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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.

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 [

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 [

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.

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 Figure

Specimen size and reinforcement drawing (unit: mm).

Specimen reinforcement (unit: mm).

Design parameters of the specimen.

No. | Beams ID | CFRP sheet layers | Concrete strength | Reinforcement ratio (%) |
---|---|---|---|---|

1 | B-0-30-4 | 0 | C30 | 0.42 |

2 | B-1-30-4 | 1 | C30 | 0.42 |

3 | B-2-30-4 | 2 | C30 | 0.42 |

4 | B-1-20-4 | 1 | C20 | 0.42 |

5 | B-1-40-4 | 1 | C40 | 0.42 |

6 | B-1-30-6 | 1 | C30 | 0.60 |

7 | B-1-30-8 | 1 | C30 | 0.82 |

Mechanical performance index of concrete.

Beams ID | Concrete strength | Modulus of elasticity (MPa) | ||
---|---|---|---|---|

Cube strength (MPa) | Axial compressive strength (MPa) | Tensile strength (MPa) | ||

B-0-30-4 | 32.95 | 22.67 | 1.72 | 16073.9 |

B-1-30-4 | 38.11 | 32.62 | 2.40 | 23023.6 |

B-2-30-4 | 36.38 | 29.63 | 2.11 | 20325.7 |

B-1-20-4 | 35.60 | 26.67 | 2.06 | 19490.8 |

B-1-40-4 | 43.85 | 38.37 | 2.69 | 24078.7 |

B-1-30-6 | 35.49 | 21.91 | 1.82 | 20898.2 |

B-1-30-8 | 32.18 | 23.59 | 2.00 | 16495.0 |

Mechanical performance index of steel bar.

Rebar grade | Diameter (mm) | Yield strength (MPa) | Tensile strength (MPa) | Modulus of elasticity (GPa) |
---|---|---|---|---|

HRB400 | 8 | 418 | 642 | 200 |

10 | 411 | 641 | 200 | |

12 | 520 | 616 | 200 | |

14 | 421 | 543 | 200 |

A 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 Figure

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 Table

Main test results of the specimen.

Beams ID | Cracking load (kN) | Yield load (kN) | Ultimate load (kN) | Failure mode |
---|---|---|---|---|

B-0-30-4 | 66.5 | 153.0 | 223.0 | Failure mode of suitable reinforced beam |

B-1-30-4 | 100.0 | 188.9 | 282.0 | CFRP sheet pulled |

B-2-30-4 | 98.0 | 190.3 | 355.0 | Boundary failure |

B-1-20-4 | 83.0 | 154.4 | 283.0 | CFRP sheet pulled |

B-1-40-4 | 115.0 | 170.5 | 303.0 | CFRP sheet pulled |

B-1-30-6 | 82.5 | 294.8 | 358.9 | Concrete crushed |

B-1-30-8 | 92.0 | 325.0 | 371.4 | Concrete crushed |

The values of curvature and stiffness for the pure bending section of beam under different bending moments were calculated by equations (

B-0-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-1-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-2-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-1-20-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-1-40-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-1-30-6. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

B-1-30-8. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

The general characteristics of bending moment-curvature and bending moment-stiffness curves are shown in Figures

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 of

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.

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.

Characteristics of moment-stiffness curve.

Characteristics of moment-curvature curve.

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 [

The basic assumption in the stage of the steel yield to the ultimate bearing state is in accordance to reference [

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 zone

Thus, the bending stiffness

The cracking moment is calculated using the following equation:

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 Figure

Cross-sectional diagram of force.

When the tensile steel yields, equations (

The yield strain

Equations (

The height of the compression zone

When the steel yields, the inertia moment of the cracked section

The inertia moment of the cracked section

In this stage, the value of average section stiffness

For the ordinary RC beam, according to the United States code ACI318-05 [

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.

When the bending moment

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 capacity

For the breakage of CFRP sheet:

For the crushing of concrete:

In the ultimate bearing state, the bending stiffness

At this stage, the value of bending stiffness

When the bending moment

The test and calculated values of cracking, yield, and ultimate bending moment are shown in Table

Experimental and calculated values of cracking moment, yield moment, and ultimate moment of member (unit: kN·m).

Beam ID | |||||||||
---|---|---|---|---|---|---|---|---|---|

Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | |

B-0-30-4 | 22.17 | 21.24 | 1.044 | 50.99 | 152.51 | 0.971 | 74.33 | 71.73 | 1.036 |

B-1-30-4 | 33.33 | 28.91 | 1.153 | 62.95 | 59.77 | 1.053 | 94.00 | 95.62 | 0.983 |

B-2-30-4 | 32.67 | 26.13 | 1.250 | 63.42 | 65.97 | 0.961 | 118.33 | 114.24 | 1.036 |

B-1-20-4 | 27.67 | 25.27 | 1.095 | 51.47 | 59.41 | 0.866 | 94.33 | 93.59 | 1.008 |

B-1-40-4 | 38.33 | 32.25 | 1.188 | 56.82 | 59.89 | 0.949 | 101.00 | 97.93 | 1.031 |

B-1-30-6 | 27.50 | 22.98 | 1.196 | 98.26 | 102.86 | 0.955 | 119.63 | 120.48 | 0.993 |

B-1-30-8 | 30.67 | 27.63 | 1.110 | 108.33 | 107.53 | 1.008 | 123.80 | 128.48 | 0.964 |

Mean value | 30.33 | 26.34 | 1.148 | 70.32 | 86.85 | 0.966 | 103.63 | 103.15 | 1.007 |

Mean-square deviation | 5.17 | 3.69 | 0.070 | 23.23 | 35.69 | 0.057 | 17.91 | 19.27 | 0.029 |

Coefficient of variation | 0.17 | 0.14 | 0.061 | 0.33 | 0.41 | 0.059 | 0.17 | 0.19 | 0.028 |

The test and calculated values of cracking, yield, and ultimate curvature are shown in Table

Experimental and calculated values of cracking curvature, yield curvature, and ultimate curvature of members.

Beam ID | |||||||||
---|---|---|---|---|---|---|---|---|---|

Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | |

B-0-30-4 | 9.98 | 8.94 | 1.116 | 0.592 | 0.553 | 1.071 | 4.965 | 4.363 | 1.138 |

B-1-30-4 | 6.71 | 7.45 | 0.900 | 0.640 | 0.532 | 1.203 | 3.313 | 3.328 | 0.995 |

B-2-30-4 | 8.82 | 7.70 | 1.144 | 0.568 | 0.593 | 0.958 | 4.302 | 3.556 | 1.210 |

B-1-20-4 | 7.52 | 7.66 | 0.982 | 0.453 | 0.578 | 0.784 | 3.367 | 3.391 | 0.993 |

B-1-40-4 | 7.29 | 8.10 | 0.900 | 0.401 | 0.468 | 0.857 | 2.970 | 3.308 | 0.898 |

B-1-30-6 | 7.68 | 7.50 | 1.024 | 0.673 | 0.833 | 0.808 | 2.442 | 2.976 | 0.821 |

B-1-30-8 | 10.60 | 11.10 | 0.955 | 0.753 | 0.783 | 0.962 | 2.686 | 2.971 | 0.904 |

Mean value | 8.37 | 8.35 | 1.003 | 0.580 | 0.62 | 0.949 | 3.44 | 3.41 | 0.994 |

Mean-square deviation | 1.47 | 1.32 | 0.098 | 0.120 | 0.14 | 0.150 | 0.90 | 0.47 | 0.138 |

Coefficient of variation | 0.18 | 0.16 | 0.097 | 0.210 | 0.22 | 0.158 | 0.26 | 0.14 | 0.139 |

The test and calculated values of cracking, yield, and ultimate bending stiffness are shown in Table

Experimental and calculated values of cracking stiffness, yield stiffness, and ultimate stiffness of members.

Beam ID | |||||||||
---|---|---|---|---|---|---|---|---|---|

Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | Exp. | Cal. | Ratio | |

B-0-30-4 | 2.22 | 2.38 | 0.933 | 0.86 | 0.95 | 0.906 | 1.50 | 1.64 | 0.915 |

B-1-30-4 | 4.97 | 3.88 | 1.281 | 0.98 | 1.12 | 0.879 | 2.84 | 2.87 | 0.990 |

B-2-30-4 | 3.71 | 3.39 | 1.094 | 1.12 | 1.11 | 1.009 | 2.75 | 3.21 | 0.857 |

B-1-20-4 | 3.68 | 3.30 | 1.115 | 1.14 | 1.03 | 1.107 | 2.80 | 2.76 | 1.014 |

B-1-40-4 | 5.26 | 3.98 | 1.322 | 1.42 | 1.28 | 1.109 | 3.40 | 2.96 | 1.149 |

B-1-30-6 | 3.58 | 3.06 | 1.170 | 1.46 | 1.23 | 1.187 | 4.90 | 4.05 | 1.210 |

B-1-30-8 | 2.89 | 2.49 | 1.161 | 1.44 | 1.37 | 1.051 | 4.61 | 4.32 | 1.067 |

Mean value | 3.76 | 3.21 | 1.154 | 1.20 | 1.16 | 1.035 | 3.26 | 3.12 | 1.029 |

Mean-square deviation | 1.07 | 0.62 | 0.128 | 0.24 | 0.15 | 0.112 | 1.17 | 0.89 | 0.124 |

Coefficient of variation | 0.28 | 0.19 | 0.111 | 0.20 | 0.13 | 0.108 | 0.36 | 0.28 | 0.121 |

The bending moment-curvature curve and bending moment-flexural rigidity curve are shown in Figures

Beam B-0-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-1-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-2-30-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-1-20-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-1-40-4. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-1-30-6. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

Beam B-1-30-8. (a) Bending moment-curvature curve. (b) Bending moment-stiffness curve.

The comparison between Tables

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.

The data used to support the findings of the study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

Wang Tingyan did the whole experiment and wrote the article, Zhang Junwei calculated the data, and Zhou Yun revised the article. All authors have read and agreed to the published version of the manuscript.

This study was supported by the National Natural Science Foundation of China (No. 50579068 and 51708514).

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