## 1 Introduction
### Analysis of Influence Range of Expansion Body Section
Through the stress cloud diagram, the stress concentration area and distribution characteristics of the soil around the ordinary anchor can be clearly observed, and the displacement cloud diagram intuitively reflects the deformation of the soil under the action of the ordinary anchor, including the size and direction of the displacement. Figure 24 further shows the quantitative analysis results of the influence of ordinary anchors on soil, which is embodied in the relationship curve between soil yield area and time. The curve reflects the dynamic change process of soil yield area with the continuous action of the anchor. The slope of the curve can be used to evaluate the development rate of soil yield, and the shape of the curve reveals the yield characteristics of the soil at different time periods. Figure 12:4 The soil displacement cloud diagram in the process of the expansion body extending into the soil (top view) (the size of the soil model is 2 m \(\times\) 1.5 m \(\times\) 1.5 m). Figure 12:5 The influence curve of expansive body on soil (yield area–time). Figure 12:6 The stress cloud diagram of the soil during the process of the ordinary anchor rod extending into the soil (top view) (the size of the soil model is 2m \(\times\) 1.5m \(\times\) 1.5m.). Figure 12:7 The displacement cloud diagram of the soil during the extension of the ordinary anchor into the soil (top view) (the size of the soil model is 2m \(\times\) 1.5m \(\times\) 1.5m.). The analysis and optimization of the umbrella-shaped hollow grouting anchor with an expansion pipe demonstrate a significantly enhanced synergistic performance between the bolt and surrounding rock/soil, validated through rigorous numerical simulations that incorporate the calibrated boundary conditions and theoretical alignment. The stiffness ratio (\(R_{k}=\frac{E_{\text{self}}\cdot A_{\text{self}}}{E_{\text{self}}\cdot A_{ \text{self}}\cdot A_{\text{self}}}\)) increased by 61% (from 6.1 to 9.8), promoting uniform stress redistribution and contributing to a 20.5% higher peak pull-out force (114.7 MPa vs. 95.2 MPa baseline), which is consistent with [PERSON]'s load-transfer theory. Additionally, the bolt's load-bearing proportion improved from 65% to 82%, reducing soil plastic deformation by 30.96% (quantified by a 1.3 reduction in the extreme value of the soil influence curve) under parametric sensitivity analysis, confirming stability (\(\pm\)5%) despite soil modulus variations. The interfacial shear stiffness (\(K_{interface}\)) increased by 50% (from 18.4 MPa/mm to 27.6 MPa/mm), suppressing differential displacement to <2 mm (vs. 5.3 mm in conventional designs), aligning with Coulomb friction principles and prior numerical studies, with mesh convergence tests (displacement variation <3% at 800k elements) and Monte Carlo robustness analysis ensuring model reliability. These advancements reflect a transition from passive load-bearing to active-passive collaborative reinforcement through stiffness matching and interfacial hardening, which is supported by theoretical validation against established rock-bolt interaction frameworks. This provides a numerically robust foundation for geotechnical design optimization, pending future experimental corroboration to further solidify practical applicability. By comparing the simulation results of the expansion anchor and the ordinary anchor in the process of extending into the soil, the following conclusions can be drawn from the analysis of stress distribution, stress diffusion, interaction between anchor and soil, anchor deformation, and bearing capacity. 1. Stress distribution: The expansion bolt shows a more uniform stress distribution in the simulation. This uniformity may be due to the fact that the expansion bolt increases the contact area with the soil during the expansion process, thus effectively dispersing the stress. In contrast, the stress concentration of the ordinary bolt is more obvious due to the small contact area, which may affect its stability and bearing capacity. 2.