison between the calculatedϕpeak_max and the corresponding OS results (b)
Comparison of the critical reinforcement ratioρsfeWhen the section curvature reachedϕpeak_max, the corresponding reinforcement ratio ρsfe is ρb_IIe (Figure 7(a)). The calculated ρb_IIe by (9) was larger than ρb_IIe in OS; the maximum error was approximately 40% (Figure 14(b)). Equation (16) is the fitted critical reinforcement ratio (ρb_II_regresse) by the regression in concrete compressive strain: (16)ρb_II_regresse=fc′Es1+εsfu/εcuεssfu-εsfy×11.379εssfy/εssfu-εsfy+1.278rsf-0.003.The comparison betweenρb_II_regresse and the reinforcement ratio in OS is illustrated in Figure 14(b). The error was ±10 percent, which indicates that (16) can be used to predict the failure mode of a designed concrete beam that is reinforced by SFCB.


### 3.6. Curvature after MPPC

After the ultimate curvature reachedϕpeak_max, the ultimate curvature will decrease with an increase in rsf, and the corresponding failure mode is concrete crushing (Mode III). Based on the force equilibrium and the simplified compression block, the tensile strain of SFCB can be calculated by(17)εsf=-γ2+γ22-8rsfγ32rsf,where γ2 and γ3 can be expressed by (18) and (19), respectively:(18)γ2=εy1-rsf+εcursf,(19)γ3=εcuεy1-rsf-fcεcuEsfeρsfe.The comparison between the calculated curvatures by (17) and the OS values is presented in Figure 15. When the curvature ductility is relatively large, the calculated curvature was similar to the OS results. When the curvature ductility is relatively small, the calculated curvature may be approximately 1.35 times the calculated curvature of the OS values, and ϕcalculated/ϕpeak_max_OS is approximately a power function of the curvature ductility (Figure 15).Figure 15
Comparison between the calculated curvatures and the OS values in Mode II.


## 3.1. Typical Moment-Curvature Curves of the SFCB Beam

According to the design code of AASHTO [26], the selected parameters are as follows: the width of the beam section b = 200 mm; the aspect ratio h/b = 2~4; ρsfe = 0.3%~1.2%; fc′ = 30~90 MPa; the rupture strain of SFCB’s outer FRP εsf = 0.015~0.025, which represent carbon fiber (0.015) to basalt fiber (0.025); and the postyield stiffness ratio of SFCB rsf = 0.001~0.95. A total of 2880 moment-curvature analyses were performed in this paper; typical moment-curvature curves are illustrated in Figure 8 by varying rsf. With an increase in rsf, the yield point remained stable, the postyield stiffness increased, and the peak curvature (ϕpeak) increased and then decreased. Before the ultimate curvature reached ϕpeak_max, the increase of ϕpeak is caused by a decrease in the concrete compression depth, and the failure mode is concrete crushing. As a result, the ultimate curvature is dominated by the tensile strain of SFCB in this failure mode. After the rupture of SFCB’s outer FRP, the moment decreased to the level in which only the inner steel bar worked, and the inner steel bar decreased with an increase in the rsf.Figure 8
Moment-curvature analyses of SFCB beams.


## 3.2. Yield Point

The yield moment increases with an increase inρsfe, whereas the yield curvature slightly decreased. With an increase in the concrete compressive strength, both the yield moment and the yield curvature increased. A semiempirical equation was proposed by Aycardi et al. [27] with consideration of the effect of yield strain and effective depth, as shown in(10)ϕy=1.7fyEsh0=1.7εsfyh0.Because the yield strain and the column depth are fixed parameters, the key parameter is the concrete strain when tensile reinforcement yielded. The fitted yield curvature is presented in (11) with consideration of the reinforcement ratio and concrete strength:(11)ϕy_regressh0=εsfy+0.0396ρsfe+0.00035.The comparison between the fitted results and the yield curvature by OS is shown in Figure 9(a). Aycardi’s empirical equation overestimated the yield curvature, the proposed (11) is consistent with the results of OS, and the coefficient of determination (R2) of the fitted equation is 0.997.Figure 9
Fitted results of yield curvature and moment.
(a)
Curvature (b)
MomentThe fitted yield moment can be obtained based on the fitted yield curvature and the sectional force balance, as shown in(12)My_regressEsbh02=0.00157ρsfe+6.233E-07,where My_regress is the fitted moment. Figure 9(b) presents the comparison between the fitted results and the OS values; R2 is 0.998.


## 3.3. Ultimate Point

The ultimate point of a conventional RC structure was defined by 80% or 85% of the peak load capacity. For SFCB-reinforced concrete structures, the rupture of SFCB’s outer FRP will cause a significant decrease in the load capacity. As a result, the rupture of SFCB’s outer FRP was defined as the ultimate point of SFCB-reinforced concrete beam in this paper.


### 3.3.1. The Influence of the Reinforcement Ratio

Before the section curvature reachedϕpeak_max, both the ultimate moment and curvature increase with an increase in ρsfe (Figure 10(a)). With an increase in rsf, the section ultimate curvature with a larger ρsfe increases faster than the section ultimate curvature with a smaller ρsfe. When the section ultimate curvature reached ϕpeak_max, the larger ρsfe is, the smaller the corresponding rsf is. As illustrated in Figure 10(b), when the ρsfe increased from 0.3% to 0.9%, ϕpeak_max decreased from 0.65 rad/m to 0.15 rad/m (reduced by 77%). After the curvature reached ϕpeak_max, with an increase in rsf, the section moment continued to increase while the curvature significantly decreased, which was caused by a decrease in the tensile strain of SFCB at the ultimate point.Figure 10
Influence of the reinforcement ratio.
(a)
r s f– moment (b)
r s f– curvature (c)
r s f– dimensionless moment (d)
r s f– dimensionless curvatureThe dimensionless ultimate moment and curvature are shown in Figures10(c) and 10(d). Before reaching the critical point (ϕpeak_max), the slopes of the dimensionless moments almost overlap. When rsf was fixed as 0.1, the ultimate curvatures were approximately ten times the yield curvature when other parameters changed. Before the ultimate curvature reached ϕpeak_max, with an increase in rsf, both the slope and the dimensionless value of the section moment with larger ρsfe decreased. The trend of the dimensionless curvature (Figure 10(d)) was similar to the trend of the original curvature (Figure 10(b)). As a result, only the dimensionless curvature was discussed when changing other parameters.


### 3.3.2. The Influence of SFCB’s Rupture Strain

The rupture strain of SFCB’s outer FRP has no effect on the yield curvature or moment. The dimensionless moments and curvatures of the SFCB beam section by changing the rupture strain of SFCB were presented in Figure11; the failure modes are Mode II and Mode III. When failure Mode II occurred, the ultimate point was determined by the concrete crush strain. As a result, the ultimate moment and curvature with different SFCB rupture strains were equivalent. In Mode III, the larger the SFCB’s rupture strain was, the larger the ultimate moment and curvature were. With an increase in rsf, the ultimate curvature increased at a faster rate than the rate of increase in the ultimate moment, which was caused by the nonlinear property of concrete stress-strain in the compression zone.Figure 11
Influence of SFCB’s rupture strain.
(a)
r s f– dimensionless moment (b)
r s f– dimensionless curvature


### 3.3.3. The Influence of Effective Depth

When the effective depth (h0) increases and other parameters remain unchanged, both the ultimate moment and the ultimate curvature will increase with an increase in h0. However, rsf (approximately to 0.25) does not change when the ultimate point reached critical point ϕpeak_max. The dimensionless moment and curvature are also listed in Figure 12, and the dimensionless curves almost overlap.Figure 12
Influence of effective depth.
(a)
r s f–moment (b)
r s f–curvature


## 3.3.1. The Influence of the Reinforcement Ratio

Before the section curvature reachedϕpeak_max, both the ultimate moment and curvature increase with an increase in ρsfe (Figure 10(a)). With an increase in rsf, the section ultimate curvature with a larger ρsfe increases faster than the section ultimate curvature with a smaller ρsfe. When the section ultimate curvature reached ϕpeak_max, the larger ρsfe is, the smaller the corresponding rsf is. As illustrated in Figure 10(b), when the ρsfe increased from 0.3% to 0.9%, ϕpeak_max decreased from 0.65 rad/m to 0.15 rad/m (reduced by 77%). After the curvature reached ϕpeak_max, with an increase in rsf, the section moment continued to increase while the curvature significantly decreased, which was caused by a decrease in the tensile strain of SFCB at the ultimate point.Figure 10
Influence of the reinforcement ratio.
(a)
r s f– moment (b)
r s f– curvature (c)
r s f– dimensionless moment (d)
r s f– dimensionless curvatureThe dimensionless ultimate moment and curvature are shown in Figures10(c) and 10(d). Before reaching the critical point (ϕpeak_max), the slopes of the dimensionless moments almost overlap. When rsf was fixed as 0.1, the ultimate curvatures were approximately ten times the yield curvature when other parameters changed. Before the ultimate curvature reached ϕpeak_max, with an increase in rsf, both the slope and the dimensionless value of the section moment with larger ρsfe decreased. The trend of the dimensionless curvature (Figure 10(d)) was similar to the trend of the original curvature (Figure 10(b)). As a result, only the dimensionless curvature was discussed when changing other parameters.


## 3.3.2. The Influence of SFCB’s Rupture Strain

The rupture strain of SFCB’s outer FRP has no effect on the yield curvature or moment. The dimensionless moments and curvatures of the SFCB beam section by changing the rupture strain of SFCB were presented in Figure11; the failure modes are Mode II and Mode III. When failure Mode II occurred, the ultimate point was determined by the concrete crush strain. As a result, the ultimate moment and curvature with different SFCB rupture strains were equivalent. In Mode III, the larger the SFCB’s rupture strain was, the larger the ultimate moment and curvature were. With an increase in rsf, the ultimate curvature increased at a faster rate than the rate of increase in the ultimate moment, which was caused by the nonlinear property of concrete stress-strain in the compression zone.Figure 11
Influence of SFCB’s rupture strain.
(a)
r s f– dimensionless moment (b)
r s f– dimensionless curvature


## 3.3.3. The Influence of Effective Depth

When the effective depth (h0) increases and other parameters remain unchanged, both the ultimate moment and the ultimate curvature will increase with an increase in h0. However, rsf (approximately to 0.25) does not change when the ultimate point reached critical point ϕpeak_max. The dimensionless moment and curvature are also listed in Figure 12, and the dimensionless curves almost overlap.Figure 12
Influence of effective depth.
(a)
r s f–moment (b)
r s f–curvature


## 3.4. Curvature before MPPC

The failure is determined by the rupture strain of SFCB’s outer FRP in failure Mode III, and the concrete in the compression zone kept undamaged. Based on the force equilibrium and elastic assumption, the concrete strain can be obtained as shown in(13)εc=ϕuh0-εsfu=γ1+γ12+4γ1εsfu2,where γ1 can be expressed as(14)γ1=2EsfeEcρsfeεy+rsfεsfu-εy.The comparison between the calculated ultimate curvature and the corresponding OS curvature is shown in Figure13.Figure 13
Comparison between the calculated curvatures and the corresponding OS results in failure Mode III.
(a)
All data (b)
Different reinforcement ratio (c)
Different SFCB’s rupture strain (d)
Differentfc′The calculated curvatures correspond with the corresponding OS results whenrsf is relatively small. The calculated curvatures became smaller than the calculated curvatures in OS when rsf is large (Figure 13(a)). The error was caused by the elastic assumption of compressive concrete, which enlarges the contribution of compressive concrete, and, therefore, the calculated ultimate curvature was underestimated when rsf is relatively large. The compressive strain of concrete will increase with an increase in the reinforcement ratio (Figure 13(b)), an increase in SFCB’s rupture strain (Figure 13(c)), or a decrease in fc′ (Figure 13(d)), which will produce a larger error between the calculated values and OS values; the maximum error is approximately 30%.


## 3.5. The Maximum Possible Curvature (MPPC)

A total of 98 sets of ultimate points were observed around the maximum possible peak curvature. The comparison between the calculatedϕpeak_max using (7) and the corresponding curvature in OS is shown in Figure 14(a), where the former was generally larger than the latter. The maximum error was approximately 20%, which was primarily caused by the error in concrete compressive strain. The fitted ϕpeak_max is listed in (15), and the corresponding comparison between the fitted value and the OS results is presented in Figure 14(a), which indicates that (15) can yield a better prediction of ϕpeak_max:(15)ϕpeak_max=εsfu+0.985εcu-0.0045h0.Figure 14
Maximum possible peak curvature and critical reinforcement ratio.
(a)
Comparison between the calculatedϕpeak_max and the corresponding OS results (b)
Comparison of the critical reinforcement ratioρsfeWhen the section curvature reachedϕpeak_max, the corresponding reinforcement ratio ρsfe is ρb_IIe (Figure 7(a)). The calculated ρb_IIe by (9) was larger than ρb_IIe in OS; the maximum error was approximately 40% (Figure 14(b)). Equation (16) is the fitted critical reinforcement ratio (ρb_II_regresse) by the regression in concrete compressive strain: (16)ρb_II_regresse=fc′Es1+εsfu/εcuεssfu-εsfy×11.379εssfy/εssfu-εsfy+1.278rsf-0.003.The comparison betweenρb_II_regresse and the reinforcement ratio in OS is illustrated in Figure 14(b). The error was ±10 percent, which indicates that (16) can be used to predict the failure mode of a designed concrete beam that is reinforced by SFCB.


## 3.6. Curvature after MPPC

After the ultimate curvature reachedϕpeak_max, the ultimate curvature will decrease with an increase in rsf, and the corresponding failure mode is concrete crushing (Mode III). Based on the force equilibrium and the simplified compression block, the tensile strain of SFCB can be calculated by(17)εsf=-γ2+γ22-8rsfγ32rsf,where γ2 and γ3 can be expressed by (18) and (19), respectively:(18)γ2=εy1-rsf+εcursf,(19)γ3=εcuεy1-rsf-fcεcuEsfeρsfe.The comparison between the calculated curvatures by (17) and the OS values is presented in Figure 15. When the curvature ductility is relatively large, the calculated curvature was similar to the OS results. When the curvature ductility is relatively small, the calculated curvature may be approximately 1.35 times the calculated curvature of the OS values, and ϕcalculated/ϕpeak_max_OS is approximately a power function of the curvature ductility (Figure 15).Figure 15
Comparison between the calculated curvatures and the OS values in Mode II.


## 4. Discussion of Curvature Ductility

Many indexes were proposed for the performance evaluation of a conventional RC structure; ductility is a main index but the ductility substantially varied because the yield curvature (displacement) that was obtained using different methods significantly varied [28, 29]. The yield point can be determined by the graphing method and the equal energy method, as shown in Figure 16. For a SFCB beam, the yield curvature is determined by the yield of the inner steel bar, and the ultimate curvature is defined by the rupture of SFCB’s outer FRP or the crushing of concrete in the compression zone.Figure 16
Different methods for defining ductility.
(a)
Different methods to determine yield point and ultimate point (b)
Ductility defined by energyThe differences among different ductility factors are calculated using (20) to (22), where μ1 is the ratio between the yield curvature and ultimate curvature (see (20)), μ2 is a ductility index that considers the effect of the ultimate moment (see (21)), and μ3 is the ratio of the total envelope area and the area before yielding (see (22)): (20)μ1=ϕuϕy,(21)μ2=MuϕuMyϕy,(22)μ3=EtotEy=ϕuϕy-MuMy+MuϕuMyϕy.The development trend ofμ1, μ2, and μ3 is presented in Figure 17(a): when the failure mode was SFCB’s outer FRP rupture, the three ductility indexes increased with an increase in rsf, and μ3 has the largest absolute value. When the failure mode was concrete crushing, the three ductility indexes decreased with an increase in rsf; the largest ductility was observed at the point of the MPPC. Figure 17(b) presents the development trend of “dimensionless” ductility (μ/μrsf=0.05): compared with the slopes before MPPC, the decreasing slope of “dimensionless” μ1 was the largest, and the decreasing slope of “dimensionless” μ2 was comparatively flat. For “dimensionless” μ3, both the ultimate moment and the ultimate curvature were considered, and the decreasing slope of “dimensionless” μ3 ranged between “dimensionless” μ1 and “dimensionless” μ2.Figure 17
Comparison ofμ1, μ2, and μ3.
(a)
Differences amongμ1, μ2, and μ3 (b)
“Dimensionless” ductilityThe influence ofρsfe, εsfu, fc′, and h0 on μ3 is shown in Figure 18. When the failure mode is the rupture of SFCB’s outer FRP (Mode III), ρsfe, fc′, and h0 have a minimal influence on μ3. By changing the rupture strain of SFCB εsfu, the slope of μ3 increases with an increase in rsf, which is caused by the increased ultimate moment and ultimate curvature. When the failure mode was concrete crushing after the yielding of SFCB’s inner steel (Mode II), the ductility varies within a relatively small range, μ3 decreases with an increase in rsf, and fc′ and εsfu had minimal influence on the softening branch. The “dimensionless” μ3 (γμ=μ3/μ3_rsf=0.05) was not affected by changes in h0 (Figure 18(d)), and μ3 was approximately 4.3 times μ1 when the MPPC was attained.Figure 18
Comparison ofμ3 with different parameters.
(a)
Change reinforcement ratio (b)
Change SFCB rupture strain (c)
Changefc′ (d)
Changeh0


## 5. Conclusions

Based on the simulation of the test results of a concrete beam reinforced by SFCB, the parametric analysis of moment-curvature behavior of singly reinforced concrete beams was conducted according to AASHTO design code. The main conclusions are as follows:(1)
The failure modes of a concrete beam reinforced by SFCB including concrete crushing before the tensile reinforcement reached yield strain (Mode I), concrete crushed after SFCB’s inner steel bar yielded (without outer FRP fiber ruptured, Mode II), and the SFCB’s outer FRP ruptured after the inner steel yielded without concrete crushing (Mode III). In addition, the ultimate curvature increases with an increase inrsf in failure Mode III, whereas the ultimate curvature decreases with an increase in rsf in failure Mode II.(2)
When the rupture of SFCB’s outer FRP accompanied the crushing of concrete in the compression zone, the section reached the MPPC and the corresponding parameters were critical for the ductility of a concrete beam. Before reaching the MPPC, the calculated ultimate curvature was consistent with the OpenSees (OS) results whenrsf was relatively small, and the ultimate curvature was underestimated with an increase in rsf. After reaching the MPPC, the calculated curvature corresponds with the OS results when the curvature ductility is large, and the curvature will be overestimated when the curvature ductility is comparatively small.(3)
The curvature ductility with consideration of the envelope area can reflect an increase in the ultimate moment whenrsf is large, and the sectional curvature ductility can be justified by changing rsf, εsfu, ρsfe, fc′, and h0. Before reaching the MPPC, ρsfe, fc′, and h0 have a minimal influence on the “dimensionless” μ3. After reaching the MPPC, μ3 slightly decreased with an increase in rsf. For the design of SFCB-reinforced concrete beams, Mode II is preferred without SFCB’s outer FRP rupture, and the bearing capacity and ductility of concrete beams are considered synthetically.



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*Source: 1309629-2017-02-15.xml*