## 1 Introduction
### Mechanical Model and Basic Assumptions  
We take the side wall near the opening as the research object, as shown in Figure 3. There are effective restraints on the side wall at the end, considering the reinforcement of hidden columns at the side wall's entrance and the side wall's box-section form. Therefore, we can simplify the research object's restraint boundaries as three fixed boundaries and one free boundary. Currently, the system comprises three members: the wall columns, the walls, and the frame beams. In the analysis of underground structures, the boundary problem of soil-structure interaction is also a crucial factor affecting the accuracy of structural calculations [19]. The soil properties, changes in the groundwater level, and ground loads can all impact the underground structure. The commonly used calculation methods for underground systems include the load-structure and stratum-structure methods [20]. The load-structure method considers that the action of soil on underground systems is only to generate loads acting on them, including active soil pressure and passive soil resistance. It is commonly used in theoretical analysis of the static characteristics of underground structures and is currently widely used in engineering. The commonly used elastic methods for underground systems, such as the elastic continuous frame method, assumed resistance method, and elastic foundation beam method, are based on this, and this paper considers the earth pressure as a linearly distributed load acting on the side wall based on the load-structure method. Only the earth pressure load factor needs to be adjusted when encountering changes in the groundwater level or ground surcharge. Moreover, the soil can be considered homogeneous in theoretical analysis, which has good applicability for the buried depth of  
Figure 3: Side walls of the opening. the underground space structure of about 15 m. Now, external loads such as lateral water and soil pressure outside the side wall are applied to the system. Side walls, beams, and wall columns next to the soil should meet the serviceability requirements (crack width control), satisfying the antiseepage requirement, i.e., the system can be considered elastic. Therefore, the bending moment at the bottom of the wall column in the middle of the large opening is the control bending moment in the structural design and the necessary condition for calculating the structural section size and reinforcement. The theoretical analysis aims to obtain the control bending moment of the wall column using the sub-zone and sub-item mixed-energy principle. The system can be divided into different sub-zones, i.e., the \(i\)-th wall column belongs to the \(a_{i}\left(I=1-n\right)\) zone, the \(j\)-th wall belongs to the \(b_{i}\left(j=1-n\right)\) zone, and the \(k\)-th frame beam belongs to the \(c_{k}\left(k  
at the beam-column joint. The complementary energy functional of the system can be simplified as Equation (4). \[\Pi=-\sum_{V_{a_{i}}}\Pi_{c}^{a_{i}}-\sum_{V_{b_{j}}}\Pi_{c}^{b_{j}}+\sum_{V_{ab} }\Pi_{cc}^{a_{i}b_{j}} \tag{4}\]  
According to the sub-zone complementary energy principle, the complementary energy functional of the system satisfies Equation (5) as long as it meets all the equilibrium equations both in the elastic sub-zones and at their interfaces. \[\delta\Pi=-\sum_{V_{a_{i}}}\delta\Pi_{c}^{a_{i}}-\sum_{V_{b_{j}}}\delta\Pi_{c }^{b_{j}}+\sum_{V_{ab}}\delta\Pi_{cc}^{a_{i}b_{j}}=0 \tag{5}\]  
where \(\delta\) represents the variational operation. In addition, the side wall's restraint on the adjacent wall column is weak enough to neglect, since the rigidity of the side wall is far less than that of the wall column. The variation of additional complementary energy functional on the interface between the wall and the wall column is approximately zero, shown as Equation (6). \[\delta\Pi_{cc}^{a_{i}b_{j}}=0 \tag{6}\]  
The above assumption ignores the interaction between walls and columns. Then, we bring Equation (6) into Equation (5) and obtain Equation (7). \[\left\{