Study on the Control Parameter of the Complex Underground Space Engineering Based on the Complementary Energy Principle with Mixed Variables

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Complementary energy principle with mixed variables, Complex underground space structure with the Track Panel Tunnel.

Abstract

The control parameter of the complex underground space structure with the Track Panel Tunnel and wall columns was studied. An analysis method of the control parameter was established based on the complementary energy principle with mixed variables. The general analytical solution of the structure under the trapezoidal load was obtained. Then, the correctness of the solution was verified by two-dimensional finite element simulation. The three-dimensional global model is built to analyze models with different story heights, the distance of two adjacent wall columns, and the thickness of the earth covering; the consequences prove the assumptions’ rationality and the engineering applicability of the analytical solution. The sensitive region of the control parameter is found through the analytical solution, which is meaningful in determining the reasonable stiffness ratio of column and beam for structural design optimization with cost savings. It can be the reference for complex underground space engineering designs.

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 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. Then, it verifies the correctness of the solution by the two-dimensional finite element simulation and proves the rationality of the assumption and the engineering applicability of this solution by three-dimensional global model analysis. It obtains a control parameter of the wall column about the column–beam stiffness ratio after an in-depth analysis of the structural stress mechanism through the analytical result. This parameter is crucial to determine the reasonable column–beam stiffness ratio and balance the openings’ convenience and the structure’s safety and economy.

2. Theoretical Analysis of the Control Parameter of Complex Underground Space Structure with Reserved Large-Size Openings

2.1. 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 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 $a_{\mathrm{i}}$ (I = 1 − n) zone, the j-th wall belongs to the $b_{\mathrm{j}}$ (j = 1 − n) zone, and the k-th frame beam belongs to the $c_{\mathrm{k}}$ (k = 1 − 2) zone. According to the complementary energy principle with mixed variables, the complementary energy functional of the system can be expressed as Equation (1).

$$\mathrm{\Pi }=-{\displaystyle \sum _{V_{a_i}}{\mathrm{\Pi }}_c^{a_i}}-{\displaystyle \sum _{V_{b_j}}{\mathrm{\Pi }}_c^{b_j}}-{\displaystyle \sum _{V_{c_k}}{\mathrm{\Pi }}_c^{c_k}}+{\displaystyle \sum _{V_{ab}}{\mathrm{H}}_{cc}^{a_i{b}_j}}+{\displaystyle \sum _{V_{ac}}{\mathrm{H}}_{cc}^{a_i{c}_k}}+{\displaystyle \sum _{V_{bc}}{\mathrm{H}}_{cc}^{b_j{c}_k}}$$

(1)

where ${\displaystyle \sum _{V_{a_i}}{\mathrm{\Pi }}_c^{a_i}}$, ${\displaystyle \sum _{V_{b_j}}{\mathrm{\Pi }}_c^{b_j}}$, and ${\displaystyle \sum _{V_{c_k}}{\mathrm{\Pi }}_c^{c_k}}$ represent the sum of the complementary energy functional of wall columns, walls, and frame beams, respectively.

${\displaystyle \sum _{V_{ab}}{\mathrm{H}}_{cc}^{a_i{b}_j}}$, ${\displaystyle \sum _{V_{ac}}{\mathrm{H}}_{cc}^{a_i{c}_k}}$, and ${\displaystyle \sum _{V_{bc}}{\mathrm{H}}_{cc}^{b_j{c}_k}}$ represent the sum of additional complementary energy functional on the interface of the adjacent partition between wall columns and their adjacent wall, wall columns and their adjacent frame beams, and frame beams and their adjacent walls, respectively.

The complementary energy of any region $\lambda$ is expressed as Equation (2).

$${\mathrm{\Pi }}_c^{\lambda }={\displaystyle \underset{V_{\lambda }}{\iiint }\frac{1}{2}{\left\{\sigma \right\}}^T[a]{\left\{\sigma \right\}}^{T}dv}-{\displaystyle \underset{s_{u\lambda }}{\iint }{\left\{\overline{u}\right\}}^T[L]{\left\{\sigma \right\}}^{T}ds}$$

(2)

where $\lambda =a_{\mathrm{i}},b_{\mathrm{j}}\mathrm{and} c_{\mathrm{k}}$ represent any zone of wall columns, walls, and beams, respectively.

• $\left\{\sigma \right\}$ represents the possible static stress of the elastomer in the specified zone.
• $[a]$ represents the generalized flexibility of the specified zone and satisfies the stress–strain relationship $\left\{\epsilon \right\}=[a]\left\{\sigma \right\}$.
• $[L]$ is the static boundary condition on the free boundary and satisfies the equation of $[L]\left\{\sigma \right\}=\left\{\overline{\mathrm{T}}\right\}$.
• where $\left\{\overline{\mathrm{T}}\right\}$ is the surface force on the free boundary.

The complementary energy functional on the boundary $s_{ab}$ of adjacent zones a and b can be expressed as Equation (3).

$${\mathrm{H}}_{cc}^{ab}={\displaystyle \underset{s_{ab}}{\iint }{({\left\{T\right\}}^{(a)}+{[T]}^{(b)})}^T{\left\{u\right\}}^{(a)}ds}$$

(3)

where $S_{ab}$ represents the boundary of two zones a and b.

${\left\{u\right\}}^{(a)}$ represents the displacement vector on the boundary of zone a.

Furthermore, we can regard the frame beam as the elastic support of the wall column in the elastic deformation range because of the coordinated deformation of the frame beam and the wall column at their beam–column joint. In addition, the stiffness of flexible support depends on the resistance to deformation of frame beams. Therefore, the system comprises two parts of side walls and wall columns, with elastic support constraints of frame beams at the beam–column joint. The complementary energy functional of the system can be simplified as Equation (4).

$$\mathrm{\Pi }=-{\displaystyle \sum _{V_{a_i}}{\mathrm{\Pi }}_c^{a_i}}-{\displaystyle \sum _{V_{b_j}}{\mathrm{\Pi }}_c^{b_j}}+{\displaystyle \sum _{V_{ab}}{\mathrm{\Pi }}_{cc}^{a_i{b}_j}}$$

(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.

$$\delta \mathrm{\Pi }=-{\displaystyle \sum _{V_{a_i}}\delta {\mathrm{\Pi }}_c^{a_i}}-{\displaystyle \sum _{V_{b_j}}\delta {\mathrm{\Pi }}_c^{b_j}}+{\displaystyle \sum _{V_{ab}}\delta {\mathrm{\Pi }}_{cc}^{a_i{b}_j}}=0$$

(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).

$$\delta {\mathrm{\Pi }}_{cc}^{a_i{b}_j}=0$$

(6)

The above assumption ignores the interaction between walls and columns. Then, we bring Equation (6) into Equation (5) and obtain Equation (7).

$$\{\begin{array}{l}{\displaystyle \sum _{V_{a_i}}\delta {\mathrm{\Pi }}_c^{a_i}}=0\\ {\displaystyle \sum _{V_{b_j}}\delta {\mathrm{\Pi }}_c^{b_j}}=0\end{array}$$

(7)

Due to the coordination of overall deformation of the system, the variation of residual energy of each wall column is positive or negative under external load, so the residual energy variation of each pillar is zero, shown as Equation (8).

$$\delta {\mathrm{\Pi }}_c^{a_i}=0$$

(8)

The frame beam’s deformation is the largest in the middle of the hole, so the bearing reaction of the wall column in the middle of the hole is the largest, as shown in the simplified model of Figure 4. The corresponding bending moment at the bottom of this wall column is the maximum value of all wall columns, and it is the control parameter for the structural design of the wall column. Therefore, the middle wall column of the hole is the crucial point of this research, and it can be further simplified as an elastic column, as shown in Figure 5.

Because of the actual functional requirements of the metro station, there is usually little difference in height between the two layers [21]. The following basic assumptions are made to facilitate the analysis and derivation:

• The influence of wall thickness is ignored;
• The cross sections of the upper and lower frame beams are the same;
• The heights of the upper and lower layers of the station are the same;
• The connection between the wall column and the bottom plate of the station is rigid. The beam and the end of the wall are also rigidly connected.
• The bending stiffness of the wall column is $EI$, the shear stiffness is $\frac{GA}{K}$, and the compressive modulus of elasticity is $E$.
• The rigidity of elastic support provided by the frame beam is $k_{eq}$.

On condition that the rigid ability of elastic support depends on the frame beam’s deformation-resistant capacity, the displacement of the fixed-supported beam at both ends under the action of unit concentrated force in the middle can be expressed as Equation (9), and the free-body diagram is shown in Figure 6.

$${\delta }_{\mathrm{b}}=L_{\mathrm{b}}^3/192E_{\mathrm{b}}{I}_{\mathrm{b}}$$

(9)

where

$E_{\mathrm{b}}$ is the elastic modulus of the frame beam.

$I_{\mathrm{b}}$ is the section moment of the frame beam.

$L_{\mathrm{b}}$ is the calculated length of the frame beam.

Figure 6. Free-body diagram of the fixed-supported beam.

Figure 6. Free-body diagram of the fixed-supported beam.

The stiffness of elastic support is the reciprocal of its flexibility; that is, the reciprocal of displacement under the action of the unit concentrated force is calculated in Equation (10), and the mechanical model of the wall column is shown in Figure 7, in which the constraints of the column is simplified as a fixed constraint at the bottom and elastic bearings both in the middle and on the top. The deformation of the wall column depends on its deformation characteristics and elastic support constraints.

$$k_{\mathrm{eq}}=k=\frac{1}{{\delta }_{\mathrm{b}}}=\frac{192E_{\mathrm{b}}{I}_{\mathrm{b}}}{L_{\mathrm{b}}^3}\cdots$$

(10)

2.2. Theoretical Analysis of the Control Bending Moment at the Bottom of the Column

Considering various loads such as soil weight [22], water pressure, and surface vehicle loads [23,24], we can simplify the distribution of these pressures on the side walls of the underground metro stations as the trapezoidal distributed load [25]. In addition, we can facilitate the control bending moment of the target-wall column as a mechanical problem in which the column is fixed-supported at the end and elastic-supported both in the middle and on the top under a trapezoidal distributed load with a specific gradient [26].

Based on the Code for metro design, the impermeability grade of the side wall of the metro station is P10, and the tolerated range of deformation is 0.2 mm, accordingly. Under this condition, the inner forces of side walls satisfy the linear superposition principle when the element is in the elastic stage [27]. Therefore, the trapezoidal distributed load can be decomposed into uniformly distributed load q and triangularly distributed load Q, as shown in Figure 8.

Under the uniformly distributed load, elastic supports at the top and the middle can be removed and replaced by unknown forces X1 and X2, respectively, as shown in Figure 9. Thus, the statically indeterminate structure is simplified as a statically determinate structure with additional unknown forces X1 and X2 [28], and then the basic structure of force method under the uniformly distributed load is obtained (shown as Figure 9).

Because the deformation of the basic system is in the elastic stage, the internal forces of the basic system satisfy linear superposition, which means that the internal forces (axial force, shear force, and bending moment) can be expressed as the sum of the internal forces of the loads X1, X2, and q acting alone, as shown in Equation (11).

$$\left\{\begin{array}{l}{F}_{\mathrm{N}}\\ F_{\mathrm{Q}}\\ M\end{array}\right\}=[\begin{array}{l}{\overline{F}}_{\mathrm{N}1}\hspace{1em}{\overline{F}}_{\mathrm{N}2}\hspace{1em}\\ {\overline{F}}_{\mathrm{Q}1}\hspace{1em}{\overline{F}}_{\mathrm{Q}2}\\ {\overline{M}}_1\hspace{1em}{\overline{M}}_2\end{array}\begin{array}{l}{F}_{\mathrm{Nq}}\\ F_{\mathrm{Qq}}\\ M_{\mathrm{q}}\end{array}]\left\{\begin{array}{l}{X}_1\\ X_2\\ 1\end{array}\right\}$$

(11)

where $F_{\mathrm{N}}$, $F_{\mathrm{Q}}$, and $M$ represent the axial force, the shear force, and the bending moment of the system, respectively.

${\overline{F}}_{\mathrm{N}1}$, ${\overline{F}}_{\mathrm{Q}1}$, ${\overline{M}}_1$, ${\overline{F}}_{\mathrm{N}2}$, ${\overline{F}}_{\mathrm{Q}2}$, and ${\overline{M}}_2$ represent the axial force, the shear force, and the bending moment on conditions of $X_1=1$ and $X_2=1$, respectively.

$F_{\mathrm{Nq}}$, $F_{\mathrm{Qq}}$, and $M_{\mathrm{q}}$ represent the axial force, the shear force, and the bending moment caused by load q.

It is assumed that the initial normal strain, the shear strain, and bending strain of the system are ${\epsilon }_0,{\gamma }_0 \mathrm{and} {\kappa }_0$, respectively.

Then, the residual energy of the system under the state of static possible internal force is expressed as Equation (12).

$${\mathrm{\Pi }}_c^{}{=\mathrm{V}+\mathrm{V}}_{\mathrm{d}}$$

(12)

where $\mathrm{V}$ indicates the strain residual energy of the system under the possible internal force state, which is related to the internal force, expressed as Equation (13).

${\mathrm{V}}_{\mathrm{d}}$ indicates the residual energy of the support displacement and is the negative value of the sum of the virtual work of the corresponding support reactions on the support seat, expressed as Equation (14).

$$\mathrm{V}={\displaystyle {\int }_L[{\epsilon }_0{F}_{\mathrm{N}}+{\gamma }_0{F}_{\mathrm{Q}}+{\kappa }_{0}M+\frac{1}{2EA}{F}_{\mathrm{N}}^2}+\frac{K}{2GA}{F}_{\mathrm{Q}}^2+\frac{1}{2EI}{M}^2]ds$$

(13)

$${\mathrm{V}}_{\mathrm{d}}=-R_{\mathrm{X}1}-R_{\mathrm{X}2}$$

(14)

where $R_{\mathrm{X}1}$ and $R_{\mathrm{X}2}$ represent the virtual work by the support reaction.

Bring Formula (11) into Formula (13), and the residual energy of the statically indeterminate structure is shown as Equation (15).

$$\mathrm{V}=\frac{1}{2}{\delta }_{11}{X}_1^2+{\delta }_{12}{X}_1{X}_2+\frac{1}{2}{\delta }_{22}{X}_2^2+{\Delta }_{1q}{X}_1+{\Delta }_{2q}{X}_2+{\Delta }_{10}{X}_1+{\Delta }_{20}{X}_2+const$$

(15)

where

• ${\delta }_{\mathrm{ii}}={\displaystyle {\int }_L[\frac{1}{EA}{\overline{F}}_{\mathrm{Ni}}^2}+\frac{K}{GA}{\overline{F}}_{\mathrm{Qi}}^2+\frac{1}{EI}{\overline{M}}_{\mathrm{i}}^2]ds$ indicates the virtual displacement of the i-th support of the system when $X_{\mathrm{i}}=1$ is acting alone ($i=1, 2$).
• ${\delta }_{\mathrm{ij}}={\delta }_{\mathrm{ji}}={\displaystyle {\int }_L[\frac{1}{EA}{\overline{F}}_{\mathrm{Ni}}}{\overline{F}}_{\mathrm{Nj}}+\frac{K}{GA}{\overline{F}}_{\mathrm{Qi}}{\overline{F}}_{\mathrm{Qj}}+\frac{1}{EI}{\overline{M}}_{\mathrm{i}}{\overline{M}}_{\mathrm{j}}]ds$ indicates the virtual displacement of the j-th support of the system when $X_{\mathrm{i}}=1$ is acting alone ($i=1, j=2$; or $i=2, j=1$).
• ${\Delta }_{\mathrm{iq}}={\displaystyle {\int }_L[\frac{1}{EA}{\overline{F}}_{\mathrm{Ni}}}{F}_{\mathrm{Nq}}+\frac{K}{GA}{\overline{F}}_{\mathrm{Qi}}{F}_{\mathrm{Qq}}+\frac{1}{EI}{\overline{M}}_i{M}_q]ds$ indicates the displacement of the i-th support under the load $q$($i=1, 2$).
• ${\Delta }_{\mathrm{i}0}={\displaystyle {\int }_L[{\overline{F}}_{\mathrm{Ni}}}{\epsilon }_0+{\overline{F}}_{\mathrm{Qi}}{\gamma }_0+{\overline{M}}_i{\kappa }_0]ds$ indicates the displacement of the i-th support during initial deformation ($i=1, 2$).

The fixed end of the structure may experience displacement due to some poor fixation under the action of external forces. Assume that the horizontal bearing displacement, the vertical bearing displacement, and the angular displacement of the structure fixed end are $C_1$, $C_2$, and $C_3$, respectively, and corresponding support reactions are $F_{\mathrm{C}1}$, $F_{\mathrm{C}2}$, and $F_{\mathrm{C}3}$, respectively.

$$\left\{\begin{array}{l}{F}_{\mathrm{C}1}\\ F_{\mathrm{C}2}\\ F_{\mathrm{C}3}\end{array}\right\}=[\begin{array}{l}{\overline{F}}_{\left(\mathrm{C}1\right)1}\hspace{1em}{\overline{F}}_{\left(\mathrm{C}1\right)2}\hspace{1em}\\ {\overline{F}}_{\left(\mathrm{C}2\right)1}\hspace{1em}{\overline{F}}_{\left(\mathrm{C}2\right)2}\hspace{1em}\\ {\overline{F}}_{\left(\mathrm{C}3\right)1}\hspace{1em}{\overline{F}}_{\left(\mathrm{C}3\right)2}\hspace{1em}\end{array}\begin{array}{l}{F}_{\left(\mathrm{C}1\right)\mathrm{q}}\\ F_{\left(\mathrm{C}2\right)\mathrm{q}}\\ F_{\left(\mathrm{C}3\right)\mathrm{q}}\end{array}]\left\{\begin{array}{l}{X}_1\\ X_2\\ 1\end{array}\right\}$$

(16)

where ${\overline{F}}_{\left(\mathrm{C}1\right)1}$, ${\overline{F}}_{\left(\mathrm{C}2\right)1}$, ${\overline{F}}_{\left(\mathrm{C}3\right)1}$, ${\overline{F}}_{\left(\mathrm{C}1\right)2}$, ${\overline{F}}_{\left(\mathrm{C}2\right)2}$, and ${\overline{F}}_{\left(\mathrm{C}3\right)2}$ represent support reactions in $C_1$, $C_2$, and $C_3$ directions on conditions $X_1=1$ and $X_2=1$, respectively.

$F_{\left(\mathrm{C}1\right)\mathrm{q}}$, $F_{\left(\mathrm{C}2\right)\mathrm{q}}$, $F_{\left(\mathrm{C}3\right)\mathrm{q}}$ represent support reactions by the load q.

The residual energy of the support displacements can be expressed as Equation (17).

$$\begin{array}{l}{\mathrm{V}}_{\mathrm{d}}=-(F_{\mathrm{C}1}{C}_1+F_{\mathrm{C}2}{C}_2+F_{\mathrm{C}3}{C}_3)\\ =-[{\overline{F}}_{\left(\mathrm{C}1\right)1}{C}_1+{\overline{F}}_{\left(\mathrm{C}2\right)1}{C}_2+{\overline{F}}_{\left(\mathrm{C}3\right)1}{C}_3]X_1-[{\overline{F}}_{\left(\mathrm{C}1\right)2}{C}_1+{\overline{F}}_{\left(\mathrm{C}2\right)2}{C}_2+{\overline{F}}_{\left(\mathrm{C}3\right)2}{C}_3]X_2-[{\overline{F}}_{\left(\mathrm{C}1\right)\mathrm{q}}{C}_1+{\overline{F}}_{\left(\mathrm{C}2\right)\mathrm{q}}{C}_2+{\overline{F}}_{\left(\mathrm{C}3\right)\mathrm{q}}{C}_3]\\ =-{\displaystyle \sum _{i=1}^2[{\overline{F}}_{\left(\mathrm{C}1\right)\mathrm{i}}{C}_1+{\overline{F}}_{\left(\mathrm{C}2\right)\mathrm{i}}{C}_2+{\overline{F}}_{\left(\mathrm{C}3\right)\mathrm{i}}{C}_3]X_{\mathrm{i}}+const}\\ ={\displaystyle \sum _{i=1}^2{\Delta }_{\mathrm{ic}}{X}_{\mathrm{i}}+const}\end{array}$$

(17)

where

${\Delta }_{\mathrm{ic}}=-[{\overline{F}}_{\left(\mathrm{C}1\right)\mathrm{i}}{C}_1+{\overline{F}}_{\left(\mathrm{C}2\right)\mathrm{i}}{C}_2+{\overline{F}}_{\left(\mathrm{C}3\right)\mathrm{i}}{C}_3]$ equals the displacement of $C_1$, $C_2$, and $C_3$ in the direction $X_{\mathrm{i}}$.

The residual energy of the structure is shown as Equation (18).

$$\begin{array}{l}{\mathrm{\Pi }}_c^{}{=\mathrm{V}+\mathrm{V}}_{\mathrm{d}}\\ \hspace{1em}=\frac{1}{2}{\displaystyle \sum _{i=1}^2}{\displaystyle \sum _{j=1}^2{\delta }_{ij}{X}_i{X}_j}+{\displaystyle \sum _{i=1}^2({\Delta }_{\mathrm{ip}}+{\Delta }_{\mathrm{i}0}+{\Delta }_{\mathrm{ic}})}{X}_{\mathrm{i}}+const\end{array}$$

(18)

Apply the residual energy stationary condition of Equation (19).

$$\frac{\partial {\mathrm{\Pi }}_{\mathrm{c}}^{}}{\partial X_{\mathrm{i}}}=0$$

(19)

We can obtain Equation (20).

$$\left[\begin{array}{l}{\delta }_{11}\hspace{1em}{\delta }_{12}\hspace{1em}{\Delta }_{1q}\hspace{1em}{\Delta }_{10}\hspace{1em}{\Delta }_{1c}\\ {\delta }_{21}\hspace{1em}{\delta }_{22}\hspace{1em}{\Delta }_{2q}\hspace{1em}{\Delta }_{20}\hspace{1em}{\Delta }_{2c}\end{array}\right]\left\{\begin{array}{l}{X}_1\\ X_2\\ 1\\ 1\\ 1\end{array}\right\}=0$$

(20)

Equation (21) can be concluded by the reciprocal theorem between the reaction and displacement.

$$\{\begin{array}{l}{\Delta }_{1c}=\frac{X_1}{k_{eq}}\\ {\Delta }_{2c}=\frac{X_2}{k_{eq}}\end{array}$$

(21)

The deformation of the structure under the action of its weight is minimal and negligible, so it can be assumed that the system has no initial strain, that is, the initial normal strain, shear strain, and bending strain are all zero, shown as Equation (22).

$${\epsilon }_0=0,{\gamma }_0=0,{\kappa }_0=0$$

(22)

Then ${\Delta }_{10}={\Delta }_{20}=0$.

Then we can obtain ${\Delta }_{10}={\Delta }_{20}=0$. Then Equation (20) can be simplified as Equation (23).

$$\left[\begin{array}{l}{\delta }_{11}\hspace{1em}{\delta }_{12}\hspace{1em}{\Delta }_{1q}\\ {\delta }_{21}\hspace{1em}{\delta }_{22}\hspace{1em}{\Delta }_{2q}\end{array}\right]\left\{\begin{array}{l}{X}_1\\ X_2\\ 1\end{array}\right\}=-\frac{1}{k_{eq}}\left\{\begin{array}{l}{X}_1\\ X_2\end{array}\right\}$$

(23)

When the bending deformation is the main form of the wall column, we can ignore the influence of axial force and shear force, and conclude Equations (24) and (25).

$${\delta }_{11}=\frac{L^3}{3EI}, {\delta }_{22}=\frac{L^3}{24EI}, {\delta }_{12}={\delta }_{21}=\frac{5L^3}{48EI}$$

(24)

$${\Delta }_{1p}=-\frac{q{L}^4}{8EI}, {\Delta }_{2p}=-\frac{17q{L}^4}{384EI}$$

(25)

where $E$, $I$, and $L$ are the elastic modulus, the section moment of inertia, and the calculated length of the wall column, respectively.

Let $t=L^3/EI$, $t_b=L_b{}^3/E_b{I}_b$, and $\alpha =t_b/t$.

Then, the rigidity k can be obtained, shown as Equation (26).

$$k=1/{\delta }_b=192E_b{I}_b/L_b{}^3=192/\alpha t$$

(26)

Substituting Equation (26) into Equation (23), we obtain the solutions of $X_1$ and $X_2$, respectively, shown as Equations (27) and (28).

$$X_1=\frac{2(12\alpha +11)qL}{({\alpha }^2+72\alpha +112)}$$

(27)

$$X_2=\frac{(17\alpha +128)qL}{2({\alpha }^2+72\alpha +112)}$$

(28)

The bending moment at the bottom of the wall column is obtained as Equations (29) and (30):

$$M=X_1{\overline{M}}_1+X_2{\overline{M}}_2+M_P$$

(29)

$$M_p=-\frac{q{L}^2}{2}$$

(30)

where ${\overline{M}}_1$ and ${\overline{M}}_2$ are bending moments at the wall column bottom for basic structure under the unit forces on points X₁ and X₂, respectively. $M_p$ is the bending moment under the action of $q$.

X₁ and X₂ are brought into Formula (29), and the analytical solution of the bending moment at the wall column bottom under uniformly distributed load can be obtained, as shown in Equation (31).

$$M=-\frac{(2{\alpha }^2+31\alpha +8)q{L}^2}{4({\alpha }^2+72\alpha +112)}$$

(31)

where ${\Delta }_{1\mathrm{p}}$, ${\Delta }_{2\mathrm{p}}$, and $M_{\mathrm{p}}$ under the action of triangular load Q can be obtained in the same way, shown as Equations (32) and (33), respectively.

$${\Delta }_{1p}=-\frac{Q{L}^4}{30EI}, {\Delta }_{2p}=-\frac{49Q{L}^4}{3840EI}$$

(32)

$$M_p=-\frac{Q{L}^2}{6}$$

(33)

Formulas (32) and (33) are brought into Formula (23), and the reaction forces at the top and middle elastic bearings on the wall column are obtained under triangular load Q, respectively, shown as Equations (34) and (35).

$$X_1=\frac{(32\alpha +11)QL}{5({\alpha }^2+72\alpha +112)}$$

(34)

$$X_2=\frac{(49\alpha +576)QL}{20({\alpha }^2+72\alpha +112)}$$

(35)

X₁ and X₂ are brought into Formula (29), and the analytical solution of the bending moment at the wall column bottom under triangular load Q can be obtained, as shown in Equation (36).

$$M=-\frac{(20{\alpha }^2+525\alpha +248)Q{L}^2}{120({\alpha }^2+72\alpha +112)}$$

(36)

Let

$$\beta =Q/q$$

(37)

Then, under the combined action of uniformly distributed load and triangular load, the analytical solution of the bending moment at the wall column bottom is obtained by linear superposition of bending moments, as shown in Equation (38).

$$M=\frac{(20{\alpha }^2\beta +60{\alpha }^2+525\alpha \beta +930\alpha +248\beta +240)q{L}^2}{-120({\alpha }^2+72\alpha +112)}$$

(38)

where

• $\alpha$ is the column–beam stiffness ratio, as shown in Equation (26).
• $\beta$ is the ratio of triangularly distributed load $Q$ and uniformly distributed load $q$, indicates the gradient change degree of the distributed load.
• $L$ is the calculated length of the wall column.

3. Simulations for the Model Verification

3.1. Basic Parameters of the Model

Taking a newly built metro station as an example, a reserved large-size opening with a cross section of 5 m × 28 m is set up on the roof of the metro station. The cross-section size of the wall column is 600 mm × 1800 mm, the distance between two adjacent wall columns is 3 m, and the strength grade of concrete used for the columns is C50. In addition, the cross-section size of the upper and lower frame beams is 800 mm × 1850 mm, and the thickness of the side wall is 600 mm with the C35 grade of concrete. The thickness of the earth covering above the roof is 1.5 m. The coefficient of active earth pressure is supposed to be 0.38, the groundwater is located below the bottom board, the ground vehicle load is supposed to be 20 kPa, the unit weight of the earth is 20 kN/m3, and other parameters are shown in Figure 10 and Figure 11.

3.2. Two-Dimensional Model for Verification

The two-dimensional model is built by the finite element software Midas gen [29], in which the beam element is used to simulate frame beams and the wall columns [30], and fixed constraints are applied at the bottom of the wall column. The wall column model with the frame beam is shown in Figure 12a (both ends of the beam and the bottom of the wall column are fixed constraints, and trapezoidal distributed load is exerted). Figure 12b–d shows the wall column models with elastic supports at the top and middle and fixed constraints at the bottom of the wall column. The stiffness of elastic supports is calculated by Equation (27). Moreover, Figure 12b–d represents the trapezoidal distributed load, the uniformly distributed load, and the triangularly distributed load acting alone on the wall column, respectively. Figure 13a–d shows the calculation results of bending moments under the corresponding model, respectively.

Based on the analytical solution of Equation (38), control bending moments at the bottom of the column and some significant parameters under various working conditions are obtained and shown in

Table 1. Calculation sheet of analytic solutions by theoretical analysis.

Project	Calculation Value	Project	Calculation Value
Uniformly distributed load $q$ (kN/m)	34.20	Load ratio β	8.33
Triangularly distributed load $Q$ (kN/m)	285.00	The bending moment under uniformly distributed load $M$ (kN·m)	688.67
Stiffness of the elastic support $k$ (kN/m)	100,460.12	The bending moment under triangularly distributed load $M$ (kN·m)	2844.01
Ratio $\alpha$	8.20	The bending moment under trapezoidal distributed load $M$ (kN·m)	3532.68

. In addition, soil pressure within the 3 m range in the horizontal direction is taken into account and converted into line load on the wall column, as the distance in the horizontal direction between two adjacent wall columns is 3 m.

As displayed in Figure 13a,b, after the beams are simplified as elastic support, the control bending moment at the bottom of the column is almost the same by theoretical equations and two-dimensional simulation, which indicates that regarding beams as elastic supports is rational and feasible. Meanwhile, it also verifies the correctness of the stiffness of the support deduced in this paper. Combined with the results in Table 1 and Figure 13, we can see that the analytical solution of control bending moments at the bottom of the wall column is the same as the finite element software simulation results. Thus, it certifies the validity of the analytical solution of control bending moments at the bottom of the wall column deduced in this paper.

3.3. Three-Dimensional Model for Verification

The two-dimensional model’s calculation only considers the structure’s deformation in the stress plane, ignoring the out-of-plane deformation, finite stiffness constraints of boundaries, and the interaction of walls, beams, and wall columns. The impact of the above factors on the accuracy of the structural calculation needs to be further discussed. Therefore, the rationality and accuracy of the two-dimensional model will be verified through three-dimensional modeling calculations.

The three-dimensional global model is built by the finite element software Midas gen [31]. The beam element is used to simulate frame beams and wall columns, and the shell element simulates the wall in the model. The structure can deform both in and out of plane, and the interaction of walls, beams, and columns is simulated through common node coupling. The essential parameters of this model are the same as those in Section 2.1. In addition, the impact of envelopes on the main construction of the metro station is ignored. The overall deformation of the three-dimensional simulation and the bending moment of the wall column is shown in Figure 14 and Figure 15, respectively.

We can see from the overall deformation (Figure 14) that the deformation of the upper frame beam is greater than that of the middle frame beam. Meanwhile, the wall column’s deformation is the largest in the middle of the opening and decreases successively to both sides. The whole deformation characteristics of the side wall at the opening are similar to the plate: freedom at the top and consolidation at the other three sides. As illustrated in Figure 12, the maximum bending moment at the bottom of the central wall column (3717 kN·m) is the control bending moment of the wall column. The control bending moment of the wall column deduced by the analytical solution in this paper is 3533 kN·m. The error between the two methods is 4.95%, which agrees with the range of allowable errors in engineering. Thus, it verifies that the assumption of the analytical solution for the bending moment of the wall column is rational. Furthermore, it indicates that analytical solutions have better applicability and better predictive validity in determining cross-section sizes of beams and columns without three-dimensional analysis. Using this solution will significantly improve work efficiency in preliminary structural design engineering.

In addition, the main reason that the analytical solution is slightly less than the three-dimensional calculation result is that the limited rotational restriction of side walls at both ends of the frame beam and boundary columns cannot meet the hypothesis of absolute rigid connection. Therefore, the stiffness of the elastic support in the analytical model is slightly larger than that in the actual frame beam, and the corresponding bending moment at the bottom of the column is marginally smaller than that in reality.

The three-dimensional model that takes into account solid structures can simulate actual projects in a better way. The consistency between the calculation results of the two-dimensional model and the three-dimensional model further validates the basic assumptions of the two-dimensional calculation. The two-dimensional model can be used for efficient calculation during the preliminary structural selection analysis. Using three-dimensional models is more accurate when conducting quantitative structural analysis.

4. Discussions of Engineering Applicability of Theoretical Solutions

To fully demonstrate the engineering applicability of approximate analytical solutions derived in this paper, more simulations with different vital parameters such as distance between two adjacent wall columns, story height of the metro station, and the thickness of earth covering are completed. Moreover, the analytical result is compelling enough to predict the bending moment at the bottom of the wall column in possible practical engineering.

4.1. The Influence of the Story Height of the Metro Station

Based on parameters of the example in Section 3.1, only the story height of the station is changed. As height differences between two adjacent layers cannot exceed 2 m in subway station, height difference values of 2 m, 1 m, 0 m, −1 m, and −2 m are considered. In addition, the control-bending moments at the bottom of column under these height differences are obtained by three-dimensional simulation via software. Meanwhile, the analytical solution is still computed on the assumption of 0m height difference via Equation (38) (i.e. the upper and the lower story heights are equal). Bending moments at the bottom of the column under different conditions are shown in Figure 16. The comparison between simulation and analytical solution is shown in

Table 2. The comparison between simulation and analytical solution within different heights of two layers.

Height Difference between Two Adjacent Layers	Analytical Solution	Simulation	The Error between Them
2	3532.68	3784	−6.64%
1	3532.68	3760	−6.05%
0	3532.68	3717	−4.96%
−1	3532.68	3663	−3.56%
−2	3532.68	3605	−2.01%

.

It can be seen from Figure 16 and Table 2 that the error between analytical solutions and the simulation results ranges from 2.0% to 6.6%, which indicates that the slight changes in the height difference of the metro station cannot play a decisive role in the control bending moment at the column bottom. In addition, it can be certified that the assumption of the same story height of the upper and lower layers of the metro station is rational in theoretical analysis.

4.2. The Influence of the Distance between Two Adjacent Wall Columns

Based on the parameters of the example in Section 3.1, the distance between two adjacent wall columns is adjusted to 2.5 m, 3 m, 3.5 m, 4 m, and 4.5 m, respectively. The relationships between the bending moment at the bottom of the column and the distance of two adjacent wall columns are calculated by three-dimensional Midas gen software, and analytical solutions are obtained via Equation (38), as shown in Figure 17.

We can see in Figure 17 and

Table 3. The comparison between simulation and analytical solution with different distances between two wall columns.

Distance between Two Wall Columns	Analytical Solution	Simulation	The Error between Them
2.5	2943.9	3369	−12.6%
3	3532.68	3717	−4.96%
3.5	4121.46	4101	0.49%
4	4710.24	4373	7.71%
4.5	5299.02	4918	7.7%

that analytical solutions are well coincident with the results obtained by software under the different distances between the wall columns, and the maximum error is less than 13%. Hence, it shows that the analytical solution of Equation (38) applies to various column spacing.

4.3. The Influence of the Thickness of Earth Covering

Different soil types affect the structure differently. In structural analysis, the impact of soil is mainly reflected by soil pressure. Different types of soil produce different soil pressure distributions. Still, the distribution rules of soil pressure have similar characteristics, which can be simplified as trapezoidal distributed loads or segmented trapezoidal distributed loads. Furthermore, the influence of segmented trapezoidal distributed loads on structural response can be considered the superposition of trapezoidal distributed loads in structural analysis.

This study focuses on analyzing control parameters for the location of TPT openings in complex underground spaces. The research object is a two-layer subway station with a buried depth of no more than 21 m. In the practical engineering of subway stations, the soil properties within this range are almost uniform. This article discusses the primary working conditions for the soil to be homogeneous and focus on the relationship between the opening position and the control bending moment under different thicknesses of earth covering trapezoidal distributed earth pressure.

Let us take parameters in Section 3.1 as an example, and only the values of thickness of the earth covering are considered as 1.5 m, 2 m, 3 m, 4 m, 5 m, and 6 m. The relationships between the bending moment at the bottom of the column and the thickness of the earth covering are gained by three-dimensional Midas gen software, and analytical solutions are obtained via Equation (38), as shown in Figure 18.

From

Table 4. The comparison between simulation and analytical solution within different thickness of earth covering.

The Thickness of Earth Covering	Analytical Solution	Simulation	The Error between Them
1.5	3717	3532.68	4.96%
2	4032	3762.3	6.69%
3	4645	4221.3	9.12%
4	5268	4668.3	11.38%
5	5886	5139	12.69%
6	6494	5598.6	13.79%

, the error between analytical solutions and the simulation results ranges from 4.9% to 13.7% under the different thicknesses of the earth covering. The error increases with the thickness of the earth covering in the acceptable scope, and the analytical solution has the advantage of high goodness of fit and satisfactory effect.

In addition, the approximate analytical solutions deduced in this paper are very close to three-dimensional simulation results compared to the above key factors, such as the distance between wall columns, the story height of the metro station, and the thickness of the earth covering in the actual engineering. Equation (38) is engineering applicable and can be quickly and easily applied to most similar projects.

5. Discussions about the Control Parameter of the Complex Underground Structure System Design

As the floor height and overburden thickness of the station are fixed values after the determination of the functional requirements of the subway station and the external environment, parameters $q$, $\beta$, and $L$ are determined in Equation (38). Engineers can actively adjust the section sizes of the wall columns and the frame beams, that is, the stiffness ratio between the wall column and the frame beam $\alpha$, which can be named as the column–beam stiffness ratio. A correct and comprehensive understanding of the influence law of the parameter $\alpha$ is essential to improve the personal initiative of engineers in engineering design. Now, we will try to discuss $\alpha$.

(1) Let us suppose that the column–beam stiffness ratio $\alpha$ approaches infinity, which means that the cross section of the frame beam of the side wall is so tiny that the stiffness can be ignored compared with the column. Then, the limit of the analytical solution formula can be expressed as Equation (39) when $\alpha$ approaches infinity.

$$\underset{\alpha \to \infty }{\lim }M=-\frac{(\beta +3)q{L}^2}{6}$$

(39)

If we let $\beta =0$, that is, there is only uniform linear load applied, Equation (39) is simplified to Equation (40).

$$M=-\frac{q{L}^2}{2}$$

(40)

(2) On the contrary, if the opening is far away from the frame beam of the side wall, the cross section of the frame beam of the side wall is large enough to bear any deformation. The column–beam stiffness ratio $\alpha$ decreases to infinitely small consequently, and Equation (38) can be expressed as Equation (41) by taking the limit $\alpha \to 0$.

$$\underset{\alpha \to 0}{\lim }M=-\frac{(31\beta +30)q{L}^2}{1680}$$

(41)

Set $\beta =0$, i.e., there are only uniform distributed linear loads applied, then Equation (40) can be simplified to Equation (42):

$$M=-\frac{q{L}^2}{56}$$

(42)

which agrees with the calculation of the hogging bending moment value at the root of the beam with the fixed support at one end, the hinged support at the other end, and the mid-span under the uniform distributed load. In addition, the correctness of the derivation in this paper is also verified from the perspective of the limit.

(3) Taking the parameters in Section 3.1 as an example, let $\beta =0$ and substitute different values $\alpha$ into Equation (38). The relationship diagram between the column bottom bending moment of the wall column and the stiffness ratio between the wall column and the frame beam is obtained, as shown in Figure 19.

The wall column’s control bending moment increases remarkably with the increase of $\alpha$ in the range of 1–64 from Figure 19, which means that there is a sensitive region of the column’s control bending moment to the column–beam stiffness ratio $\alpha$, and the change of the bending moment is not noticeable out of this region. Therefore, it is beneficial to reduce the concentration of the bending moment at the bottom of the wall column by taking the Track Panel Tunnel (TPT) openings a little closer to the middle longitudinal beam under the condition of meeting the functional requirements. However, the mechanical performance of the structure cannot be remarkably improved if the opening of the track panel well is too close to the middle longitudinal beam blindly, and this action will make the track panel inconvenient to be lifted. The determination of the reasonable column–beam stiffness ratio in the sensitive region is critical for structure designers to create a balance between the convenience and the mechanical performance of the structure.

Taking the metro station in Section 3.1 as an example, when the distance between the edge of the opening and the outside wall of the metro station is 3 m, the column–beam stiffness ratio $\alpha$ is 1.92. The corresponding bending moment at the bottom of the column is 131.3 kN·m, with 42.8% fall compared with the original design distance of 1.85 m (α = 8.2, M = 229.5 kN·m). It is observed that a clear understanding of the relationship between the control bending moment of the wall column and the position of openings is essential for rational structure design, which is vital to improve the mechanical performance of the structure and save on the project cost for engineering.

(4) Because the story height of the station is 5.5~7 m, determined by the subway station’s functional requirements, the opening’s size is 27~30 m because of the standard length of the rail, the length-to-height ratio of the side wall in the opening range is generally greater than 2. Therefore, the mechanical characteristic of a double-story station’s side wall in the middle with an opening is similar to the one-way slab. The simplified model in the paper can be established and in good agreement with the simulation. Whether this implicit condition is satisfied affects the accuracy of the analytic solution directly and should be paid attention to in design.

(5) Variability of soil properties is a major source of uncertainty in assessing the seismic response of geotechnical systems [19], which makes the dynamic characteristics of the system more complex and needs to be analyzed in the future. This paper uses the load-structure method to consider the soil impact, which is appropriate in the static calculation results of this paper. The stratum-structure rule is more suitable for the numerical calculation of dynamic analysis, which treats the underground structure and soil as a whole under the unified stress and deformation and calculates the deformation of the underground structure and surrounding soil based on continuum mechanics. In dynamic analysis, the range of the soil is often taken as the calculation area 5 times the size of the underground structure if the fixed boundary is used, which can eliminate the impact of boundary effects. Moreover, the research results of Zhang Zhiying et al. show that the soil–structure interaction system exhibits significant classical damping system characteristics under small magnitude loads, which means that the system can be solved by classical dynamic analysis methods such as the mode superposition method [32].

6. Conclusions

The design of the reserved large-size openings is a crucial point in complex underground space structure design, because this opening leads to its surroundings being the weakest part of the structure. The convenience, safety, and economy of the station with large-openings design cannot be considered comprehensively because of limited research about the relationship between the control bending moment at the bottom of the wall column and the position of the openings.

In this paper, the simplified calculation model of the control bending moment of the wall column under rational assumptions is abstracted, and an approximate analytical solution of the control bending moment of the wall column under soil and water pressure loadings is deduced. Then, the correctness of the formula is certified by two-dimensional finite element simulation. The three-dimensional simulation proves the assumptions’ rationality and the engineering applicability of the analytical solution. Meanwhile, the sensitive region of the control bending moment to the column–beam stiffness ratio $\alpha$ is found through the analytical solution, which can be the critical point in determining the column–beam stiffness ratio reasonably while meeting the requirements of the actual design. It is useful for designers to determine the reasonable column–beam stiffness ratio to provide the basis for structural design optimization with cost savings in the preliminary design stage.

In addition, it is worth noting that basic assumptions and corresponding analytical solutions are obtained on the basis that the length-to-height ratio of the side is engineering applicable wall in the opening range should be generally greater than 2.