# Calculation on Bending Stiffness of RC Short Beam Strengthened by CFRP

Authors: Wang Tingyan; Zhou Yun; Zhang Junwei
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Based on the bending tests of seven reinforced concrete (RC) short beams strengthened with carbon fiber reinforced polymer (CFRP), the bending stiffness curves of the whole process of the short beams strengthened with CFRP were obtained. The variation law of bending stiffness curve of short beam in the whole loading process was analyzed. Based on the reasonable calculation assumption, the calculation method of flexural rigidity of short reinforced concrete beams strengthened with CFRP sheets in the whole loading process was put forward. The comparison between the calculated value and the test value of bending stiffness showed that the calculation method of bending stiffness was reasonable and had high calculation accuracy. This calculation method can be used not only in the calculation of flexural rigidity of short reinforced concrete beams strengthened with CFRP sheets but also in the calculation of flexural rigidity of ordinary short reinforced concrete beams. The calculation method in this paper can provide a theoretical basis for the deformation calculation of reinforced concrete short beams strengthened with CFRP sheets.


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## Body

## 1. Introduction

In China, a large number of structural engineering has been damaged or destroyed due to the damage of natural disasters such as earthquake, the increase of service load, the long construction time, or the lower original design standard. It is necessary to reinforce and reconstruct the existing engineering structure. In the existing engineering structure, there is a kind of simply supported beam whose span height ratio is between 2 and 5, which is called short beam. Short beam is a common horizontal component in the existing engineering structure. It is not only widely used in construction engineering but also widely used in hydraulic engineering, port engineering, railway, highway, municipal engineering, and other fields [1]. Because of the small span height ratio, the short beam has a large bearing capacity. The failure of reinforced concrete short beam belongs to bad brittle failure. In order to improve the ductility and bearing capacity of reinforced concrete short beams, it can be realized by adding a certain amount of short fiber or by fiber reinforced polymer (FRP) reinforcement. Steel fiber, polypropylene fiber, and polyvinyl alcohol fiber are the short fibers to be added. This method is often used in new structure [2–4]. FRP is more suitable for the reinforcement of existing reinforced concrete structures. Among the commonly used FRP materials, the CFRP material has the best performance and is most commonly used in structural strengthening [5].Scholars at home and abroad have done a lot of experimental and theoretical research on the flexural behavior of reinforced concrete shallow beams strengthened with FRP materials and achieved many research results [6–10]. There are many research studies on the calculation method of bearing capacity of beams strengthened with CFRP in the existing literature [11–15]. There are few studies on the stiffness of beams strengthened with CFRP in the existing literature [16–18]. In the existing literature, the research object of flexural rigidity is mostly reinforced concrete shallow beam [19–21]. The stiffness analysis method is often used in the existing literature [22]. The obtained stiffness calculation formula is cumbersome or needs to be discussed [23–25]. The stress stage of the existing literature is the stage before the steel bars yield, and the stiffness after the steel bars yield has not been studied. In view of the importance of beam stiffness to deformation calculation and the lack of current research, based on the effective moment of inertia method, this paper further discusses the calculation method of short-term stiffness of reinforced concrete short beams strengthened with carbon fiber reinforced plastic sheets in the whole loading process.In order to study the flexural rigidity of short RC beams strengthened with CFRP sheets, the flexural rigidity expression consistent with that of ordinary reinforced concrete beams is established. In this paper, the influence of CFRP layers, concrete strength grade, and longitudinal reinforcement ratio on the flexural rigidity of reinforced concrete short beams is studied through 7 members. In this paper, the calculation formula of flexural rigidity for short reinforced concrete beams strengthened with CFRP sheets and ordinary short reinforced concrete beams is put forward.


## 2. Experimental Program

Seven beams were designed in this experiment. The cross section of the beams was 150 mm in width and 500 mm in height. The clear span was 2000 mm with the span ratio of 4. The geometrical dimensions and reinforcement of the beams are shown in Figure1. One beam was an ordinary RC short beam without CFRP sheets and the rest were RC short beams with CFRP sheets. As shown in Figure 2, CFRP sheets were pasted on the beam bottom to improve bending strength, and two strips of CFRP sheet with 100 mm width were set up in the bending shear zone of each side to prevent the occurrence of stripping at the ends. The detailed design parameters of beams are shown in Table 1. The measured elastic modulus of CFRP sheet was 246 GPa, the tensile strength was 3512 MPa, the thickness was 0.167 mm, and the elongation was 1.71%. The measured mechanical properties of concrete and reinforcement are shown in Tables 2 and 3.Figure 1
Specimen size and reinforcement drawing (unit: mm).Figure 2
Specimen reinforcement (unit: mm).Table 1
Design parameters of the specimen.
No.Beams IDCFRP sheet layersConcrete strengthReinforcement ratio (%)1B-0-30-40C300.422B-1-30-41C300.423B-2-30-42C300.424B-1-20-41C200.425B-1-40-41C400.426B-1-30-61C300.607B-1-30-81C300.82Table 2
Mechanical performance index of concrete.
Beams IDConcrete strengthModulus of elasticity (MPa)Cube strength (MPa)Axial compressive strength (MPa)Tensile strength (MPa)B-0-30-432.9522.671.7216073.9B-1-30-438.1132.622.4023023.6B-2-30-436.3829.632.1120325.7B-1-20-435.6026.672.0619490.8B-1-40-443.8538.372.6924078.7B-1-30-635.4921.911.8220898.2B-1-30-832.1823.592.0016495.0Table 3
Mechanical performance index of steel bar.
Rebar gradeDiameter (mm)Yield strength (MPa)Tensile strength (MPa)Modulus of elasticity (GPa)HRB4008418642200104116412001252061620014421543200A monotone static loading test was carried out for all beams. The concrete strain, deflection, inclination, and bearing capacity of beams are measured. The arrangement of test loading and measuring equipment is shown in Figure3. The concrete strains in the compression zone, tension zone, and beam height range of the pure bending span were measured using the strain gauges of π shape. Strain-type displacement sensors were arranged in the upper part of the support, the lower part of the loading point, and the middle of the span to measure the deflection. Ten inclinometers were placed at 25 mm intervals on the top of the beam to measure the rotation angle of the top of the beam. All test data were collected by a data acquisition instrument.Figure 3
Layout of test measuring equipment (unit: mm).The cracking load, yield load, and ultimate load of 7 test pieces measured in the test are listed in Table4. The failure modes of CFRP strengthened beams include three modes: first tensile failure of CFRP, first crushing failure of concrete, and almost simultaneous boundary failure of both. The specific failure modes are also listed in Table 4.Table 4
Main test results of the specimen.
Beams IDCracking load (kN)Yield load (kN)Ultimate load (kN)Failure modeB-0-30-466.5153.0223.0Failure mode of suitable reinforced beamB-1-30-4100.0188.9282.0CFRP sheet pulledB-2-30-498.0190.3355.0Boundary failureB-1-20-483.0154.4283.0CFRP sheet pulledB-1-40-4115.0170.5303.0CFRP sheet pulledB-1-30-682.5294.8358.9Concrete crushedB-1-30-892.0325.0371.4Concrete crushedThe values of curvature and stiffness for the pure bending section of beam under different bending moments were calculated by equations (1) and (2). All bending moment-curvature curves and bending moment-stiffness curves are drawn in Figures 4–10.(1)ϕ=1ρ=ε¯c+ε¯sh,(2)B=EI=Mϕ,where ρ is the radius of curvature; ε¯c is the average concrete strain in the compression zone of pure bending section; ε¯s is average concrete strain in the tension zone of pure bending section; h is the average distance between strain gauge in the compression zone and the tension zone of the pure bending section; E is the elastic modulus of concrete; I is inertia moment of beam; and M is the bending moment of the midspan section.Figure 4
B-0-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 5
B-1-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 6
B-2-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 7
B-1-20-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 8
B-1-40-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 9
B-1-30-6. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 10
B-1-30-8. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)The general characteristics of bending moment-curvature and bending moment-stiffness curves are shown in Figures11 and 12, respectively. Curves are divided into three stages by the cracking moment, yielding moment, and ultimate bending moment:(1)
Stage of precracking of concrete: the curvature is linear, and linear slope is a maximum value. The stiffness is a constant value in general. However, the measured bending stiffness changes greatly because of the small curvature in this stage, which is affected by the precision ofπ strain gauge, and there will be small fluctuation in curvature. From formula (2), it can be seen that the small fluctuation of curvature will cause great fluctuation in bending stiffness. Therefore, the bending stiffness value close to the cracking load should be taken as the fixed value in this stage as far as possible.(2)
Stage of concrete cracking to the yield-steel: the curvature is approximately linear, the slope decreases, and the stiffness decreases sharply. A nonlinear regulation is presented in this stage.(3)
Stage of the yield-steel to the ultimate bearing state: the curvature is also approximately linear, and the slope and stiffness continuously decrease. A nonlinear change regulation is also presented in this stage.Figure 11
Characteristics of moment-stiffness curve.Figure 12
Characteristics of moment-curvature curve.


## 3. Calculation of Bending Stiffness

### 3.1. Calculation Assumption

There are three assumptions for the calculation of bending stiffness:The concrete is elastic in the precracking stage.In the stage of concrete cracking to the yield-steel, the compressive strain of concrete is smaller than the peak strain. Concrete is assumed as a linear elastic material, and the tensile steel bar is still elastic. The stress-strain relationship conforms to Hooke’s law. The average strain of the beam section is assumed to be in line with the plane section assumption. The correction coefficient of internal force arm is taken into account for the short beam, and the calculation method is conformed to reference [16]. The resistance to tension of concrete in the tensile zone of cracking section is ignored.The basic assumption in the stage of the steel yield to the ultimate bearing state is in accordance to reference [16].


### 3.2. Calculation of the Precracking Stage

In this stage, as the concrete has not cracked yet, the steel bar and CFRP sheet can be equivalent to concrete according to the elastic theory. So, the calculation equations of parameters for beam section are established.The height of neutral axis from the edge of concrete in the compressive zonex0 and the inertia moment of section I0 can be calculated by the following two equations:(3)x0=bh2/2+αE−1Ash0+αfAfhbh+αE−1As+αfAf,(4)I0=bx033+bh−x033+αE−1Ash0−x02+αfAfh−x02,where b, h, and h0 are beam width, height, and the distance from the joint action point of tensile steel bars to the compression edge of the beam, respectively. As and Af are section areas of tensile steel bar and CFRP sheet, respectively. αE is the ratio of elastic modulus of steel bar to the elastic modulus of concrete Es/Ec. αf is the ratio of elastic modulus of CFRP sheet to the elastic modulus of concrete Ef/Ec. Es, Ec, and Ef are elastic moduli of steel bar, concrete, and CFRP sheet, respectively. I0 is the inertia moment of the uncracked beam section.Thus, the bending stiffnessB0 of the uncracked beam section is calculated using the following equation:(5)B=B0=EcI0.The cracking moment is calculated using the following equation:(6)Mcr=γmftI0h−x0,where ft is tensile strength of concrete and γm is the plastic coefficient of section resistance moment. The value of γm is 1.75, referring to the ordinary RC beam.


### 3.3. Calculation of Concrete Cracking to the Yield-Steel

It is assumed that the concrete in the compression zone is a linear elastic material, and the cross-sectional force diagram at this stage is shown in Figure13. C is the resultant force of compression concrete. σs and As are tensile stress and area of steel bars, respectively. σf and Afe are the tensile stress and effective area of CFRP sheets. Afe is the actual area of the CFRP sheet multiplied by the thickness reduction coefficient Km. The value of Km is determined according to Chinese specification GB50367-2013 [26].Figure 13
Cross-sectional diagram of force.When the tensile steel yields, equations (7) and (8) can be obtained according to the equilibrium condition of the force.(7)fyAs+σfAfe−12bxcσc=0,(8)EsεyAs+EfεfAfe−12bxcEcεc=0,where fy, σc, and xc are yield strengths of steel bar, compressive stress of concrete, and height of the concrete compression zone, respectively.The yield strainεy of steel bar is known from εy=fy/Es. As shown in equations (9) and (10), the concrete strain εc and CFRP sheet strain εf can be obtained from the similar triangle relation of strain.(9)εc=εyxch0−xc,(10)εf=εyh−xch0−xc.Equations (9) and (10) are substituted into equation (8) to obtain the following equation:(11)EsεyAs+Efεyh−xch0−xcAfe−12bxcEcεyxch0−xc=0.The height of the compression zonexc can be obtained by solving equation (11), and then the yield bending moment is obtained as follows:(12)My=EsεyAsαdh0−xc3+EfεfAfαdh−xc3.When the steel yields, the inertia moment of the cracked sectionIcr can be calculated by the following equation:(13)Icr=13bxc3+αE−1Ash0−xc2+αfAfh−xc2.The inertia moment of the cracked sectionIcr is the minimum inertia moment of each section along the component axis. The stiffness of the cracked section is calculated using the following equation:(14)Bcr=EcIcr.In this stage, the value of average section stiffnessB is a value between B0 and Bcr, which decreases with the increase of bending moment.For the ordinary RC beam, according to the United States code ACI318-05 [27], the effective stiffness of the beam section Beff in normal service stage is calculated using the following equation:(15)Beff=EcIeff=McrM3EcI0+1−McrM3EcIcr.According to the test data of bending moment-bending stiffness curve in this experiment, the equation of bending stiffness for RC short beams strengthened by CFRP sheet is obtained using the nonlinear regression method.(16)B=McrM4B0+1−McrM4Bcr.When the bending momentM is the yield bending moment My, the calculated yield stiffness By of the beam is calculated using the following equation:(17)By=McrMy4B0+1−McrMy4Bcr.


### 3.4. Calculation of Longitudinal Bars after Yielding

Two bending failure modes as breakage of CFRP sheet and crushing of concrete are observed in this experiment. Two theoretical equations for calculating the bending capacityMu in corresponding failure modes have been established in literature [16]. The calculation method in this paper is the same as that in reference [16]. Equations (18) and (19) are calculation equations of section curvature ϕu, respectively.For the breakage of CFRP sheet:(18)ϕu=0.9εfu/h−xcαd.For the crushing of concrete:(19)ϕu=εcu/xcαd.In the ultimate bearing state, the bending stiffnessBu is calculated using the following equation:(20)Bu=Muϕu.At this stage, the value of bending stiffnessB of the beam section is a value between By and Bu and decreases with the increase of bending moment. According to the experimental data of bending moment-bending stiffness, the nonlinear regression method is adopted to obtain the calculation equation of bending stiffness of RC beams reinforced by CFRP sheet in the stage of yield-steel to the ultimate bearing state.(21)B=By+Bu−ByM−My0.7Mu−My0.7.When the bending momentM is equal to the yield bending moment Mu, the bending stiffness of the beam is calculated using the following equation:(22)B=Bu.


## 3.1. Calculation Assumption

There are three assumptions for the calculation of bending stiffness:The concrete is elastic in the precracking stage.In the stage of concrete cracking to the yield-steel, the compressive strain of concrete is smaller than the peak strain. Concrete is assumed as a linear elastic material, and the tensile steel bar is still elastic. The stress-strain relationship conforms to Hooke’s law. The average strain of the beam section is assumed to be in line with the plane section assumption. The correction coefficient of internal force arm is taken into account for the short beam, and the calculation method is conformed to reference [16]. The resistance to tension of concrete in the tensile zone of cracking section is ignored.The basic assumption in the stage of the steel yield to the ultimate bearing state is in accordance to reference [16].


## 3.2. Calculation of the Precracking Stage

In this stage, as the concrete has not cracked yet, the steel bar and CFRP sheet can be equivalent to concrete according to the elastic theory. So, the calculation equations of parameters for beam section are established.The height of neutral axis from the edge of concrete in the compressive zonex0 and the inertia moment of section I0 can be calculated by the following two equations:(3)x0=bh2/2+αE−1Ash0+αfAfhbh+αE−1As+αfAf,(4)I0=bx033+bh−x033+αE−1Ash0−x02+αfAfh−x02,where b, h, and h0 are beam width, height, and the distance from the joint action point of tensile steel bars to the compression edge of the beam, respectively. As and Af are section areas of tensile steel bar and CFRP sheet, respectively. αE is the ratio of elastic modulus of steel bar to the elastic modulus of concrete Es/Ec. αf is the ratio of elastic modulus of CFRP sheet to the elastic modulus of concrete Ef/Ec. Es, Ec, and Ef are elastic moduli of steel bar, concrete, and CFRP sheet, respectively. I0 is the inertia moment of the uncracked beam section.Thus, the bending stiffnessB0 of the uncracked beam section is calculated using the following equation:(5)B=B0=EcI0.The cracking moment is calculated using the following equation:(6)Mcr=γmftI0h−x0,where ft is tensile strength of concrete and γm is the plastic coefficient of section resistance moment. The value of γm is 1.75, referring to the ordinary RC beam.


## 3.3. Calculation of Concrete Cracking to the Yield-Steel

It is assumed that the concrete in the compression zone is a linear elastic material, and the cross-sectional force diagram at this stage is shown in Figure13. C is the resultant force of compression concrete. σs and As are tensile stress and area of steel bars, respectively. σf and Afe are the tensile stress and effective area of CFRP sheets. Afe is the actual area of the CFRP sheet multiplied by the thickness reduction coefficient Km. The value of Km is determined according to Chinese specification GB50367-2013 [26].Figure 13
Cross-sectional diagram of force.When the tensile steel yields, equations (7) and (8) can be obtained according to the equilibrium condition of the force.(7)fyAs+σfAfe−12bxcσc=0,(8)EsεyAs+EfεfAfe−12bxcEcεc=0,where fy, σc, and xc are yield strengths of steel bar, compressive stress of concrete, and height of the concrete compression zone, respectively.The yield strainεy of steel bar is known from εy=fy/Es. As shown in equations (9) and (10), the concrete strain εc and CFRP sheet strain εf can be obtained from the similar triangle relation of strain.(9)εc=εyxch0−xc,(10)εf=εyh−xch0−xc.Equations (9) and (10) are substituted into equation (8) to obtain the following equation:(11)EsεyAs+Efεyh−xch0−xcAfe−12bxcEcεyxch0−xc=0.The height of the compression zonexc can be obtained by solving equation (11), and then the yield bending moment is obtained as follows:(12)My=EsεyAsαdh0−xc3+EfεfAfαdh−xc3.When the steel yields, the inertia moment of the cracked sectionIcr can be calculated by the following equation:(13)Icr=13bxc3+αE−1Ash0−xc2+αfAfh−xc2.The inertia moment of the cracked sectionIcr is the minimum inertia moment of each section along the component axis. The stiffness of the cracked section is calculated using the following equation:(14)Bcr=EcIcr.In this stage, the value of average section stiffnessB is a value between B0 and Bcr, which decreases with the increase of bending moment.For the ordinary RC beam, according to the United States code ACI318-05 [27], the effective stiffness of the beam section Beff in normal service stage is calculated using the following equation:(15)Beff=EcIeff=McrM3EcI0+1−McrM3EcIcr.According to the test data of bending moment-bending stiffness curve in this experiment, the equation of bending stiffness for RC short beams strengthened by CFRP sheet is obtained using the nonlinear regression method.(16)B=McrM4B0+1−McrM4Bcr.When the bending momentM is the yield bending moment My, the calculated yield stiffness By of the beam is calculated using the following equation:(17)By=McrMy4B0+1−McrMy4Bcr.


## 3.4. Calculation of Longitudinal Bars after Yielding

Two bending failure modes as breakage of CFRP sheet and crushing of concrete are observed in this experiment. Two theoretical equations for calculating the bending capacityMu in corresponding failure modes have been established in literature [16]. The calculation method in this paper is the same as that in reference [16]. Equations (18) and (19) are calculation equations of section curvature ϕu, respectively.For the breakage of CFRP sheet:(18)ϕu=0.9εfu/h−xcαd.For the crushing of concrete:(19)ϕu=εcu/xcαd.In the ultimate bearing state, the bending stiffnessBu is calculated using the following equation:(20)Bu=Muϕu.At this stage, the value of bending stiffnessB of the beam section is a value between By and Bu and decreases with the increase of bending moment. According to the experimental data of bending moment-bending stiffness, the nonlinear regression method is adopted to obtain the calculation equation of bending stiffness of RC beams reinforced by CFRP sheet in the stage of yield-steel to the ultimate bearing state.(21)B=By+Bu−ByM−My0.7Mu−My0.7.When the bending momentM is equal to the yield bending moment Mu, the bending stiffness of the beam is calculated using the following equation:(22)B=Bu.


## 4. Comparison of Calculated and Experimental Data

The test and calculated values of cracking, yield, and ultimate bending moment are shown in Table5. The ratio in the table is the calculated value divided by the test value. The average values of the ratio of cracking, yield, and ultimate bending moment are 1.148, 0.996, and 1.007, respectively. The coefficient of variation of the ratio of cracking, yield, and ultimate bending moment is 0.061, 0.059, and 0.028, respectively.Table 5
Experimental and calculated values of cracking moment, yield moment, and ultimate moment of member (unit: kN·m).
Beam IDMcrMyMuExp.Cal.RatioExp.Cal.RatioExp.Cal.RatioB-0-30-422.1721.241.04450.99152.510.97174.3371.731.036B-1-30-433.3328.911.15362.9559.771.05394.0095.620.983B-2-30-432.6726.131.25063.4265.970.961118.33114.241.036B-1-20-427.6725.271.09551.4759.410.86694.3393.591.008B-1-40-438.3332.251.18856.8259.890.949101.0097.931.031B-1-30-627.5022.981.19698.26102.860.955119.63120.480.993B-1-30-830.6727.631.110108.33107.531.008123.80128.480.964Mean value30.3326.341.14870.3286.850.966103.63103.151.007Mean-square deviation5.173.690.07023.2335.690.05717.9119.270.029Coefficient of variation0.170.140.0610.330.410.0590.170.190.028Mcr denotes cracking moment. My denotes yield moment. Mu denotes ultimate moment.The test and calculated values of cracking, yield, and ultimate curvature are shown in Table6. The ratio in the table is the calculated value divided by the test value. The average values of the ratio of cracking, yield, and ultimate curvature are 1.003, 0.949, and 0.994, respectively. The coefficients of variation of the ratio of cracking, yield, and ultimate curvature are 0.097, 0.158, and 0.139, respectively.Table 6
Experimental and calculated values of cracking curvature, yield curvature, and ultimate curvature of members.
Beam IDϕcrϕyϕuExp.Cal.RatioExp.Cal.RatioExp.Cal.RatioB-0-30-49.988.941.1160.5920.5531.0714.9654.3631.138B-1-30-46.717.450.9000.6400.5321.2033.3133.3280.995B-2-30-48.827.701.1440.5680.5930.9584.3023.5561.210B-1-20-47.527.660.9820.4530.5780.7843.3673.3910.993B-1-40-47.298.100.9000.4010.4680.8572.9703.3080.898B-1-30-67.687.501.0240.6730.8330.8082.4422.9760.821B-1-30-810.6011.100.9550.7530.7830.9622.6862.9710.904Mean value8.378.351.0030.5800.620.9493.443.410.994Mean-square deviation1.471.320.0980.1200.140.1500.900.470.138Coefficient of variation0.180.160.0970.2100.220.1580.260.140.139ϕcr is cracking curvature. ϕy is yield curvature. ϕu is ultimate curvature.The test and calculated values of cracking, yield, and ultimate bending stiffness are shown in Table7. The ratio in the table is the calculated value divided by the test value. The average values of the ratio of cracking, yield, and ultimate flexural rigidity are 1.154, 1.035, and 1.029, respectively. The coefficient of variation of the ratio of cracking, yield, and ultimate flexural rigidity is 0.111, 0.108, and 0.121, respectively.Table 7
Experimental and calculated values of cracking stiffness, yield stiffness, and ultimate stiffness of members.
Beam IDB0ByBuExp.Cal.RatioExp.Cal.RatioExp.Cal.RatioB-0-30-42.222.380.9330.860.950.9061.501.640.915B-1-30-44.973.881.2810.981.120.8792.842.870.990B-2-30-43.713.391.0941.121.111.0092.753.210.857B-1-20-43.683.301.1151.141.031.1072.802.761.014B-1-40-45.263.981.3221.421.281.1093.402.961.149B-1-30-63.583.061.1701.461.231.1874.904.051.210B-1-30-82.892.491.1611.441.371.0514.614.321.067Mean value3.763.211.1541.201.161.0353.263.121.029Mean-square deviation1.070.620.1280.240.150.1121.170.890.124Coefficient of variation0.280.190.1110.200.130.1080.360.280.121B0 is cracking stiffness. By is yield stiffness. Bu is ultimate stiffness.The bending moment-curvature curve and bending moment-flexural rigidity curve are shown in Figures14–20. It can be seen from the figures that the dotted line is the calculated value and the solid line is the test value. Except for the difference between the calculated value and the test value at the cracking moment, the calculated value and the test value at the rest of the moment are basically consistent.Figure 14
Beam B-0-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 15
Beam B-1-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 16
Beam B-2-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 17
Beam B-1-20-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 18
Beam B-1-40-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 19
Beam B-1-30-6. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)Figure 20
Beam B-1-30-8. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.
(a)(b)The comparison between Tables5–7 and Figures 14–20 shows that the bending rigidity calculation method proposed in this paper is reasonable, with high calculation accuracy, and is suitable for the bending rigidity calculation of reinforced concrete short beams strengthened with carbon fiber sheet.


## 5. Conclusions

Based on the experiment of flexural performance of short reinforced concrete beams strengthened with CFRP sheets, the calculation method of flexural rigidity of short beams in the whole loading process was proposed. The calculated flexural rigidity value was compared with the experimental value, and the following conclusions are drawn:The characteristics of the bending moment-curvature curve and the bending moment-stiffness curve of the pure bending section in the span of 7 specimens were obtained. The curve of flexural stiffness was divided into three sections by cracking load and yield load. The bending stiffness curve after cracking moment was nonlinear.According to the characteristics of the three stress stages, the reasonable calculation assumption was adopted. Based on the effective moment of inertia method, the calculation formula of flexural rigidity of reinforced concrete short beams strengthened with CFRP sheets was proposed. The formula was applicable to both short beams strengthened by CFRP and ordinary reinforced concrete beams without reinforcement, and the calculation accuracy was high.The calculation method of flexural rigidity proposed in this paper can provide theoretical basis for the calculation of the corner of the reinforced concrete short beam strengthened by CFRP and also provide data support and reference for other research studies in the future.



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