## 1 Introduction  
The construction of large-scale subway stations and tunnel integration projects further promotes the research of complex underground structure engineering. The rationality of design, durability of structures, and safety of construction [1, 2] have been widely concerned. Different soil parameters, including the soil-water characteristic [3], nonlinear stress-strain relationship of soils [4], and structured soil types [5], can have an impact on the design of subway stations and the shield construction [6]. This paper will analyze the control parameters of the underground space structure with the Track Panel Tunnel and wall columns using the complementary energy principle with mixed variables. The track panel, shown in Figure 1, is assembled by two rails on the ground first and hoisted to the metro station for track laying in the process of metro construction. Hence, using the central station's top plates with one or several large openings offers an engineering solution for track laying called the Track Panel Tunnel (TPT) [7], as shown in Figure 2. In addition, the openings are usually set right above the metro line for safety and high hoisting efficiency. As the length of each rail section is almost 25 m, the length and the width of the net size of openings are about 27-30 m and 3.5-5 m, respectively. Therefore, the number of openings is determined according to the needs of the track panel lifting. Side walls near the openings lose the support of the structural slabs and become the weak region [8]. The combination of frame beams around the openings and wall columns of the side wall is widely used in metro station design to improve the out-of-plane stiffness of the structure [9; 10; 11; 12]. The system can resist high water and soil pressures on the outside side walls and satisfy the normal usage requirements of the metro station [1]. The maximal bending moment at the bottom of the column in the middle of the Track Panel Tunnel (TPT) opening plays a controlling role in the design of the cross-section size and the reinforcement area of the wall column [13]. There are two extreme tendencies in determining the position of the opening in engineering projects generally. One is setting the opening close to the side wall simply for the convenience of track panel lifting. However, such an approach will commonly make the stiffness of the frame beam insufficient, as the opening limits its height and results in an enormous waste of the cross section or the quantity of reinforcement of the wall column further. The other is establishing the opening away from the side wall for effectively supporting without considering the convenience of lifting [14]. The lack of theoretical research about the relationship between the controlling bending moment and the position of reserved openings is due to a weak understanding of this changing law and makes the design of TPT difficult in balancing the convenience of using and the safety and economy of designing. The complementary energy principle with mixed variables is an unconditional variational principle based on the classical minimum complementary energy principle [15], which takes stress and displacement as the auto-variable functions [16]. The complex  
Figure 1: Picture of track panel. Figure 2: Picture of metro station with reserved large-size openings. structural system is reduced to a variational problem of finding the extreme value of the function through the energy conservation idea. It applies to analyzing complex structural systems coupled with various structural components with different performances [17; 18]. This paper draws out the mechanical model for calculating the control parameter of the TPT structure system under the premise of appropriate assumptions. It derives the general analytical solution of the control bending moment for the TPT column under trapezoidal loadings by the complementary energy principle with mixed variables.