# Effect of Support Conditions on Performance of Continuous Reinforced Concrete Deep Beams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generation of the 3D Numerical Model

#### 2.1. Test Results

#### 2.2. Mesh Generation of Test Beams

_{cu}) was the only required input, which was 37.8 MPa. This corresponded to cylinder compressive strength (f′

_{c}) of 32.1 MPa, which was identical to the concrete strength reported by Yang et al. [9]; Table 1. The tensile strength (f

_{t}) was taken as half of f

_{t}generated by the software (1.35 MPa) [35]. The concrete compressive and tensile stress-strain relation began with a linear relationship, with a slope equal to the concrete modulus of elasticity (E

_{c}), until reaching a strain value equal to the localized strain (ε

_{loc}). The value of ε

_{loc}was generated automatically by the software as a function of the concrete cube strength. After reaching ε

_{loc}, the strain is localized to the finite element by the software. The value of ε

_{loc}was equal to zero and 0.00097 under tension and compression, respectively. The post-peak localized strain is affected by the L/L

_{ch}ratio, where L = crushing band size for the compression behavior or crack band size for the tension behavior, and L

_{ch}= characteristic length, having a default value of 0.1 generated by the software for the compression behavior. The value of L

_{ch}under tension was set to be equal to the mesh size, as recommended by the software manual [31]. The compressive and tensile behavior after strain localization as generated by the software for the given set of inputs is shown in Figure 2 [31].

_{c}) and shear strength (f

_{sh}) are presented in Figure 3. The reduction factor for compressive strength is affected by the fracturing strain (ε

_{f}) determined from the strain tensor at the finite element integration points. The reduction factor for shear strength is affected by fracturing strain after localization ($\stackrel{~}{{\epsilon}_{f}}$) determined by Equation (1) [31].

_{y}) is 420 MPa. The modulus of elasticity (E) for steel is 200 GPa and the strain at yielding (ε

_{y}) is 0.0021. The chosen concrete and steel material models were used to simulate the performance of the continuous deep beams tested by Yang et al. [9] using ATENA software [31]. A mesh sensitivity study was carried out to warrant solution convergence during the numerical analysis of the tested beam while minimizing the computational time. To find the best model mesh, different element sizes were tried prior to choosing the mesh layout exhibited in Figure 5.

#### 2.3. Verification of the Generated 3D Numerical Model

## 3. Parametric Study

#### 3.1. Sets of Support Conditions Considered in This Study

- Hinged-Roller-Hinged support set—HRH (two exterior hinged supports; interior roller support)
- Hinged-Hinged-Hinged support set—HHH (all three supports are hinged)
- Fixed-Hinged-Fixed support set—FHF (two exterior fixed supports; interior hinged support)

#### 3.2. Changing Specimen L10-40 Support Conditions

#### 3.3. Changing Specimen L10-60 Support Conditions

#### 3.4. Changing Specimen L10-72 Support Conditions

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Details of beam geometry and support conditions; (

**b**) experimental total load-midspan displacement relations [9].

**Figure 3.**Functions of concrete strength reduction factors: (

**a**) Compressive strength; (

**b**) shear strength.

**Figure 5.**Generated 3D finite element modeling of beams: (

**a**) Mesh configuration; (

**b**) geometry and reinforcement bars.

**Figure 9.**Responses of specimen L10-40 with three different sets of support conditions: (

**a**) Experimental (RHR, [9]) and numerical Load-deflection curves for specimen L10-40; (

**b**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—HRH support conditions; (

**c**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (bottom)—HHH support conditions; (

**d**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—FHF support conditions.

**Figure 10.**Responses of specimen L10-60 with three different sets of support conditions: (

**a**) Experimental (RHR, [9]) and numerical Load-deflection curves for specimen L10-60; (

**b**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—HRH support conditions; (

**c**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—HHH support conditions; (

**d**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—FHF support conditions.

**Figure 11.**Responses of specimen L10-72 with three different sets of support conditions: (

**a**) Experimental (RHR, [9]) and numerical Load-deflection curves for specimen L10-72; (

**b**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—HRH support conditions; (

**c**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—HHH support conditions; (

**d**) crack pattern (

**top**), crack width (

**middle**), and stress distribution (MPa) (

**bottom**)—FHF support conditions.

**Table 1.**Test specimen specifics (Yang et al. [9]).

Specimen | f_{c}′: MPa | a/h | h | a | d | b_{w} | L | A_{st} = A′_{st} | ${\mathit{\rho}}_{\mathit{s}}={\mathit{\rho}}_{\mathit{s}}^{\prime}$ |
---|---|---|---|---|---|---|---|---|---|

L10-40 | 32.1 | 1.0 | 400 | 400 | 355 | 160 | 800 | 574 | 0.01 |

L10-60 | 32.1 | 1.0 | 600 | 600 | 555 | 160 | 1200 | 861 | 0.01 |

L10-72 | 32.1 | 1.0 | 720 | 720 | 653 | 160 | 1440 | 1148 | 0.01 |

_{c}′ = compressive strength of concrete, a/h = shear span to depth ratio, h = section depth, a = shear span, d = section effective depth, b

_{w}= beam width, L = beam span, A

_{st}= bottom longitudinal steel area, A′

_{st}= top longitudinal steel area, ${\rho}_{s}$ = ratio of bottom longitudinal steel area, ${\rho}_{s}^{\prime}$ = ratio of top longitudinal steel area; dimensions in mm.

Parameter | Description | Value |
---|---|---|

f_{cu} | Cube compressive strength | 37.8 MPa |

f′_{c} | Cylinder compressive strength | 32.1 MPa |

f_{t} | Tensile strength | 1.4 MPa |

E_{c} | Elastic modulus | 3.3 × 10^{4} MPa |

μ | Poisson’s ratio | 0.2 |

ε_{loc}_{,} _{f} | Localized fracturing strain | ≈0 |

L_{ch}_{,} _{t} | Tensile characteristic length | 0.03 m |

ε_{loc}_{,} _{p} | Localized plastic strain | 9.7 × 10^{−4} |

L_{ch}_{,} _{c} | Compressive characteristic length | 0.1 m |

Specimen | Support Type * | Numerical Results | Experimental Results | Ratios | |||
---|---|---|---|---|---|---|---|

Total Load | Mid-Span Deflection ‴ | Total Load | Mid-Span Deflection ‴ | ||||

P_{FE} | Δ_{FE} | P_{Exp} | Δ_{Exp} | P_{FE}/P_{Exp} | Δ_{FE}/Δ_{Exp} | ||

(kN) | (mm) | (kN) | (mm) | ||||

L10-40 | RHR | 781 | 1.4 | 724 | 1.3 | 1.08 | 1.08 |

HRH | 1394 | 0.8 | 1.93 | 0.62 | |||

HHH | 1394 | 0.8 | 1.93 | 0.62 | |||

FHF | 1441 | 0.9 | 1.99 | 0.69 | |||

L10-60 | RHR | 871 | 1.3 | 883 | 1.6 | 0.99 | 0.81 |

HRH | 1212 | 1.0 | 1.37 | 0.63 | |||

HHH | 1219 | 1.0 | 1.38 | 0.63 | |||

FHF | 1254 | 1.0 | 1.42 | 0.63 | |||

L10-72 | RHR | 994 | 1.8 | 1008 | 2.1 | 0.99 | 0.86 |

HRH | 1311 | 0.9 | 1.30 | 0.43 | |||

HHH | 1311 | 0.9 | 1.30 | 0.43 | |||

FHF | 1400 | 1.0 | 1.39 | 0.48 |

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**MDPI and ACS Style**

Mansour, M.; El-Ariss, B.; El-Maaddawy, T.
Effect of Support Conditions on Performance of Continuous Reinforced Concrete Deep Beams. *Buildings* **2020**, *10*, 212.
https://doi.org/10.3390/buildings10110212

**AMA Style**

Mansour M, El-Ariss B, El-Maaddawy T.
Effect of Support Conditions on Performance of Continuous Reinforced Concrete Deep Beams. *Buildings*. 2020; 10(11):212.
https://doi.org/10.3390/buildings10110212

**Chicago/Turabian Style**

Mansour, Moustafa, Bilal El-Ariss, and Tamer El-Maaddawy.
2020. "Effect of Support Conditions on Performance of Continuous Reinforced Concrete Deep Beams" *Buildings* 10, no. 11: 212.
https://doi.org/10.3390/buildings10110212