## 2 Engineering Background  
This study used a stope in an underground hard rock metal mine as the engineering background, utilizing theoretical analytical methods to study its surrounding stress field and the mine pillar parameter design. The mining area consists of multiple ore belts, among which, the northern ore belt has reached detailed exploration, with a significant amount of controlled level ore. Both the hanging wall and footwall of the ore body in this belt are made of conglomerate, sandstone silty sandstone, with fine- to medium-grain dense blocky structure, exhibiting hard and stable characteristics without the need for support. Fractures within the ore body are not well-developed and have poor penetration; the ore is of sedimentary metamorphic type relatively hard. To improve the initial economic benefits of the mine and shorten the investment payback period, the mine initially aims to recover ore bodies with greater thickness and higher grades, adhering to a bottom-up mining sequence. The design of the first mining section includes a substantial part of the ore body in the northern ore belt at Level 650 and involves sublevel open stoping followed by backfilling.  
Figures 1 and 2 show a schematic diagram of Level 650 stope layout and cross-section of Stope 101 at Level 650 in the northern ore belt, respectively, where the red dashed lines denote the boundaries of the stopes. The designed length of Stope 101 is 116 m. At 51 m from the headentry, there is a gangue with a thickness of about 8 m. To improve the recovery rate and reduce mining standards, it is planned to reserve the gangue area as a  
mine pillar. In this study, we used Stope 101 as the research object, employing theoretical analytical solutions for the surrounding stress field of the stope to analyze the mine pillar's design parameters and stope stability. As the mechanical properties of the ore body and surrounding rock in the study area are very similar and exhibit good integrity, they can be analyzed as homogeneous materials. The uniaxial compressive strength of the rock mass in this section is 157.0 MPa. The original stress field of Level 650 was taken as the external stress field in the model, with three principal stresses of 12.3 MPa, 11.6 MPa 9.8 MPa. The direction of the maximum principal stress is essentially consistent with the direction of gravity; hence, in the two-dimensional mathematical model, the rotation of the opening or shear stress need not be considered. The horizontal stress was taken as the average value, the vertical stress was 12.3 MPa, and the lateral pressure coefficient \(\lambda\) was 0.86. According to the actual engineering needs, three sets totaling 27 models were established. The first set of the models was used to study the characteristics of surrounding rock stress around the roadways of a single stope. The second set of the models was used to study the influence of the adjacent roadway on the existing roadway. The third set of the models, based on the distribution characteristics of the gangue in the studied stope, was used for the pillar's parameter optimization design. The above three sets of mathematical calculation models were analyzed from the stress field perspective for the stope stability and pillar parameter design in accordance with the actual operational sequence. These mathematical calculation models were solved using self-developed Python 3.11 and MATLAB 2021 programs.