# Crack Propagation Law of Reinforced Concrete Beams

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## Abstract

**:**

## 1. Introduction

## 2. Comparison of Finite Element Model and Experiment Results

#### 2.1. Overview of the Test

Species | Effective Span/mm | Shear Span to Effective Depth Ratio | Concentrated Load Span/mm | Stirrup Spacing | Stirrup Ratio/% |
---|---|---|---|---|---|

JL-1.5-6-150 | 1500 | 1.5 | 338 | 6@150 | 0.25 |

JL-1.5-8-200 | 1500 | 1.5 | 338 | 8@200 | 0.36 |

JL-1.5-8-150 | 1500 | 1.5 | 338 | 8@150 | 0.45 |

JL-2.0-6-150 | 1500 | 2.0 | 450 | 6@150 | 0.25 |

JL-2.0-8-200 | 1500 | 2.0 | 450 | 8@200 | 0.36 |

JL-2.0-8-150 | 1500 | 2.0 | 450 | 8@150 | 0.45 |

JL-2.5-6-150 | 1500 | 2.5 | 563 | 6@150 | 0.25 |

JL-2.5-8-200 | 1500 | 2.5 | 563 | 8@200 | 0.36 |

JL-2.5-8-150 | 1500 | 2.5 | 563 | 8@150 | 0.45 |

**Figure 3.**Reinforced concrete beam loading test device from reference [19].

#### 2.2. Material Constitute of Concrete in the FE Model

#### 2.3. Finite Element Model Building

#### 2.4. Comparison of the Experimental and Finite Element Modeling of Strains

#### 2.5. Comparison of the Experimental and Finite Element Modeling of Cracks

## 3. Simulation of Frame Joints with Floor Slabs

#### 3.1. Analysis Model

#### 3.2. Loading Mode

#### 3.3. Loading Regime

^{4}kN with an axial compression ratio of 0.4 is slowly applied to the top of the column by a vertical hydraulic jack to simulate the axial compression transmitted by the superstructure to the column. The method of reciprocating loading was used in the analysis. According to Building Seismic Test Method Regulations JGJ/T101-2015, the test specimen is preloaded with no more than 30% of the calculated cracking load value; then, the reciprocating load is applied by using the mixed load and displacement control loading system. Before cracking, the test specimen is loaded by load control and the load is repeated according to 0.5 times the estimated cracking load. The corresponding displacement during cracking is Δ = 1.79 mm. After cracking, reciprocating loading is adopted to apply vertical displacement Δ

_{total}(Δ

_{total}= 4Δ) to the coupling point of the beam end for displacement loading. Each stage of loading reciprocates three times; the loading system is shown in Figure 14.

## 4. Calculation Results and Analysis

#### 4.1. Stress Distribution

#### 4.2. Hysteretic Properties and Skeleton Curves

#### 4.3. Reinforcement Strain

^{−6}, which is lower than the yield strain, and the reinforcement has no yield.

^{−6}, reaching the yield strain approximately at the final stage of loading.

#### 4.4. Concrete Damage

## 5. Conclusions

- (1)
- Calculation of the nonlinear behavior of concrete based on the constitutive formula in GB 50010-2010 has been proven to be more appropriate. The obtained axial tensile/compressive stress–strain curves, as well as the axial tensile/compressive damage–strain curves of concrete, can more accurately simulate the concrete stresses in the tests;
- (2)
- The simply supported test beams using over-reinforced beams all experienced shear and compressive failure and the development of oblique cracks in reinforced concrete beams was studied. The strain changes of reinforcement during the failure process of test beams were qualitatively analyzed. The strain of the longitudinal bar at the bottom of the beam shows an approximately linear increase in both test and simulation results, with no bending failure observed in the test beam;
- (3)
- The Abaqus concrete damage plastic model was unable to quantitatively reflect the width, spacing, and depth of macroscopic cracks. However, it could qualitatively illustrate the development direction and propagation process of cracks through the DAMAGET diagram;
- (4)
- Reinforced concrete joints were modeled using Abaqus to predict their forces and crack development. The cracks in the joint between the frame and the floor slab mainly showed bending cracks, with less compressive damage and more tensile damage to the concrete in the core area of the joint. With the increase in load, the crack developed outward from near the end of the column to form a vertical bending crack and the depth and width of the crack decreased gradually. This indicates that the design is relatively safe and meets the requirements of the bearing capacity of the oblique section.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Test results of reinforced concrete beams from reference [19].

**Figure 5.**Stress–strain and damage relationship curves of concrete under axial compression. (

**a**) stress–strain curve; (

**b**) damage–strain curve.

**Figure 6.**Stress–strain and damage relationship curves of concrete under axial tensile. (

**a**) stress–strain curve; (

**b**) damage–strain curve.

**Figure 8.**Load–strain comparison curve between test and finite element. (

**a**) λ = 1.5; (

**b**) λ = 2.0; (

**c**) λ = 2.5.

**Figure 15.**Stress distribution in reinforced concrete joints with floor slabs. (

**a**) Stress distribution of the concrete joints; (

**b**) Stress distribution of the steel reinforcement.

**Figure 19.**Concrete damage distribution of the joint. (

**a**) Concrete compression damage distribution; (

**b**) Concrete tension damage distribution.

Component | Diameter/mm | Material | Yield Strength/MPa |
---|---|---|---|

Longitudinal | 22 | HRB335 | 379.35 |

Strut | 12 | HPB300 | 312.55 |

Stirrup | 6/8 | HPB300 | 312.55 |

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**MDPI and ACS Style**

Yang, Y.; Yang, H.; Fan, Z.; Mu, Z.
Crack Propagation Law of Reinforced Concrete Beams. *Appl. Sci.* **2024**, *14*, 409.
https://doi.org/10.3390/app14010409

**AMA Style**

Yang Y, Yang H, Fan Z, Mu Z.
Crack Propagation Law of Reinforced Concrete Beams. *Applied Sciences*. 2024; 14(1):409.
https://doi.org/10.3390/app14010409

**Chicago/Turabian Style**

Yang, Yuqing, Hongyue Yang, Zhong Fan, and Zaigen Mu.
2024. "Crack Propagation Law of Reinforced Concrete Beams" *Applied Sciences* 14, no. 1: 409.
https://doi.org/10.3390/app14010409