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Predictive simulation of anchor pullout from concrete structures is not only a serious problem in structural mechanics but also very important in structural design safety. In the finite element method (FEM), the crack paths or the points of crack initiation usually need to be assumed in advance. Otherwise, some special crack growth treatment or adaptive remeshing algorithm is normally used. In this paper, an extended peridynamic method was introduced to avoid the difficulties found in FEM, and its application on anchor bolt pullout in plain concrete is studied. In the analysis, the interaction between the anchor bolt and concrete is represented by a modified short-range force and an extended bond-level model for concrete is developed. Numerical analysis results indicate that the peak pullout load obtained and the crack branching of the anchoring system agreed well with the experimental investigations.

Anchor bolts are very important components of load transfer in a wide range of civil engineering structures such as dams, nuclear power plants, highways, and bridges. A better understanding of the pullout behavior of anchor bolts can contribute not only to the optimization of the design of the anchor system, but also to the improvement of the durability and stability of a structure. Therefore, the pullout behavior of anchor bolts in concrete structures has become a major concern in the past three decades and a lot of experimental studies have been performed [

Among previous works, most of the researchers focused on the finite element method (FEM) [

Etse [

Recent developments of mesh-free (or meshless) methods such as diffuse element method (DEM), material point method (MPM), and element free Galerkin method (EFGM) are invented to circumvent the mesh-dependence problem and relieve the volumetric locking for suitable choice of support size of shape function [

Silling and Askari [

In this paper, an extended bond-level model for concrete is proposed and the interaction between the anchor bolt and concrete is represented by a modified short-range force. The analysis of anchor bolt pullout problem is carried out using the bond-based peridynamic (PD) model. Moreover, the capabilities of the improved numerical method to capture the progressive damage process and the extreme load of the anchor bolt are validated. Finally, a parametric study was performed to investigate the influence of the size of the horizon and the embedded length. Comparison of the experiments and simulations with those in the literature is also carried out.

The PD theory may be viewed as a special version of particle method or mesh-free method. It is based on assumptions that an object possesses a spatial domain

Pairwise interaction between

The direction of the pairwise force vector can be expressed as

A two-dimensional plate subjected to uniform deformation.

Evaluation of fracture energy.

As peridynamics do not need the continuous displacement field, there are no concepts of stress or strain required in the model. Thus, the constitutive model is defined through the relationship between the bond stretch and the pairwise force among material particles or, in other words, the material damage is introduced at the bond level. Analogous to the softening function proposed by Gerstle et al. [

Constitutive model of concrete and steel.

Concrete

Steel

As shown in Figure

Stress is transferred mainly by adhesion, mechanical interaction, and friction between steel and concrete. Adhesion comes from chemical bonding and stresses are generated during curing of concrete and experimentally it is very tough to measure them. Due to the limited experimental information, it is difficult to determine the friction coefficient. As stresses due to adhesion and friction are relatively small, therefore, only mechanical interaction is considered in the present simulation. In this paper, we introduce a short-range force [

Peridynamic equations of motion (see (

Here,

Finally, with the assumptions that

Both the numerical algorithm and constitutive modeling are implemented in Fortran-90 language based on Visual Studio using an in-house peridynamic code.

Because of the distinctive advantages in solving crack propagation problems, the peridynamic method is adopted in the present study to model the anchor bolt pullout in concrete. The anchor bolt pullout experiment conducted by Vervuurt et al. [

Geometrical parameters used in the simulation.

Group ID | | | | | | |
---|---|---|---|---|---|---|

1 | 300 | 300 | 100 | 50 | 15 | 5 |

2 | 600 | 600 | 200 | 100 | 30 | 10 |

3 | 900 | 900 | 300 | 150 | 45 | 15 |

Specimen geometry and the pullout test setup.

Material parameters used in the simulation.

Material type | Concrete | Steel |
---|---|---|

Young’s modulus | 30 | 200 |

Yield or tensile strength | 3 | 400 |

Compressive strength | 30 | - |

Fracture energy | 100 | - |

Density ^{3}) | 2500 | 7890 |

Numerical model.

To investigate the influence of the size of material horizon on the load-displacement response of the anchor bolt, three different material horizon sizes (i.e.,

Load-displacement response with different horizon sizes.

Comparison of the numerical load-displacement curve with past research work.

Figure

Comparison of the peak loads.

Group ID | This paper (KN) | Experimental results (KN) | Relative error (%) |
---|---|---|---|

1 | 17.04 | 13.4 | 27.2% |

2 | 28.86 | 24.5 | 17.8% |

3 | 37.48 | 33.6 | 11.6% |

Load-displacement response for different groups.

The values of the predicted peak loads in the present study are a little higher than the experimental values which may be because of the higher Poisson’s ratio; 0.33 instead of normal 0.3, based on the peridynamic theory [

Although the precision obtained by the current model is not very high as compared to EFGM in peak load prediction, it was based on many simplifications [

Comparison of crack patterns (PD versus EFGM).

Group 1 (

Group 2 (

Group 3 (

The crack pattern from experimental results is different from most of the previous numerical methods results [

Comparison of failure modes (PD versus EXP).

Group 1 (

Group 2 (

Group 3 (

In this paper, the problem of anchor bolt pullout in plain concrete was investigated with an improved peridynamic model. An extended constitutive model was used to refine the behaviors of concrete material simulation. A short-range force was introduced to simulate anchor bolt and concrete interaction in 2D. The numerical discretization and iteration algorithms were implemented with an in-house peridynamic FORTRAN code. It was observed that the crack propagation of concrete was exposed in more detail by the proposed approach as compared to conventional FEM or EFGM. Compared with the results from the literatures and experiments, it can be concluded that the extreme failure load and the final failure mode of the anchor bolt by analysis of the peridynamic approach match well those of the experimental observations. From all the results and comparisons, this approach was proved to be a promising method for solving the problem of anchor bolt pullout in plain concrete.

The mass density

The material horizon

The prescribed external body force density

The displacement vector

The relative position vector

The relative displacement vector

The pairwise force function

The pairwise potential function

The influence function

The micromodulus function

The fracture energy

The damage of material point

The tensile stretch limit of steel

The compressive stretch limit of concrete

The tensile stretch limit of concrete

The adjustment coefficients

The equivalent calculation volume

The resultant force density vector

The damping coefficient at the

The modified density at the point

The time step size

The diagonal stiffness matrix.

The authors declare that there are no conflicts of interest regarding the publication of this paper.