# Numerical Investigations on Non-Rectangular Anchor Groups under Shear Loads Applied Perpendicular or Parallel to an Edge

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Investigation

#### 2.1. Description of Cases Analyzed

_{ef}= 120 mm and their nominal diameter was d = 20 mm. The base plates used for triangular groups were of size 200 mm × 200 mm and the dimensions of the plates used for hexagonal groups were 250 mm × 250 mm. The thickness of the base plate was 50 mm in every case.

#### 2.2. Numerical Modeling

_{ij}. On the microplane, one normal (σ

_{N}; ε

_{N}) and two shear stress-strain components (σ

_{K}, σ

_{M}; ε

_{K}, ε

_{M}) are considered. The normal microplane stress and strain components are further decomposed into volumetric and deviatoric parts (σ

_{N}= σ

_{V}+ σ

_{D}, ε

_{N}= ε

_{V}+ ε

_{D}) as shown in Figure 3. The microplane stresses are calculated using uniaxial constitutive laws of each microplane component (volumetric, deviatoric, and shear). In the model, the tensorial invariance restrictions do not need to be directly enforced. By employing virtual work approach, the macroscopic stress tensor is obtained as an integral over all predefined 21 microplane directions (symmetric part of the unit sphere shown in Figure 3 using Equation (1).

_{ij}denotes Kronecker delta and k

_{i}and m

_{i}are directions of shear microplane components. It has been shown that this is an optimal number of integration points that still yields acceptable accuracy [19]. Further details of the model can be found in [18]. Damage and cracking are modeled in the framework of the smeared crack continuum. To assure the objectivity of the analysis with respect to the size of the finite elements, the crack band method is used according to Bažant and Oh [20]. For creating the 3D finite element model and for the evaluation of the numerical results, the commercial pre- and post-processing software FEMAP (Siemens) was used.

#### 2.3. Material Parameters

_{c}= 20 MPa. The compressive strength of concrete corresponded to standard cubic compressive strength of 25 MPa and the other parameters were derived using the following expressions.

_{F}is assumed to be 7.

_{max}= 32.5 MPa, residual bond stress τ

_{R}= 8 N/mm², values of slip s

_{1}= 0.01 mm, s

_{2}= 0.2 mm, and s

_{3}= 2 mm. Note that the bond stresses remained within the linear range for all the analysis cases.

## 3. Results and Discussion

#### 3.1. Triangular Anchor Groups

#### 3.1.1. Group Tri-A

#### 3.1.2. Group Tri-B

#### 3.2. Hexagonal Anchor Groups

#### 3.2.1. Group Hex-A

#### 3.2.2. Group Hex-B

## 4. Comparison with Analytical Models

_{1}/c

_{1,1}) less than 1.0 and loaded in shear perpendicular to the concrete edge, the crack origination from the front row is suppressed by the compression field that originates from the back anchor row. Therefore, it is attempted to calculate the failure load of the anchorages analyzed in this work by adapting the models given in EN1992-4 [1] (considering the failure crack from the front anchor row) and fib Bulletin 58 [3] (considering the failure crack from back anchor row).

_{cc}

_{,150}(mean cube compressive strength with side length of a = 150 mm) ${f}_{cm}=0.8\xb7{f}_{cc,150}$;

_{1}= edge distance of the front row according to EN1992-4 [1] and or edge distance of back row according to fib Bulletin 58 [3]);

_{1}is taken as 120mm (corresponding to front anchor row, see Table 1 for calculations according to EN1992-4, while for calculations according to fib Bulletin 58, c

_{1}is taken as 240 mm (corresponding to back anchor row, see Table 1).

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Test setup used by Bokor et al. [12] for conducting shear tests on anchorages of (

**a**) triangular configuration and (

**b**) hexagonal configuration (photos by author).

**Figure 3.**Concept of the microplane model: integration point with microplanes (

**left**) and decomposition of strain vector on the specified microplane (

**right**)—Figure taken from [16].

**Figure 4.**(

**a**) 3D finite element model of GS-Tri-A, (

**b**) base plate and anchos, (

**c**) contact between anchor and concrete and contact between the base plate and concrete surface, (

**d**) bond type bar elements along the anchor shaft and contact type bar elements below the base plate.

**Figure 7.**Crack pattern obtained from numerical analysis for Group Tri-A loaded perpendicular to the edge (

**a**) at peak load, and (

**b**) in post-peak phase.

**Figure 8.**Crack pattern obtained from numerical analysis for Group Tri-A loaded parallel to the edge (

**a**) first cracks, (

**b**) at peak load, and (

**c**) post-peak phase.

**Figure 9.**Crack pattern obtained from numerical analysis for Group Tri-B loaded perpendicular to the edge (

**a**) at peak load, and (

**b**) post-peak phase.

**Figure 10.**Crack pattern obtained from numerical analysis for Group Tri-B loaded parallel to the edge (

**a**) first cracks, (

**b**) at peak load, and (

**c**) post-peak phase.

**Figure 12.**Crack pattern obtained from numerical analysis for Group Hex-A (

**a**) at peak load, and (

**b**) Crack pattern in the post-peak phase.

**Figure 13.**Crack pattern obtained from numerical analysis for Group Hex-A loaded parallel to the edge (

**a**) first cracks, (

**b**) at peak load, and (

**c**) post-peak phase.

**Figure 14.**Crack pattern obtained from numerical analysis for Group Hex-B (

**a**) at peak load, and (

**b**) Crack pattern in the post-peak phase.

**Figure 15.**Crack pattern obtained from numerical analysis for Group Hex-B loaded parallel to the edge (

**a**) first cracks, (

**b**) at peak load, and (

**c**) post-peak phase.

Series ID | Anchor Configuration | Edge Distance of the Corresponding Anchor Row [mm] | Anchor Spacing, s [mm] | |||
---|---|---|---|---|---|---|

c_{11} | c_{12} | c_{13} | c_{14} | |||

Tri-A | 120 | 240 | - | - | 139 | |

Tri-B | 120 | 240 | - | - | 139 | |

Hex-A | 101 | 171 | 240 | - | 80 | |

Hex-B | 80 | 120 | 200 | 240 | 80 |

**Table 2.**Comparison of numerical results with analytically obtained failure loads for the investigated anchor groups loaded perpendicular to the edge.

Series ID | Failure Load from Numerical Analysis, V _{u,num}[kN] | Analytical Mean Failure Loads, V_{u,calc} | Ratio V _{u,num}/V_{u,calc} | ||
---|---|---|---|---|---|

EN1992-4 [1] [kN] | Fib Bull. 58 [3] [kN] | EN1992-4 [1] [-] | Fib Bull. 58 [3] [-] | ||

Tri-A | 96.8 | 49.5 | 88.4 | 1.96 | 1.10 |

Tri-B | 107.9 | 35.7 | 105.4 | 3.02 | 1.02 |

Hex-A | 109.2 | 21.4 | 88.4 | 5.10 | 1.24 |

Hex-B | 97.8 | 36.3 | 98.2 | 2.69 | 1.00 |

**Table 3.**Comparison of numerical results with analytically obtained failure loads for the investigated anchor groups loaded parallel to the edge.

Series ID | Failure Load From Numerical Analysis, V _{u,num}[kN] | Analytical Mean Failure Loads, V_{u,calc} | Ratio V _{u,num}/V_{u,calc} | ||
---|---|---|---|---|---|

EN1992-4 [1] [kN] | Fib Bull. 58 [3] [kN] | EN1992-4 [1] [-] | Fib Bull. 58 [3] [-] | ||

Tri-A | 213.4 | 99.0 | 176.8 | 2.16 | 1.21 |

Tri-B | 230.4 | 71.4 | 210.8 | 3.23 | 1.09 |

Hex-A | 260.7 | 42.8 | 176.8 | 6.09 | 1.47 |

Hex-B | 236.2 | 72.6 | 196.4 | 3.25 | 1.20 |

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**MDPI and ACS Style**

Bokor, B.; Sharma, A.
Numerical Investigations on Non-Rectangular Anchor Groups under Shear Loads Applied Perpendicular or Parallel to an Edge. *CivilEng* **2021**, *2*, 692-711.
https://doi.org/10.3390/civileng2030038

**AMA Style**

Bokor B, Sharma A.
Numerical Investigations on Non-Rectangular Anchor Groups under Shear Loads Applied Perpendicular or Parallel to an Edge. *CivilEng*. 2021; 2(3):692-711.
https://doi.org/10.3390/civileng2030038

**Chicago/Turabian Style**

Bokor, Boglárka, and Akanshu Sharma.
2021. "Numerical Investigations on Non-Rectangular Anchor Groups under Shear Loads Applied Perpendicular or Parallel to an Edge" *CivilEng* 2, no. 3: 692-711.
https://doi.org/10.3390/civileng2030038