Influence of Freezing Tunnel Excavation on Foundation Settlement of Buildings

Abstract

In this paper, the effects of the cement content and excavation speed parameters on the improvement effect of the artificial freezing method and the characteristics of frozen walls are studied by means of a field test, numerical simulation and theoretical model. The optimization effect of the cement content in the artificial freezing method is studied. It is found that 10% is the best content, which can maximize the freezing wall thickness and grouting effect, and promote uniform distribution of the temperature field. At the same time, the influence of the excavation speed on the stress and settlement of the foundation soil is analyzed. It is pointed out that the increase of the excavation speed will aggravate the settlement and stress redistribution, which may threaten the building structure. The evaluation method proposed in this paper verifies that the construction deformation and settlement control are within the safety standards and provides theoretical support and construction guidance for tunnel engineering.

1. Introduction

With the acceleration of the urbanization process, as tunnel engineering is a key measure to solve the problem of urban traffic congestion, the safety and stability of its construction have been paid more and more attention [1,2]. As a construction technology with significant advantages in complex geological conditions, the freezing method for the tunnel curtain provides a reliable safety guarantee for tunnel excavation by freezing the surrounding soil to form a stable frozen soil curtain [3,4]. However, the tunnel construction process will inevitably cause disturbance to the surrounding strata, resulting in surface settlement and the overall or uneven settlement of the upper building, which may lead to the dislocation of the internal structure of the building and pose a threat to the safety of residents [5,6]. Therefore, it has important engineering guidance significance and application value to deeply study the formation law of the freezing temperature field of the tunnel curtain and its force characteristics, and to analyze the influence of cement improvement on the freezing temperature field and stress field, as well as the influence of these changes on the settlement of surface buildings [7,8].

Domestic and foreign scholars have carried out a lot of research on the formation law of the freezing temperature field in the tunnel curtain. For example, the formation, melting, thawing displacement field and frost heave displacement field of the frozen curtain have been studied in detail during the whole process of tunnel freezing construction through laboratory tests and numerical simulation [9,10]. ANSYS and other finite element software were used to calculate the freeze-heaving and thawing amount under the interaction of pipe shed and freezing, and to compare it with the field-measured data to verify the rationality of the reinforcement method [11,12].

Although the improvement effect of cement on the physical and mechanical properties of soil has been widely recognized, the study of its effect on the freezing temperature field and stress field is not sufficient. The effect of cement on soil improvement in a tunnel project of the Nanjing Metro was studied by laboratory tests. It was found that the cement content and curing age had significant effects on the freezing temperature and mechanical properties of soil. These studies provide a valuable reference for understanding the effect of cement improvement on the freezing temperature field and stress field [13]. The formation law of the freezing temperature field in the tunnel curtain is studied as follows: through the method of combining field monitoring and numerical simulation, the development and change law of the freezing temperature field is studied, and the changing law of temperature in different jacking tubes and soil mass temperatures is analyzed to reveal the formation mechanism of the freezing temperature field [14].

The effects of cement improvement on the freezing temperature field and stress field are assessed as follows: through laboratory tests and numerical simulation, the changes in the freezing temperature and mechanical properties of soil before and after cement improvement are studied, the effects of the cement content and curing age on the freezing temperature field and stress field are analyzed, and the strengthening effect of cement improvement is evaluated. Combined with the actual situation in the tunnel construction process, the surface settlement prediction model was established to analyze the influence of different construction parameters on settlement. At the same time, reasonable settlement control measures were put forward to ensure that the impact of tunnel construction on the surrounding buildings is minimized [15].

At present, the research on the tunnel curtain freezing method has made progress in improving construction safety and stability, but there are still limitations such as the insufficient analysis of multi-factor coupling, lack of long-term monitoring data, limited research on three-dimensional spatial effects, insufficient research on adaptability to complex geological conditions, incomplete numerical simulation and actual measurement verification, and insufficient research on cement improvement technology optimization. More research is needed to further explore and optimize the method. Domestic and foreign scholars have conducted a relatively complete study on the influence of frost heave displacement and deformation under penetration of the artificial freezing method on the surrounding environment, but the main research direction is still focused on the influence of the artificial freezing method on the deformation of surface, above-ground buildings and pipelines, as well as the influence of tunneling under penetration on existing underground buildings. The combined construction of an underground tunnel with the artificial freezing method has relatively little influence on the existing upper structure. By revealing the formation mechanism and mechanical characteristics of the freezing temperature field of the tunnel curtain, this paper provides a more scientific and reliable theoretical basis for tunnel construction, which is helpful in realizing the safety and stability of tunnel construction under complex geological conditions. The effects of the cement content and curing age on the freezing temperature and mechanical properties of soil are systematically studied, which provides an optimization scheme for the application of cement improvement technology in tunnel engineering, which is helpful in further improving the soil improvement effect and enhancing the overall stability of the tunnel structure. The prediction model of surface settlement is established, and the influence of different construction parameters on settlement is analyzed, which provides a scientific basis for formulating reasonable settlement control measures. This not only helps to reduce the potential threat of tunnel construction to surrounding buildings but also ensures the safety of residents’ lives and property.

2. Project Overview

The Class I terrace and Class II terrace across the left bank of the Yangtze River between the back and the city of Wuhan, Hubei Province, belong to a buried terrace. The Quaternary strata are widely distributed in the site, with a thickness of 42~63 m. The tunnels in the interval mainly pass through the soil layers from top to bottom in the order of mixed fill, plain fill, clay and silty clay. The ground buildings are mostly low-rise houses, which are to be demolished. The tunnel runs directly through a five-story civil house and passes through 32 civil houses on the side. The tunnel adopts the earth pressure balance shield construction method. The tunnel excavation adopts single line excavation, first excavating the left line tunnel, then excavating the right line tunnel, and finally excavating the contact channel, which is excavated by the freezing method.

(1) Model construction and grid division

Model size and grid division: The total size of the model is 44 m (length) × 15 m (width) × 35 m (height). In order to more accurately simulate the changes in the temperature and stress fields, the model is mesh-encrypted in key areas such as around the tunnel, near the freezing tube, and under the building. The size of the grid gradually increases according to the distance from the tunnel and the freezing tube to balance the calculation accuracy and calculation time.

Cell type: In the model, strata and buildings are simulated using solid units (such as zone units in FLAC3D), and tunnel segments are simulated using shell units (such as shell units in FLAC3D). The shell element can better reflect the mechanical characteristics of the segment during tunnel excavation and freezing.

(2) Material property definition

Stratum material: According to the parameters in

Table 1. Physical and mechanical parameters of the surrounding rock.

Material Category	Density (kg/m3 )	Modulus (MPa)	Poisson’s Ratio	Angle of Internal Friction (°)	Cohesion (kPa)
Miscellaneous fill	1940	5.9	0.43	10.1	31.3
Clay	1980	9.3	0.42	12.2	32.7
Silty clay	2000	11.7	0.42	18.1	28.7
Shield segments	2450	3.45 × 104	0.2

and

Table 2. Physical and mechanical parameters of the building.

Building Structure	ness Density	ness Density	ness Density	ness Density	ness Density	ness Density	ness Density
	(m)	(kg/m3)	(MPa)	(MPa)	(MPa)	(°)	(MPa)
Foundation	2.00	2500	4460	2290	1.80	48	1.80
Floor slab	0.40	2500	4460	2290	1.80	48	1.80
Wall	0.45	2500	2975	1530	1.25	47	1.25

, the stratum is subdivided into the plain fill soil layer and silty clay layer. Each soil layer defines its density, elastic modulus, Poisson’s ratio, internal friction angle, cohesion, thermal conductivity, specific heat capacity and other physical and mechanical parameters and thermal property parameters.

Tunnel segment: The tunnel segment adopts an elastic model, and its elastic modulus, Poisson’s ratio, thermal conductivity, specific heat capacity, density and other parameters have been given, which will directly affect the temperature change and stress state of the segment during the freezing process.

Air: The surface air and air in the tunnel are treated as homogeneous materials with low thermal conductivity and specific heat capacity to simulate the effect of air on the temperature field.

(3) Setting of boundary conditions

Displacement boundary conditions: The bottom of the model is completely constrained, and the left and right sides are constrained by the displacement in the X direction to simulate the formation’s fixation in the horizontal direction. The upper surface is a free surface, allowing vertical displacement.

Temperature boundary condition: The temperature of the frozen tube wall changes over time, and it is usually set to a gradually decreasing temperature curve to simulate the freezing process. The temperature of the surface air and the air in the tunnel should also be set according to the actual situation.

Thermal boundary condition: In addition to the cold source provided by the frozen tube, the heat exchange between the formation and the surrounding environment should also be considered, such as the heat exchange between the surface and the air, and the heat exchange between the air and the tube in the tunnel.

(4) Solution process

Initial condition: The initial temperature field of the model is set, usually as the original temperature of the formation.

Freezing process simulation: The freezing process is simulated by gradually reducing the temperature of the freezing tube wall. At each step, changes in the temperature field are calculated and the material’s thermo-physical parameters (such as the thermal conductivity, specific heat capacity, etc.) are updated to reflect the effect of temperature on the material’s properties.

Stress field calculation: When the temperature field changes, the stress field changes caused by the temperature change are calculated. Special attention is paid to the stress state of the tunnel segments, strata and buildings during freezing.

Result analysis: The distribution law of the temperature field and stress field is analyzed, and the influence of the freezing process on the tunnel and buildings is evaluated, including the surface settlement, tunnel deformation, building stress state, etc.

(5) Verification and verification

Field data comparison: The simulation results are compared with field monitoring data to verify the accuracy and reliability of the model.

Sensitivity analysis: Sensitivity analysis is performed on key parameters in the model, such as the freezing tube temperature, formation parameters, etc., to assess the impact of these parameters on the simulation results.

Through the description of the above details, the construction process of the numerical model, the definition of the material properties, the setting of the boundary conditions and the key steps in the solution process can be more comprehensively understood.

3. Construction of Numerical Model

3.1. Process of Numerical Simulation

Establishment of the calculation model of the freezing body temperature field proceeds as follows.

In order to facilitate the calculation, the following assumptions are made for the finite element model:

(1)

The soil is continuous and homogeneous during freezing;

(2)

When the soil is frozen, the latent heat of the phase change is continuously released at the freezing interface;

(3)

It is assumed that the moisture in the soil is completely frozen by cold, that is, the content of unfrozen water is zero;

(4)

The freezing temperature load on the freezing hole wall of the model changes with the change of temperature;

(5)

The wall of the freezing tube is negligible relative to the whole model, so the material characteristics of the freezing tube itself are not considered.

(6)

The appropriate solution algorithm and iteration step size are selected to ensure the stability and convergence of the calculation process. For complex nonlinear problems, more advanced algorithms may be required, such as implicit solvers or adaptive time step algorithms.

(7)

In the freezing process of the tunnel, the change of the temperature field will cause the change of the stress field, and there is a coupling effect between the two. In the calculation process, it is necessary to adopt a proper heat–force coupling model to ensure that the calculated results of the temperature field and stress field are compatible with each other.

(8)

The initial conditions (such as the initial temperature field, initial stress field, etc.) should be set according to the actual situation of the project. The accuracy of the initial conditions has an important influence on the subsequent calculation results.

(9)

In the calculation process, it is necessary to monitor the change of key parameters (such as the temperature, stress, displacement, etc.), and to discover and deal with abnormal conditions in a timely manner. If the calculation results are not convergent or are unreasonable, the model parameters or the solution algorithm should be adjusted.

In order to analyze the influence of the horizontal freezing of the contact channel on the existing tunnel in the upper part, according to the general situation of the project and the symmetry of the contact channel, this paper uses the FLAC3D finite difference software to establish a structural model and then realizes the three-dimensional temperature field calculation of the frozen body. The position relationship between the frozen tube and the existing tunnel is shown in Figure 1. The total size of the model is 44 m long × 15 m wide × 35 m high, as shown in Figure 2.

When calculating the model numerically, the vertical tunnel axis direction in the water intake plane is the X axis, the direction along the tunnel axis is the Y axis, and the vertical upward direction is the Z axis. In order to avoid the influence of the boundary effect on the numerical calculation results of the temperature field, the model range (X, Y, Z) is selected as (44,15,35 m) from the two aspects of the calculation accuracy and solution time.

The upper surface of the tunnel and frozen tube model is a free surface, and the displacement of the lower surface and four side surfaces is completely constrained. The load is generated from top to bottom according to the gradient of the gravity field, considering only the action of gravity. The Mohr–Coulomb elastoplastic model is used as the soil element in the model. In order to make the calculation results more realistic, the model is subdivided into two layers, namely plain filled soil and silty clay. The physical and mechanical parameters and thermophysical parameters of each soil layer are shown in Table 1 according to the in situ geotechnical investigation and sampling and testing. The elastic model is adopted for the shield tunnel segment elements. The elastic modulus of the shield segment is set at 33.5 GPa, Poisson’s ratio at 0.2, thermal conductivity at 0.5 W/(m·°C), specific heat capacity at 1 kJ/(kg·°C), and density at 2500 kg/m³. The thermal conductivity of the surface air and air in the tunnel is 0.025 W/(m·°C), and the specific heat capacity is 1 kJ/(kg·°C).

In the actual freezing method construction process, the active freezing period of the interval contact channel is 40 d, and the numerical calculation time of the temperature field is also 40 d. This is because the freezing pipe is arranged in a inclined radial shape and the freezing pipe layout on the upper and lower lines is not the same.

In order to facilitate analysis, a typical cross-section in the direction of Y = −7.5 is selected, respectively, and a total of 5 holes for measuring the settlement are set up, as shown in Figure 2. Represented by the data of the same point as the C1, C2 and C3 settlement holes in the tunnel, the settlement change value is recorded according to the actual settlement point in the numerical simulation. After 40 days of active freezing, the temperature curves of the three settlement holes in the numerical simulation during the tunnel excavation are compared with the field measured settlement, as shown in Figure 3.

As shown in Figure 3, the numerical simulation results are basically consistent with the field measured results. The numerical simulation results are generally slightly lower than the measured values because the numerical calculation assumes that the groundwater is static. In the actual project, because the groundwater is mobile, the heat exchange of the frozen curtain is more complicated, and the flow of the groundwater will take away part of the cold volume, so the numerical simulation results are slightly lower than the measured values. To sum up, the transient freezing temperature field obtained by numerical simulation using the finite difference software FLAC 3D 5.0 can reflect the actual situation of engineering.

3.2. Analysis of Numerical Simulation Results of Temperature Field

The cross-section Y = −7.5 m temperature field cloud map at the longitudinal midpoint of the frozen dug tunnel is shown in Figure 4. At the beginning of the active freezing period, with the rapid decrease in the temperature of the route brine, the temperature of the soil around the freezing tube decreases rapidly and the freezing front rapidly diffuses outward with each freezing tube at the circle center, forming a cylindrical frozen soil column. Adjacent frozen soil columns begin to intersect and form a freezing curtain.

As shown in Figure 4a, at the beginning of the artificial freezing method, the temperature of the target formation drops rapidly, a thin freezing wall begins to form around the laid freezing tube, and the freezing front begins to develop outwards. After 10 days of freezing, the frozen front adjacent to the side wall of the underground tunnel and the bottom begin to gradually intersect, and the adjacent frozen front also tends to intersect. After the freezing front gradually intersects, the development speed of the freezing temperature field gradually slows down. The freezing pipes around the tunnel are relatively dense, which leads to the rapid development of the freezing front. Because the excavation area of the two frozen underground tunnels is far away from the freezing tube, the freezing cooling trend does not affect this area for the time being, and the central area of the excavation area does not start to freeze, so the “high temperature area” of the two cuboids is formed.

As shown in Figure 4b, after 20 days of freezing, the freezing front of the side wall and the freezing tube at the bottom completes the intersection circle. As the brine temperature decreases slowly, the freezing wall develops outwards at a constant speed and steadily, and the “high temperature area” in the central part of the frozen dug tunnel begins to shrink. As shown in Figure 4d, after freezing for 40 days, the thickness of the freezing wall meets the design requirements, the freezing wall generally presents a “U” font, the freezing wall is evenly distributed at the side wall and the bottom, and no freezing dead corners are found.

3.3. Influence of Cement Improvement on Freezing Temperature Field

The distribution and development of the freezing temperature field in the artificial freezing method are affected by many factors. Under the condition that the arrangement mode and structure of the freezing pipe remain unchanged, the physical mechanical parameters and thermal physical parameters of soil become one of the main factors affecting the freezing temperature field. Under the condition of grouting with different cement content, the physical mechanical parameters and thermal physical parameters of soil will be different.

In order to obtain the distribution of the freezing temperature field after cement improvement more accurately and objectively, the Y = −7.5 m cross-section of the frozen dug tunnel is selected, which can comprehensively reflect the objective law of the development of the freezing wall thickness of the frozen temperature field in the frozen dug tunnel after cement improvement, as shown in Figure 5.

At the end of the active freezing period, when the cement content increases from 0% to 20%, the thickness of the freezing curtain first increases and then decreases. The thickness of the frozen wall at the left, right and bottom frozen walls increases from 2.95 m, 3.03 m and 2.82 m to 3.14 m, 3.22 m and 3.04 m, respectively, and then decreases to 2.97 m, 3.06 m and 2.84 m. When the cement content is 10%, the thickness of the frozen wall reaches the maximum value. It can be considered that the gain effect of the grouting improvement on the artificial freezing method is the strongest when the cement content is 10%. When the cement content is 20%, the thickness of the three frozen walls is still slightly greater than that of the frozen wall when the cement content is 0%, but according to the downward trend, when the cement content is greater than 20%, the frozen wall thickness will be significantly reduced and the cement improvement will have a negative impact on the artificial freezing method.

At this stage, the addition of cement improves the physical mechanical properties and thermal physical properties of the soil. The increase in soil density and thermal conductivity makes the heat transfer in the freezing process more efficient, thus promoting the formation and thickening of the freezing wall. In addition, the increased strength of the soil also helps to resist deformation and damage during the freezing process, further enhancing the stability of the frozen wall.

When the cement content continues to increase, although the strength and compactness of the soil may be further improved, too high a cement content may cause the soil to become too hard and brittle. This change may not be conducive to the uniform transfer and distribution of heat, causing the freeze wall to form too fast or too slowly in some areas, thus affecting the overall freezing effect. In addition, a high cement content may also increase the thermal resistance of the soil mass, making it more difficult to transfer heat over longer distances, resulting in a decrease in the thickness of the frozen wall.

The average temperature curve of the frozen wall of a frozen tunnel with a cross-section of Y = −7.5 m under different cement content is shown in Figure 6. When the cement content increases from 0% to 20%, the average temperature of the frozen wall at different positions changes slightly with the increase in the cement content, and the average temperature difference of the frozen wall at the same position is about 1 °C under different cement contents. From the trend point of view, with the increase in the cement content, the average freezing wall temperature presents the characteristics of first increasing and then decreasing, and the peak value of the average freezing wall temperature at different positions is basically between 5% and 10% of the cement content. The average temperature of the frozen wall at the left, right and bottom frozen walls increases from −15.2 °C, −16 °C and −14.3 °C to −15.1 °C, −16.5 °C and −14.0 °C, respectively, and then decreases to −15.1 °C, −16.6 °C and −13.9 °C. The variation trend of the average temperature of the freezing wall also indicates that the freezing temperature field develops more evenly with the increase in the cement content when the cement content is less than 10%. However, when the cement content is greater than 10%, the increase in the cement content hinders the development of the freezing wall, which also hinders the development of the freezing front and decreases the distribution range of the freezing temperature field.

When the cement content increases from 0%, the initial cement addition improves the physical mechanical properties and thermal physical properties of the soil. The compactness of the soil increases and the thermal conductivity improves, which makes the heat transfer in the freezing process more efficient. However, this improvement may not initially be sufficient to significantly reduce the average temperature of the freezing wall, as the heat transfer and distribution are also affected by a variety of other factors, such as the placement of the freezing tubes, brine temperature, and formation conditions. Therefore, when the cement content is low, the average temperature of the freezing wall may rise slightly, which may be due to the improvement of the heat transfer efficiency, but the overall freezing effect is not fully manifested.

As the cement content continues to increase, the thermal conductivity and strength of the soil are further improved, which helps to speed up the freezing process and promote the formation of the freezing wall. At this stage, the average temperature of the freezing wall begins to drop as more heat is efficiently transferred to the freezing wall and consumed by the freezing process. When the cement content reaches an optimal equilibrium point (about 5% to 10% in this case), the average temperature of the freezing wall reaches the lowest point and the freezing effect is most significant.

It is also important to note that the numerical simulation results can be affected by a variety of factors, such as the model assumptions, parameter settings, and calculation accuracy. Therefore, in practical application, it is necessary to verify and correct the numerical simulation results by combining field monitoring data and engineering experience.

In summary, the change law of the average temperature of the freezing wall in Figure 6, which first increases and then decreases, can be attributed to the complex effects of the cement content on the physical and mechanical properties and thermal physical properties of the soil, as well as the dynamic changes in these effects during the freezing process. When the cement content reaches the optimal balance point, the freezing effect is the most significant. However, when the cement content is excessive, it may have adverse effects.

4. Theoretical Calculation of the Influence of the Excavation of the Contact Passage on the Building

4.1. Calculation Parameters of Foundation Model

In this paper, the Livkin model is used to consider the soil–tunnel interactions. The Livkin model takes into account the variation of the foundation reaction along the length and width of the foundation and with the soil type. Its characteristic function is Equation (1).

$$p(x)=k\left[1+\beta {\mathrm{e}}^{-\alpha \left(1-{\displaystyle \frac{x}{v}}\right)}\right]w^{\prime }(x)$$

(1)

$$\phi =1+\beta {\mathrm{e}}^{-\alpha \left(1-{\displaystyle \frac{x}{U}}\right)}$$

(2)

where x is the distance between the calculation point and the center of the foundation beam; p(x) is the foundation reaction force of the Livkin model; k is the coefficient of the foundation bed; α and β are dimensionless parameters related to the properties of the foundation soil; U is 1/2 the length of the foundation beam; w′ is the amount of foundation deformation; and φ is the Livkin coefficient.

The tunnel is treated as a Timoshenko long beam, assuming that the foundation bed coefficient is constant along the tunnel width. When calculating the foundation of the plane, α is 10 and β is 1.0.

It is proposed that for a long beam resting on the ground, the Kvesic formula for calculating the foundation bed coefficient is Equation (3).

$$k={\displaystyle \frac{0.65E_{\mathrm{s}}}{D\left(1-{\nu }^2\right)}}\sqrt[12]{{\displaystyle \frac{E_{\mathrm{s}}{D}^4}{(EI)}}}$$

(3)

where Es and ν are the elastic modulus and Poisson’s ratio of the soil layer where the tunnel is located, respectively; D is the outer diameter of the tunnel; and (EI)_eq is the equivalent flexural stiffness of the tunnel.

In this paper, 2 times the Kvesic subbed coefficient is used to estimate the subbed coefficient of the foundation beam at a certain buried depth.

$$k=2k$$

(4)

4.2. Tunnel Model Calculation Parameters

Under the action of external load, the subway tunnel will produce uneven settlement, which will cause shear and misalignment deformation between the tunnel segments. In this paper, the tunnel is regarded as a Timoshenko long beam, and the amount of misalignment between the segments is calculated considering the shear action between the segments.

Based on the Timoshenko beam theory, the beam bending moment M and shear force Q can be expressed as follows:

$$M=-{(EI)}_{\mathrm{eq}}{\displaystyle \frac{\mathrm{d}\theta (x)}{\mathrm{d}x}}$$

(5)

$$Q={(\lambda C\mathsf{\Omega })}_{\mathrm{cq}}\left[{\displaystyle \frac{\mathrm{d}w(x)}{\mathrm{d}x}}-\theta (x)\right]$$

(6)

where θ(x) is the vertical angle of the tunnel; and w is the tunnel settlement.

It is assumed that there is no separation between the tunnel and the soil, and the deformation coordination element is satisfied, that is, the tunnel settlement is equal to the foundation deformation, w = w′.

Considering the influence of the transverse performance of the subway tunnel on the longitudinal equivalent flexural stiffness, the calculation formula for (EI)_eq is given.

$${(EI)}_{\mathrm{eq}}=E_{\mathrm{c}}{\lambda }_1+{\displaystyle \frac{n{l}_{\mathrm{s}}{I}_{\mathrm{b}}{\lambda }_2}{A_{\mathrm{s}}}}$$

(7)

In the formula: E_c is the elastic modulus of the segment concrete; and λ₁, λ₂ are the parameters related to the tunnel cross section. Based on the Timoshenko beam theory, the formula for calculating the dislocation δ between the tunnel segments is given.

$${(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}=\zeta {\displaystyle \frac{l_{\mathrm{s}}}{\frac{l_{\mathrm{b}}}{n{k}_{\mathrm{b}}{G}_{\mathrm{b}}{A}_{\mathrm{b}}}+\frac{l_{\mathrm{s}}-l_{\mathrm{b}}}{k_{\mathrm{s}}{G}_{\mathrm{s}}{A}_{\mathrm{s}}}}}$$

(8)

$$\delta =l_{\mathrm{s}}{\displaystyle \frac{Q}{{(\lambda C\mathsf{\Omega })}_{\mathrm{oq}}}}$$

(9)

where ζ is the correction coefficient; l_b is the length of bolt; kb and k_s are the shear coefficients of the bolt and tunnel segment, respectively; G_b and G_s are the shear modulus of the bolt and tunnel segment, respectively; and A_b is the cross-sectional area of the bolt.

4.3. The Vertical Additional Stress Exerted by the Excavation of the Connection Channel on the Foundation of Adjacent Buildings

As shown in Figure 7, the excavation of the contact channel exerts a uniform rectangular load of length L and width B. The load density is q, the horizontal distance between the center of the connection channel and the foundation axis of the adjacent building is S, and the buried depth of the tunnel axis is H.

The unit force at some point (ξ, η) in the excavation of the connection channel is qdξdη. Using the basic solution of Boussinesq, the vertical additional stress σ exerted on a certain point (x₁, S, H) of the foundation of adjacent buildings is obtained.

$$\sigma ={\displaystyle \frac{3q}{2\pi }}{\int }_{-L/2}^{L/2}{\int }_{-B/2}^{B/2}{\displaystyle \frac{H^3}{R^5}}\mathrm{d}\xi \mathrm{d}\eta$$

(10)

where $R=\sqrt{{\left(x_1-\xi \right)}^2+{(S-\eta )}^2+H^2}$.

4.4. Deformation of Building Foundation Caused by Vertical Additional Stress

The foundation of the building is simplified as a Timoshenko long beam placed on a Livkin foundation, and the force analysis of the micro-elements on the tunnel is carried out to obtain the higher-order differential equation of the tunnel settlement w and the vertical additional stress σ.

$$\begin{array}{c}{\displaystyle \frac{{\mathrm{d}}^{4}w(x)}{\mathrm{d}{x}^4}}-{\displaystyle \frac{k\phi D}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\displaystyle \frac{{\mathrm{d}}^{2}w(x)}{\mathrm{d}{x}^2}}+{\displaystyle \frac{k\phi D}{{(EI)}_{\mathrm{eq}}}}w(x)\\ ={\displaystyle \frac{D}{{(EI)}_{\mathrm{eq}}}}\sigma (x)-{\displaystyle \frac{D}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\displaystyle \frac{{\mathrm{d}}^2\sigma (x)}{\mathrm{d}{x}^2}}\end{array}$$

(11)

The building foundation is divided into n long l small units, and 2 long l virtual underestimated to the beginning and end of the building foundation. The numerical solution of Equation (11) is found using the finite difference method.

A difference transformation is performed on the derivative of w to convert Equation (11) into the finite difference form.

$$\begin{array}{c}{\displaystyle \frac{6w_i-4\left(w_{i+1}+w_{i-1}\right)+\left(w_{i+2}+w_{i-2}\right)}{l^4}}-\\ {\displaystyle \frac{k{\phi }_{i}D}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\displaystyle \frac{w_{i+1}-2w_i+w_{i-1}}{l^2}}+{\displaystyle \frac{k{\phi }_{i}D}{{(EI)}_{\mathrm{eq}}}}{w}_i\\ ={\displaystyle \frac{D}{{(EI)}_{\mathrm{eq}}}}{\sigma }_i-{\displaystyle \frac{D}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\displaystyle \frac{{\sigma }_{i+1}-2{\sigma }_i+{\sigma }_{i-1}}{l^2}}\end{array}$$

(12)

where w_i, σ_i are the settlement amount of node element i and the vertical additional stress they are subjected to, respectively; and φ_i is the Livkin coefficient at node element i.

There is no fixed constraint at both ends of the building foundation, so the bending moment M and shear force Q at both ends of the building foundation are 0. The displacement expressions of the four virtual nodes of the building foundation are obtained by connecting the vertical (5), Equations (6) and (11).

$$\begin{array}{c}{w}_{-2}=\left[{\displaystyle \frac{k^2{\phi }^2{D}^2{l}^4}{{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}^2}}+{\displaystyle \frac{4k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+4\right]w_0-\\ \left[4+{\displaystyle \frac{2k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}\right]w_1+w_2+\\ {\displaystyle \frac{D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}\left({\sigma }_1-{\sigma }_{-1}\right)-\\ \left[{\displaystyle \frac{2D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+{\displaystyle \frac{k\phi D^2{l}^4}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}^2}}\right]{\sigma }_0\end{array}$$

(13)

$$\begin{array}{c}{w}_{n+2}=\left[{\displaystyle \frac{4k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+{\displaystyle \frac{k^2{\phi }^2{D}^2{l}^4}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}^2}}+4\right]w_n-\\ \left[4+{\displaystyle \frac{2k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}\right]w_{n-1}+w_{n-2}+\\ {\displaystyle \frac{D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}\left({\sigma }_{n-1}-{\sigma }_{n+1}\right)-\\ \left[{\displaystyle \frac{2D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+{\displaystyle \frac{k\phi D^2{l}^4}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}^2}}\right]{\sigma }_n\end{array}$$

(14)

$$\begin{array}{c}{w}_{-1}=\left[{\displaystyle \frac{k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+2\right]w_0-w_1-\\ {\displaystyle \frac{D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\sigma }_0\end{array}$$

(15)

$$\begin{array}{c}{w}_{n+1}=\left[{\displaystyle \frac{k\phi D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}+2\right]w_n-w_{n-1}-\\ {\displaystyle \frac{D{l}^2}{{(\lambda C\mathsf{\Omega })}_{\mathrm{eq}}}}{\sigma }_n\end{array}$$

(16)

4.5. Engineering Verification of Calculation Method

The settlement of the existing building above is due to the construction. Figure 8 compares the calculated results of this method with the measured data in document [13] and the calculation results of the Euler–Bernoulli beam method. It can be seen that the calculated results of the present method have better consistency with the measured longitudinal settlement of the tunnel, which is less than the calculated results of the E-B beam method and the calculation method in reference [15]. The maximum settlement obtained by the method presented in this paper and by the E-B beam method is 20.1 mm and 25.0 mm, respectively, both exceeding the norm limit of 20 mm.

The calculated result obtained by the E-B beam method is much higher than the measured value. If this method is used to predict the longitudinal deformation of the building, excessive protective measures will be taken to reduce the settlement of the tunnel, which will cause unnecessary economic losses. The method presented in this paper can well reflect the deformation effect of the excavation of the contact passage on the building.

4.6. Influence of Different Excavation Speed on Building Foundation

The influence of different excavation speeds on the building foundation is calculated, as shown in Figure 9. It is not difficult to see from Figure 9a that with the increase in the excavation rate, the settlement rate may be accelerated: a faster excavation rate means that the soil support structure is removed faster, which may lead to a faster redistribution of the soil stress, which in turn causes an increase in the settlement rate of the foundation of the building above. When the settling rate reaches 60 mm per day, the settling rate reaches −34 mm, which is 1.5 times that at 10 mm per day. Although the specific settlement amount is also affected by a variety of factors, such as the soil properties, freezing effect, and support measures, in general, a faster excavation rate may lead to a greater total settlement amount because the soil does not have enough time to gradually adapt to the stress changes caused by the excavation. During excavation, the soil will undergo a complicated stress redistribution process. With the advance of the excavation face, the stress state in the surrounding soil will change, resulting in the deformation and settlement of the soil. Due to the heterogeneity of the soil and the asymmetry of the excavation face (for example, the excavation face may be tilted to one side or irregularly shaped), the stress redistribution process will also show asymmetry, which will affect the distribution of the settlement rate.

It Is not difficult to find from Figure 9b that a faster excavation speed may lead to more drastic changes in the lateral pressure of the soil on the building foundation, and such changes in the lateral pressure will be transformed into the bending moment of the building foundation. Therefore, as the excavation speed increases, the foundation of the building may bear a greater bending moment. Rapid excavation may also cause the distribution of the bending moments across the foundation of the building to become more uneven, especially if the excavation face approaches or passes under the building.

It is not difficult to see from Figure 9c that the shear force may increase with the increase in the excavation speed: similar to the bending moment, the faster excavation speed may also cause the soil to increase the shear force on the building foundation. This is because after the soil loses support quickly, its internal stress state will change rapidly and then produce greater shear force on the building foundation. At a faster excavation speed, the redistribution and transfer process of the soil stress may be more rapid, so the peak of the shear may have reached a higher level in the early stage of the excavation.

4.7. Influence of Different Elastic Moduli of Frozen Soil on Building Foundation

The influence of different elastic moduli of frozen soil on the building foundation is calculated, as shown in Figure 10. When the influence of different elastic moduli of frozen soil on the settlement, shear force and bending moment of the foundation of the building above is discussed in relation to Figure 10a during the tunnel freezing excavation, it is not difficult to find from Figure 10a that when the elastic modulus of frozen soil is less than 7 MPa, as the elastic modulus of the frozen soil decreases (e.g., from 4 MPa to 2 MPa), the settlement of the building foundation above the tunnel increases significantly. This is because the low elastic modulus means that the soil has a weak resistance to deformation. In the process of tunnel excavation and freezing, the compressibility of the soil is enhanced, resulting in the large subsidence of the foundation above. Specifically, when reducing from 4 MPa to 2 MPa, the settlement amount may increase sharply from a small negative value (such as −10 mm) to −40 mm, indicating that the settlement change is very significant.

It can be seen from Figure 10b that when the elastic modulus of frozen soil reaches or exceeds 7 MPa, a significant change is that the settlement changes to uplift. This is because the soil with a high elastic modulus has strong stiffness and stability. Under the action of freezing, the soil around the tunnel is compressed, resulting in upward displacement, which leads to the uplift phenomenon of the foundation of the building above. This change is a new challenge for the stability of buildings and requires special attention. When the elastic modulus of frozen soil is low, with the increase in the settlement, the shear force at the bottom of the foundation of the building will also increase correspondingly. Especially when the settlement increases sharply (such as from 2 MPa to 4 MPa and then to a lower value), the shear force may reach a peak value, causing adverse effects on the structure of the building.

It can be seen from Figure 10c that during the change in the elastic modulus of frozen soil, the change in the bending moment is also affected by the change in the settlement and shear force. When the settlement increases, the building foundation will bear a greater bending moment, especially at the edge or corner of the foundation, where the bending moment may reach the maximum value, posing a threat to the overall stability of the structure.

4.8. Building Safety Evaluation

In the process of tunnel construction, the most important criterion for determining the safety of buildings and tunnel structures is whether the deformation of their structures meets the control standards. According to GB 50007-2011 [16], the settlement amount of building foundation must meet the following:

$$\begin{array}{l}L\le 30 \mathrm{mm}\\ \Delta L\le \min (0.004E,L)\end{array}$$

(17)

In the formula, L—settlement of building foundation, mm; ΔL—uneven settlement of building foundation, mm; and E—the length or width of the building foundation, m.

Different cities have different regulations governing the amount of land surface and building settlement. The settlement of the surface and the building in this project is required to be no more than 30 mm. For buildings with different structural forms, the standard of the relative settlement of the ground surface stipulated in the code is unified.

From the analysis of the measured engineering data, it can be seen that the maximum settlement of the foundation of this building ΔL = 45 mm < min{0.004 E, L} = 124 mm. It can be seen that in the process of tunnel construction, the deformation and settlement of the building meet the requirements of the control standards.

5. Conclusions

Through the grouting improvement technology, we find that the freezing wall thickness is optimal when the cement content is 10%, indicating that the gain effect of the grouting on the artificial freezing method is most significant at this content. However, when the cement content exceeds this value, the freezing wall thickness begins to decrease, indicating the possible adverse effects of excessive cement incorporation. This finding provides an important reference for the setting of grouting improvement parameters in practical engineering.

The analysis of the temperature field of the freezing wall reveals the double-sided effect of the cement content on the freezing effect. The temperature field of the freezing wall develops more evenly when the content is less than 10%. However, excessive incorporation will hinder the development of the freezing wall and freezing front and limit the expansion range of the freezing temperature field. This conclusion highlights the importance of precise control of the cement content to ensure the stability and efficiency of freezing works.

This study on the excavation speed points out that with the increase in the excavation speed, the changes in the foundation settlement speed and the soil side pressure are intensified, which may cause the shear peak to reach a higher level in advance, posing a threat to the safety of the building structure. Therefore, it is necessary to reasonably control the excavation speed and adjust the construction plan according to the real-time monitoring data to ensure the safety of the project. The method proposed in this paper is consistent with the measured data, which verifies its effectiveness in evaluating the influence of tunnel excavation on building deformation and provides a useful reference for similar projects.