# Rational Choice of Reinforcement of Reinforced Concrete Frame Corners Subjected to Opening Bending Moment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{failur}

_{e}denotes the opening bending moment causing failure of the corner and M

_{capacity}is a theoretical capacity of an adjacent member, computed as for a RC beam in pure bending. Some chosen results of the laboratory tests of frame corners with different reinforcement details are presented in Table 1. It is worth noting that the laboratory tests were only performed for corners joining elements with the same cross section heights. Example distributions of cracks and failure forms are presented in the works of Johansson [13] and Starosolski [14]. Some recent results concerning reinforced concrete corners are presented in the works of Marzec [15], Wang [16], Berglund and Holström [17], Getachew [18], Haris and Roszevak [19], Abdelwahed [20], and Abdelwahed et al. [21]. They describe laboratory tests, strut-and-tie method approaches, and numerical simulations in various FEM software (e.g., Athena, VecTor2, LS-Dyna).

- (1)
- Strut-and-tie method (S&T)—to calculate the required reinforcement and to calculate the efficiency factor.
- (2)
- Finite element method (FEM) in Abaqus software using the Concrete Damaged Plasticity (CDP) model for concrete—to calculate the efficiency factor and to recreate the history of loading, the yielding of steel and crack development.

## 2. Methodology

#### 2.1. Strut-and-Tie Method

_{Rd,max}according to Eurocode 2 [1] as follows:

- In struts: σ
_{Rd,max}= f_{cd}, where f_{cd}denotes the design compressive strength if a strut is under compression only and 0.6ν’f_{cd}if a strut is also in tension in a perpendicular direction, where (Equation (2)):$${\nu}^{\prime}=1-\frac{{f}_{ck}}{250}$$ - In nodes: CCC node—σ
_{Rd,max}= ν’f_{cd}, CCT node—σ_{Rd,max}= 0.85ν’f_{cd}, CTT node—σ_{Rd,max}= 0.75ν’f_{cd}.

#### 2.2. Concrete Damaged Plasticity Model for Concrete in FEM Analysis

_{c}and d

_{t}are damage parameters and s

_{c}and s

_{t}are stiffness recovery functions in compression and tension, respectively. The yield function in the CDP model is defined according to Equation (5):

_{c}

_{0}to the biaxial compressive strength f

_{b}

_{0}—see Equation (7):

_{b}

_{0}to f

_{c}

_{0}are in the range of 1.10 to 1.16 (Kupfer [34], Lubliner et al. [31]). Parameters β and γ are calculated as (Equations (8) and (9)):

_{T}and K

_{C}define the shape of the yield surface. Typical values of these parameters vary from 0.56 to 0.61 for K

_{T}and from 0.66 to 0.80 for K

_{C}(Szwed and Kamińska [35]). The yield surface can be presented in the meridian plane—see Figure 1.

- (1)
- The stress–strain relationship defining a compressive behavior of concrete, usually in a form of a set of points;
- (2)
- The dilatation angle ψ in the $\overline{p}-\overline{q}$ plane;
- (3)
- The flow potential eccentricity e;
- (4)
- The ratio f
_{b0}/f_{c0}of the biaxial compressive strength to the uniaxial compressive strength; - (5)
- The ratio K of the second stress invariant on the tensile meridian to that on the compressive meridian for the yield function;
- (6)
- The tension behavior of concrete in the post-critical range in Abaqus can be defined in three different ways (see Figure 3), namely, as coordinates of points on σ–ε
_{in}curve in a tabular form called STRA in Abaqus code (Figure 3a), σ–u_{cr}curve called DISP (Figure 3b), or the fracture energy G_{f}called GFTEN (Figure 3c).

_{cr}relation and u

_{cr,m}= 2G

_{f}/f

_{t}. Because the model formulation is a continuous one and defined in terms of the stress–strain relation rather than stress–displacement, in the numerical implementation the options DISP and GFTEN are transformed to the σ–ε

_{in}relation depending on the size of the given finite element based on the so-called crack band approach (Bažant and Oh [37]).

^{−5}s to 10

^{−4}s. The calibration and validation of the dilatation angle and relaxation time are described in the next section.

## 3. Calibration and Validation of CDP Model Parameters

#### 3.1. Uniaxial and Biaxial Compression Tests

_{c}= 34.30 MPa; tensile strength f

_{t}= 3.5 MPa; elastic modulus E

_{c}= 35 GPa; Poisson’s ratio ν

_{c}= 0.167; fracture energy G

_{f}= 146.5 N/m. Four values of the dilatation angle were tested, namely 0, 5, 15, and 30 degrees. The relationship between the volumetric strain ε

_{v}and linear strain ε

_{11}is shown in Figure 5 and Figure 6. For relatively small values of the dilatation angle (0 to 15 degrees) the volumetric strains obtained in numerical computations were similar to those of Kupfer [34]. In the laboratory tests the volumetric strains were negative, which means the compaction of concrete. In the numerical simulations they remained negative when the dilatation angle was in the range of 0 to 15 degrees. In the case of 30 degrees, large positive volumetric strains in the post-critical range occurred. For that reason, the authors suggest using relatively low values of the dilatation angle

**in the range of 0 to 15 degrees**. If the higher values are used the stiffness and bearing capacity of concrete elements can be overestimated in the case of confinement, e.g., in the plane strain case.

#### 3.2. Uniaxial Tension Test

_{c}= 34.30 MPa; tensile strength f

_{t}= 3.5 MPa; elastic modulus E

_{c}= 35 GPa; Poisson’s ratio ν

_{c}= 0.167. The tensile behavior of concrete is defined as a set of points on the σ–u

_{cr}curve taken from Woliński’s research—see Figure 8.

**is not the value**of the relaxation time itself but the ratio μ/t. In order to obtain results that are close enough to reality the authors recommend using

**a value of μ/t equal to 0.0001 or less**.

## 4. Strut-and-Tie and FEM Analysis of Corners under Opening Bending Moment

- Concrete: f
_{c}= 34.30 MPa, E_{c}= 35 GPa, ν = 0.167, f_{t}= 3.5 MPa, G_{f}= 146.5 N/m; - Reinforcing steel: f
_{y}= 434.8 MPa, E_{s}= 200 GPa, ν = 0.3.

_{ref}= 30 kNm modeled as a pair of forces—see Figure 14 and Figure 15. Seven different reinforcement details taken into consideration are listed in Table 2. In this section, corners are subjected to a pure bending moment, but in the Section 5.2. there is an example of a calculation where the moment is accompanied with normal and shear forces, which is a very important case in practice.

#### 4.1. Calculations in the Strut-and-Tie Method

#### 4.1.1. The Case of Elements with the Same Cross Section Heights

_{ref}= 30 kNm. The main reinforcement of the elements joined in the corner was chosen as a pair of 2ϕ20, top and bottom (the reinforcement ratio calculated for the 200-mm-high elements joined in the corner equal to 0.0262). For the main reinforcement, the carrying capacity of the RC cross section was computed according to Eurocode 2 [1] (for cross section dimensions—see Figure 12, section A-A and Figure 14, section B-B) leading to M

_{capacity}= 32.7 kNm and the ratio (Equation (12)):

_{i}is the proportionality factor and the subscript i denotes a number of the S&T scheme element (i = 1, 2, 3 … N, where N is a number of the elements). Because, by definition, the S&T method is linear, for the bending moment at the point of failure the same relation is valid (Equation (15)):

_{i}obtained in the S&T analysis for M

_{ref}= 30 kNm (Equation (17)):

#### 4.1.2. The Case of Elements with Different Cross Section Heights

#### 4.2. Calculations in FEM

#### 4.2.1. The Case of Elements with the Same Cross Section Heights

_{ref}= 30 kNm modeled as a pair of forces. The reinforcement was first calculated with the S&T method and then put in the FEM model. The corners are calculated in both the plane stress and the plane strain states. The applied load was defined with a load parameter λ whose value of 1 represents the reference value of the opening bending moment M

_{ref}= 30 kNm. As presented in Section 4.1 the carrying capacity value of the bending moment is M

_{capacity}= 32.7 kNm. The FE analysis, on the other hand, is performed in a similar manner to the S&T analysis using the reference value of the bending moment M

_{ref}. Therefore, the load parameter is defined as (Equation (18)):

_{f}option was set as input in Abaqus. The used values of the CDP parameters are listed in Table 6.

_{t}is a finite element width perpendicular to the direction of the crack, ε

_{cr}denotes tensile strain in the cracked element, γ

_{max}is assumed as equal to 1.5, and θ is the crack propagation angle (given in degrees).

#### 4.2.2. The Case of Elements with Different Cross Section Heights

#### 4.2.3. Influence of Reinforcement Ratio on the Efficiency Factor

#### 4.2.4. Dependence of the Results on the Diagonal Reinforcement Area

## 5. Comparison of Numerical Results with Laboratory Tests

#### 5.1. Efficiency Factors for the Case of Elements with the Same Cross Section Heights

#### 5.2. Comparison with Johansson’s Laboratory Tests

- Concrete: f
_{c}= 32.2 MPa, E_{c}= 31 GPa, ν = 0.2, f_{t}= 2.6 MPa, G_{f}= 136.4 N/m, - Reinforcing steel: f
_{y}= 570 MPa, E_{s}= 200 GPa, ν = 0.3.

## 6. Conclusions

- (1)
- It is possible to choose a rational reinforcement detail for a corner under opening moment even in the case of elements with different cross section heights using a combination of the S&T method and FEM.
- (2)
- (3)
- There are no significant differences in the obtained results when applying the different areas of the diagonal bars (2ϕ8, 2ϕ12 or 2ϕ16 mm).
- (4)
- The use of diagonal bars only is insufficient to obtain a satisfactory efficiency factor and a limited crack width; these goals can only be achieved by using diagonal stirrups.
- (5)
- The reinforcement ratio of the adjacent elements has a large influence on the efficiency factor of the corner, namely an increase in the reinforcement ratio causes a decrease in the efficiency factor.
- (6)
- It is possible to assume simpler and still appropriate S&T truss schemes than that used in Eurocode 2 [1] and handbooks for a corner under opening bending moment, even in the case of the use of corner elements with different cross section heights.
- (7)
- When using the CDP model its parameters should be assumed in a careful way. The authors recommend calibrating and validating some of these parameters. To obtain realistic results, the authors propose the following values of CDP parameters (see the discussion in Section 3):
- -
- A relaxation time of 0.0001 s or less (for the loading time 1 s);
- -
- A dilatation angle of 5 to 15 degrees.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Volumetric strain vs. longitudinal strain for different dilatation angle values in the uniaxial compression test.

**Figure 6.**Volumetric strain vs. longitudinal strain for different dilatation angle values in the biaxial 1:1 compression test.

**Figure 11.**(

**a**) Crack pattern, relaxation time equal to 0. (

**b**) Crack pattern, relaxation time equal to 0.0001 s. (

**c**) Crack pattern, relaxation time equal to 0.001 s.

**Figure 12.**Geometry and main reinforcement for the case of elements with the same cross section heights; dimensions in (mm).

**Figure 13.**Geometry and main reinforcement for the case of elements with different cross section heights; dimension in (mm).

**Figure 14.**Geometry and main reinforcement for the case of elements with different cross section heights; dimensions in (mm).

**Figure 15.**Loading scheme for the case of elements with different cross section heights; dimensions in (mm).

**Figure 16.**Truss schemes assumed for each reinforcement detail in the S&T method; dimensions in (mm).

**Figure 17.**Truss schemes assumed for each reinforcement detail in the S&T method; dimensions in (mm).

**Figure 18.**Meshing of concrete and reinforcing steel (detail NO. 7) and analyzed nodal displacement.

**Figure 20.**Equivalent plastic strains in tension PEEQT for chosen reinforcement details in the plane stress state.

**Figure 23.**Meshing of frame corner and analyzed nodal displacement for the case of elements with different cross section heights.

**Figure 24.**Equivalent plastic strains in tension PEEQT for the chosen reinforcement details in the plane stress state.

**Figure 27.**Corner efficiency factor vs. reinforcement ratio—detail No. 4, (

**a**) the same section heights, (

**b**) different section heights.

**Figure 28.**Nodal displacement vs. loading parameter for different diagonal reinforcement areas, (

**a**) in the plane stress state, (

**b**) in the plane strain state.

**Figure 29.**DAMAGET maps in the plane stress state for different areas of diagonal reinforcement: (

**a**) 2ϕ8 mm, (

**b**) 2ϕ12 mm, (

**c**) 2ϕ16 mm bars.

**Figure 30.**Crack width for different areas of diagonal reinforcement, (

**a**) the plane stress state, (

**b**) the plane strain state.

**Figure 35.**A sketch of the crack pattern obtained in Johansson’s laboratory tests (reprinted with permission from Johansson [13], Copyright 2000, M. Johansson).

Reinforcement Detail | Efficiency Factor | Reinforcement Detail | Efficiency Factor |
---|---|---|---|

0.43 (Mayfield et al. [4]) 0.66 (Kordina and Wiedemann [7]) 0.46 (Al-Khafaji et al. [11]) | 0.78 (Mayfield et al. [4]) | ||

0.59 (Mayfield et al. [4]) 0.55 (Skettrup et al. [9]) | 1.32 (Kordina and Wiedemann [7]) 1.13 (Skettrup et al. [9]) | ||

1.07 (Moretti and Tassios [12]) | 1.12 (Moretti and Tassios [12]) | ||

0.88 (Mayfield et al. [4]) |

Detail No. 1 | Detail No. 2 | Detail No. 3 |

Detail No. 4 | Detail No. 5 | Detail No. 6 |

Detail No. 7 | Detail No. 8 | Detail No. 9 |

Source | Dilatation Angle ψ [°] |
---|---|

Jankowiak [43] | 49 |

Genikomsou and Polak [40] | 38 |

Mostafiz et al. [44] | 38 |

Kmiecik and Kamiński [45] | 36 |

Malm [46] | 25–38 |

Menetrey [47] | 10 |

Mostofinejad and Saadatmand [48] | 0 |

Marzec [49] | 8 or 10 |

Rodriguez et al. [50] | 30 |

Urbański and Łabuda [51] | 15 |

**Table 4.**Efficiency factors and provided reinforcement for each reinforcement detail—the case of elements with the same cross section heights (maximal value in bold).

Detail No. | Efficiency Factor Obtained in S&T | Decisive Element (No. of Strut or Node) | Provided Reinforcement, Diameters Given in (mm) |
---|---|---|---|

1. | 0.70 | Strut No. 5 | main reinforcement: 2ϕ20 top and 2ϕ20 bottom, no loops |

2. | 0.64 | Strut No. 9 | 2 diagonal bars ϕ8 each, no loops |

3. | 0.66 | Strut No. 5 | 2 diagonal bars ϕ8 each, no loops |

4. | 0.94 | Strut No. 5 | diagonal stirrup ϕ16, looped main bars |

5. | 1.21 | Nodes N1, N2, N4, N5 | central diagonal stirrup ϕ16, outside stirrups ϕ10, looped main bars |

6. | 1.25 | Nodes N2, N3, N4, N5 | central diagonal stirrup ϕ16, outside stirrups ϕ12, looped main bars |

7. | 1.31 | Struts No. 14 and 15 | 2 diagonal bars ϕ16 each, central diagonal stirrup ϕ12, outside stirrups ϕ16, looped main bars |

**Table 5.**Efficiency factors and provided reinforcement for each reinforcement detail—the case of elements with different cross section heights (maximal value in bold).

Detail No. | Efficiency Factor Obtained in S&T | Decisive Element (No. of Strut or Node) | Provided Reinforcement, Diameters Given in (mm) |
---|---|---|---|

1. | 0.61 | Strut No. 5 | main reinforcement: 2ϕ20 top and 2ϕ20 bottom in a smaller cross section (column), and 2ϕ12 top and 2ϕ12 bottom in a larger cross section (beam), no loops |

2. | 0.58 | Strut No. 5 | 2 diagonal bars ϕ8 each, no loops |

3. | 0.60 | Strut No. 5 | 2 diagonal bars ϕ8 each, no loops |

4. | 1.44 | Nodes N1, N2, N3 | diagonal stirrup ϕ16, looped main bars |

5. | 1.23 | Nodes N1, N2, N5 | diagonal stirrups ϕ12, looped main bars |

6. | 1.15 | Nodes N2 and N3 | diagonal stirrups ϕ12, looped main bars |

7. | 1.17 | Node N7 | 2 diagonal bars ϕ16 each, central diagonal stirrup ϕ16, outside stirrups ϕ12, looped main bars |

8. | 1.07 | Nodes N3 and N4 | diagonal bar ϕ8, diagonal stirrups ϕ12, looped main bars |

9. | 0.88 | Nodes N1 and N2 | 2 horizontal stirrups ϕ8 each, diagonal bar ϕ8, looped main bars |

Dilatation Angle ψ (degree) | Flow Potential Eccentricity e | Ratio f_{b}_{0}/f_{c}_{0} | Ratio K | Viscosity Parameter μ (s) | Fracture Energy G_{f} (N/m) |
---|---|---|---|---|---|

15 | 0.1 | 1.16 | 0.667 | 0.0001 | 146.5 |

**Table 7.**Efficiency factors for all reinforcement details obtained using all methods (maximal values in bold).

Detail No. | Efficiency Factor S&T | Efficiency Factor FEM, Plane Stress | Efficiency Factor FEM, Plane Strain |
---|---|---|---|

1. | 0.70 | 0.75 | 1.01 |

2. | 0.64 | 0.79 | 1.11 |

3. | 0.66 | 0.82 | 1.17 |

4. | 0.94 | 1.23 | 1.26 |

5. | 1.21 | 1.24 | 1.29 |

6. | 1.25 | 1.27 | 1.29 |

7. | 1.31 | 1.23 | 1.32 |

**Table 8.**Efficiency factors for all reinforcement details obtained using all methods (maximal values in bold).

Detail No. | Efficiency Factor S&T | Efficiency Factor FEM, Plane Stress | Efficiency Factor FEM, Plane Strain |
---|---|---|---|

1. | 0.61 | 0.75 | 1.31 |

2. | 0.58 | 0.78 | 1.36 |

3. | 0.60 | 0.89 | 1.43 |

4. | 1.44 | 1.32 | 1.40 |

5. | 1.23 | 1.57 | 1.40 |

6. | 1.15 | 1.34 | 1.38 |

7. | 1.17 | 1.54 | 1.51 |

8. | 1.07 | 1.32 | 1.52 |

9. | 0.88 | 1.38 | 1.44 |

**Table 9.**Efficiency factors obtained in authors’ analyses and in laboratory tests—the case of elements with the same cross section heights (maximal values in bold).

Detail No. | Efficiency Factor in Laboratory Tests | Efficiency Factor S&T | Efficiency Factor FEM, Plane Stress | Efficiency Factor FEM, Plane Strain |
---|---|---|---|---|

1. | 0.43 (Mayfield et al. [4]) 0.66 (Kordina and Wiedemann [7]) 0.46 (Al-Khafaji et al. [11]) | 0.70 | 0.75 | 1.01 |

2. | 0.59 (Mayfield et al. [4]) 0.55 (Skettrup et al. [9]) | 0.64 | 0.79 | 1.11 |

3. | 1.07 (Moretti and Tassios [12]) | 0.66 | 0.82 | 1.17 |

4. | 0.88 (Mayfield et al. [4]) | 0.94 | 1.23 | 1.26 |

5. | 0.78 (Mayfield et al. [4]) | 1.21 | 1.24 | 1.29 |

6. | 1.32 (Kordina and Wiedemann [7])1.13 (Skettrup et al. [9]) | 1.25 | 1.27 | 1.29 |

7. | 1.12 (Moretti and Tassios [12]) | 1.31 | 1.23 | 1.32 |

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## Share and Cite

**MDPI and ACS Style**

Szczecina, M.; Winnicki, A. Rational Choice of Reinforcement of Reinforced Concrete Frame Corners Subjected to Opening Bending Moment. *Materials* **2021**, *14*, 3438.
https://doi.org/10.3390/ma14123438

**AMA Style**

Szczecina M, Winnicki A. Rational Choice of Reinforcement of Reinforced Concrete Frame Corners Subjected to Opening Bending Moment. *Materials*. 2021; 14(12):3438.
https://doi.org/10.3390/ma14123438

**Chicago/Turabian Style**

Szczecina, Michał, and Andrzej Winnicki. 2021. "Rational Choice of Reinforcement of Reinforced Concrete Frame Corners Subjected to Opening Bending Moment" *Materials* 14, no. 12: 3438.
https://doi.org/10.3390/ma14123438