Mathematical Problems in Engineering

Volume 2018, Article ID 8412620, 10 pages

https://doi.org/10.1155/2018/8412620

## Peridynamic Model for the Numerical Simulation of Anchor Bolt Pullout in Concrete

^{1}School of Civil Engineering & Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China^{2}Balochistan University of Information Technology, Engineering and Management Sciences, Baleli, Quetta, Pakistan

Correspondence should be addressed to Yaoting Zhang; nc.ude.tsuh@5691tyz

Received 13 September 2017; Accepted 31 January 2018; Published 21 March 2018

Academic Editor: Roman Wendner

Copyright © 2018 Jiezhi Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–6]. Likewise, various numerical methods have been adopted to analyze the failure mechanism and the progressive damage of anchor bolts in concrete structures.

Among previous works, most of the researchers focused on the finite element method (FEM) [7–11]. One of the early methods [7] includes modeling of concrete as the four-node cracked element. In this approach, the tensile fracture behavior of the concrete for a solid body containing an internal discontinuous surface is formulated by deriving an incremental formulation. As a result, the crack is distributed in each grid and the localization of microcracking cannot be obtained. In order to better simulate the mode-I fracture, Alfaiate et al. [8] utilized the interface elements which are inserted along the interelement boundaries among the concrete elements to model the cracking path. Although multiple cracks can be obtained and no special remeshing technique is required with such approach, the crack direction is still restricted and ladder-shaped crack paths are formed.

Etse [9] predicted the distribution of equivalent fracture strain at peak load of the anchor system by adopting a fracture energy-based plasticity formulation. The propagation of crack is described in terms of an equivalent plastic softening process, but still it is difficult to obtain the final cracking pattern by this approach. Xu et al. [10] simulated the crack patterns and mechanical behavior of the anchor bolt pullout in concrete by using the Mohr–Coulomb criterion with tension cut-off. In this approach, the heterogeneity of the concrete material is modeled by randomly assigning strength and elastic modulus to the elements according to Weibull’s distribution and the ongoing cracking process is represented by groups and alignments of failed elements. Feenstra [11] used the smeared crack approach to study two-dimensional pullout problems. In this approach, it is assumed that a crack can be distributed over a special band in the model and its influence on the mechanical behavior of a material can be represented by adjusting the constitutive matrix irrespective of the real displacement discontinuities within the band. It is obvious that these aforementioned approaches had overcome some deficiencies of crack propagation problems, but still these approaches require remeshing or redefining of the geometry to model the progressive crack growth. Furthermore, the accuracy of results heavily relies on the complex adaptive meshing algorithms [12]. Therefore, the low efficiency and accuracy of simulation are still a major problem [12–14].

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 [15, 16]. Soparat and Nanakorn [14] and Coetzee et al. [17] used mesh-free methods for pullout problems. Though many discussions on the advantages of the mesh-free methods have been reported [15–18], it should still be noted that the treatment of essential boundary conditions is not straightforward as the conventional FEM and the use of shape functions of any desired order of continuity may lead to computational difficulties and complexity in deriving the coefficients of the stiffness matrix [18, 19]. Other contributions can be found in the literature regarding anchors’ modeling using the microplane model [20, 21].

Silling and Askari [22, 23] proposed peridynamics as an alternative technique for the solution of crack related problems where direct displacement is used in the formulations instead of displacement derivatives, and deformation continuity is not based on assumptions. Mainly, bonds containing constitutive information of the materials are used to reproduce the nonlocal interacting forces between particles over a certain distance. Moreover, in contrast to the partial differential equations used in the classical formulation, this theory uses spatial integral equations which permit spontaneous crack occurring at multiple sites and freely extend along an arbitrary path without extra remedial techniques [24]. Therefore, the peridynamic theory has obvious advantages in handling cracks problems and predicting the progressive failure process in solid mechanics. Numerous achievements have been obtained using this model during the past few years including deformation of one-dimensional bar [25, 26], progressive damage of composite laminates [27, 28], damage and fracture of membranes and fibers [29], static and dynamic fracture of plain and reinforced concrete structure [30–34], dynamic fracture in functionally graded materials [35], and coupling with classical continuum mechanics [36].

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.

#### 2. Introduction of the Peridynamic Theory (PD)

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 modeled as a discrete set of particles and each particle owns a subregion within a certain radius called the material horizon as shown in Figure 1. The peridynamic equation of motion at any time is given by Silling [22] as follows:where denotes the mass density; is the prescribed external body force density; and are the displacement vector and acceleration vector, respectively. Also, is a pairwise force vector in the peridynamic bond that represents the nonlocal interactions between the particle and the rest of the particles. Unlike MD, in peridynamics, the magnitude of depends upon the initial reference configuration and the state of the bond, that is, the relative position vector and the relative displacement vector , as follows: