# Numerical Simulation of Anchor Pullout and Shear Tests Using a Regularized Damage Model
## 3 Simulation of Single-Stud Tests  
### Simulation with Deformable Stud  
The first step of the validation process is the simulation of single-stud tests to validate the constitutive laws and verify that various failure mechanisms are accurately represented. The simulation is first performed on a test described in [3]. The test consists of a pullout test of a headed stud in reinforced concrete (Figure 2). The dimensions of the concrete block are approximately 2 m \(\times\) 2 m \(\times\) 0.6 m. The diameter of the stud head is 60 mm, and the diameter of the rod is 20 mm. The numerical boundary conditions are the same as in the experimental setup. Experimentally, the boundary conditions are created by metal bars anchored to the ground. Figure 1: Constitutive law for concrete (tension (**a**) and compression (**b**)). Figure 2: Setup of the single-stud test [3]. For the simulation, the boundary conditions will be applied to the concrete mesh, using the same geometry as in the experiment, by applying a constraint to the relevant nodes. To reduce the computational time, the simulations utilize symmetry axes, representing a quarter of the experiments. Additionally, perfect bond conditions are assumed between the reinforcement and the concrete that impose the same displacements between both the materials through additional cinematic relations [36]. The forces are mainly transmitted through the head of the stud, given the size of the stud head and the low adhesion between the rod and the concrete, considering the friction between the stud and the concrete to be negligible. Therefore, the interaction between the anchor and the concrete will be modeled as a frictionless sliding contact (unilateral contact condition). As discussed in the previous section, to ensure the model's optimal \"regularization\", the mesh should be as cubic as possible. The elements that risk being damaged and are therefore close to the stud are cubic elements of approximately 1 cm. The more distant elements are less regular and coarser. The concrete elements are represented as linear solid elements (eight nodes per element), the reinforcement elements as 1D truss elements, and the stud elements as quadratic to better model their deformation. Displacement is imposed at the top of the stud (red arrow) (Figure 3). The simulations are performed using the Cast3M software 2024 version [36]. The characteristics of the concrete model are summarized in Table 1; the material parameters of the concrete are derived from tests on specimens made from the concrete used in the experiments. \(G_{\beta}\) was calculated using the formula from the FIB Model Code [41], and \(G\!f_{c}\) was calculated based on the article [35]. Figure 4 presents the comparison between experimental results and simulation in terms of maximum forces versus embedment depth (length of the stud). The magnitudes of the breaking forces are well represented despite a slight overestimation (Figure 4). However, due to the experimental setup, it is observed that the studs undergo random bending, which reduces the steel strength. Simulations, which impose pure tension (without bending), cannot reproduce this behavior, which is highly dependent on the experimental conditions (for each case). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline E (GPa) & \(\mathbf{f_{c}}\) (MPa) & \(\mathbf{f_{t}}\) (MPa) & \(\mathbf{G_{fc}}\) (N/m) & \(\mathbf{G_{ft}}\) (N/m) & \(\mathbf{\epsilon_{c1}}\) & \(\mathbf{\epsilon_{c2}}\) & Mesh Size \\ \hline
43 & 50 & 3 & 60000 & 117 & 0.002 & 0.0023 & \(\simeq\)1 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Concrete parameters. Figure 3: Mesh used for the simulation (**a**), zoom around the stud (**b**). Experimentally, two types of failure are observed. Concrete can fail in the form of a cone, or the stud can break when the steel's ultimate strength is exceeded. Both cases are well represented by the simulations.