# Numerical Simulation of Anchor Pullout and Shear Tests Using a Regularized Damage Model
## 1 Introduction  
In certain sensitive installations, pre-installed anchors embedded in reinforced concrete structures support equipment. These anchors transfer the load from the equipment into the concrete, ensuring stability and structural safety. An anchor typically consists of a plate and one or several studs. There are several types of anchoring systems, including cast-in-place anchors [1; 2; 3] and post-installed anchors [4; 5; 6]. This study focuses specifically on cast-in-place anchors in reinforced concrete, as these are most commonly used in French nuclear power plants. The failure modes of anchors have been extensively studied in technical literature. The three main failure modes include concrete failure, often in the form of a cone shape; steel stub rupture; and stud slip [7; 8] in the case of a pullout loading. Understanding these failure modes is critical for the sizing and safe design of anchors. Various standards for anchor design, such as Eurocode 1992-4 [9] and ACI 318 [10], take into account these failure modes with the assumption of a rigid baseplate. They provide specific recommendations and requirements to ensure that anchors can withstand the applied loads without failure. These standards include safety factors and calculation methods to determine the load-bearing capacity of anchors based on concrete and steel characteristics, as well as loading conditions. Anchor design is based on the Concrete Capacity Design (CCD) method for concrete failure [11], initially developed for single-stud design and later adapted for multi-stud anchor design. An important consideration is the group effect that influences the overall load-bearing capacity of a complete anchor system. This effect occurs when studs are too close together, causing the mechanical forces transmitted through the studs to create cone-shaped stress patterns in the concrete. When these stress cones overlap, the anchor's overall resistance can decrease significantly. Thisphenomenon occurs during concrete failure. Therefore, the resistance of an anchor \(N_{Rk,c}\) can be calculated as follows:  
\[N_{Rk,c}=N_{Rk,c}^{0}\frac{A_{c,N}}{A_{c,N}^{0}}\cdot\psi_{s,N}\cdot\psi_{re,N} \cdot\psi_{ec,N}\cdot\psi_{M,N}\]  
where,  
\(N_{Rk,c}\) is the concrete breakout strength for a single anchor bolt,  
\(\frac{A_{c,N}}{A_{c,N}^{0}}\) is the ratio of the actual projected area of all studs (overlap) to the projected area of a single stud,  
\(\psi_{s,N}\) is a coefficient accounting for the edge disturbance effect,  
\(\psi_{re,N}\) is a coefficient taking into account dense reinforcement,  
\(\psi_{ec,N}\) is a coefficient accounting for the potential eccentricity of the loading,  
\(\psi_{M,N}\) is a coefficient accounting for potential bending in the anchor. The anchor can also be associated with steel rupture, in which case the group effect has no influence and the anchor strength is directly proportional to the strength of a single stud. This design approach is generally complemented by experimental campaigns associated with the development of numerical solutions, which provide more refined information on the behavior of the anchorage systems. Experimentally, the anchor pullout test is the most often used test. Many anchor pullout tests can be found in the literature, including various failure modes and different parameters that affect anchor strength (anchor length and stud dimensions, for example). These studies especially highlight the cone-shaped concrete failure and led to the development of the CCD method. However, this method may prove too conservative in cases where the head of the bolt becomes relatively large compared to the rod diameter. Studying the key parameters of the behavior of anchorage systems, [PERSON] [12; 13] demonstrates the influence of the ratio between anchor length (h\({}_{\text{ef}}\)) and head diameter (d\({}_{\text{h}}\)). The failure mode depends on this ratio. For a ratio of approximately d\({}_{\text{h}}\) > 0.5 h\({}_{\text{ef}}\), failure occurs in the form of a cone shape.