## 1 Introduction
### Analysis of Influence Range of Expansion Body Section
Numerical Simulation of the Hollow Grouting Bolt Drawing with an Umbrella-Shaped Expansion Head and an Expansion Pipe  
In order to systematically verify the theoretical calculation results from Section 3, which show that the anti-pulling performance of the hollow grouting anchor with an expansion tube is significantly better than that of the ordinary anchor, this study adopts a multi-software collaborative simulation method based on SOLIDWORKS v2024 to establish an accurate geometric model, which is combined with Abaqus CAE v2023 for numerical analysis. Additionally, a comparative pull-out test of an ordinary hollow grouting anchor and a hollow grouting anchor with an expansion tube under anchor-soil coupling was carried out. The numerical simulation is strictly controlled under the same boundary conditions, and the analysis is mainly carried out from the following three aspects: (1) Identification and verification of mechanical parameters of surrounding rock; (2) Ultimate uplift bearing capacity evaluation of umbrella-shaped enlarged head section; (3) Analysis of the maximum pull-out resistance of the expansive body section. This multi-software co-simulation strategy effectively ensures the reliability and accuracy of the research results. Numerical Simulation of Pull-Out Comparison Between Umbrella-Shaped Enlarged Head Anchorage Section and Ordinary Bolt Anchorage Section  
This study adopts the method of step-by-step modeling and numerical analysis. Firstly, the geometric models of ordinary hollow grouting anchors and umbrella-shaped enlarged-head hollow grouting anchors with expansion tubes were established by SOLIDWORKS v2024 software. The two kinds of anchors are completely consistent in terms of material properties, diameter size, and anchorage length. Subsequently, a borehole and surrounding soil model considering the initial in situ stress field was established in Abaqus CAE v2023 [42; 43]. Because the elastic-plastic model of Abaqus CAE v2023 adopts the calculation framework of the separation of elastic and plastic parts, and this study mainly focuses on the failure mode of the soil around the enlarged head end, the bolt itself remains in the elastic state and so it is described by the isotropic linear elastic constitutive model following [PERSON]'s law. The matrix expression is shown in Equation (40). Considering that the research focus is on the mechanical behavior of the bolt, the soil model does not consider the significant rheological effect and its elastic behavior is also based on a generalized Hookean law, which is characterized by the anisotropic elastic model. The specific constitutive matrix is shown in Equation (41), where the plastic constitutive uses the Mohr-Coulomb yield criterion and simulates the crack propagation and material damage evolution process by embedding cohesive damage particles in the soil model. Equations (40) and (41) are as follows:  
\[\left\{ \begin{array}{c}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{13}\\ \gamma_{23}\end{array} \right\}=\left[ \begin{array}{cccccc}\frac{1}{E}&- \frac{p}{E}&-\frac{p}{E}&0&0&0\\ -\frac{p}{E}&\frac{1}{E}&-\frac{p}{E}&0&0&0\\ -\frac{p}{E}&-\frac{p}{E}&\frac{1}{E}&0&0&0\\ 0&0&0&\frac{1}{C}&0&0\\ 0&0&0&0&\frac{1}{C}&0\\ 0&0&0&0&\frac{1}{C}\end{array} \right]\left\{ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\end{array} \right\} \tag{40}\]\[\left\{ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\end{array} \right\}=\left[ \begin{array}{cccccc}D_{1111}&D_{1111}&D_{ 1111}&D_{1111}&D_{1111}&D_{1111}\\ 0&D_{2222}&D_{2233}&D_{2212}&D_{2213}&D_{2223}\\ 0&0&D_{3333}&D_{3312}&D_{3313}&D_{3323}\\ 0&0&0&D_{1212}&D_{1213}&D_{1223}\\ 0&0&0&0&D_{1313}&D_{1323}\\ 0&0&0&0&0&D_{2323}\end{array} \right\}\left\{ \begin{array}{c}\varepsilon_{11} \\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{13}\\ \gamma_{23}\end{array}