Keywords

1 Introduction

Rail transit shield tunnel with shallow buried depth often passes through the urban core area. The site stratum is seriously weathered, the underground pipe network is dense, and there are many ground buildings. When the soil disturbance caused by tunnel construction is transmitted to the surface, it will lead to surface deformation. And if the stratum disturbance is large, it will bring risks to the adjacent existing rail transit lines and buildings (structures). Therefore, mastering the law of surface deformation caused by shield tunnel construction is the key link to realize project safety risk control. For the problem of stratum deformation caused by tunnel construction, Many researches have been carried out by domestic and abroad researchers, but they mostly ignore the influence of spatial variability of geotechnical parameters. Random field theory [1] provides an effective means to describe the spatial variation characteristics of geotechnical parameters. In recent years, it is widely used in the reliability analysis of tunnel engineering. Based on the random field theory, Cheng et al. [2] studied the effects of elastic modulus variation coefficient and scales of fluctuation on surface settlement. Considering the spatial variability of parameters, Wen et al. [3] analyzed the mechanical response of surrounding rock after large section tunnel excavation. Li et al. [4] studied the influence of scales of fluctuation and variation coefficient of soil elastic modulus on stratum deformation during twin shield tunnel construction. Miro et al. [5] studied the influence of parameter distribution type on surface deformation and pointed out that when the variability is small, the influence of parameter probability distribution type is very small. Xiao et al. [6] studied the influence of low stiffness random field location on surface deformation during tunnel construction.

At present, the analysis on the surface deformation response of tunnel construction considering the spatial variability of parameters mostly focuses on the influence of the spatial variability of parameters on the digital characteristics of deformation and the dispersion degree of deformation curve. There are relatively few systematic studies on the surface deformation mode of shield tunnel due to the spatial variability of geotechnical parameters, It is very important to clarify the meso characteristics and mode of surface deformation curve for the protection of shallow old houses and underground pipelines. In view of this, taking the spatial variability of soil elastic modulus as the starting point, this paper systematically studies the influence of two basic spatial variability characteristics of parameter spatial autocorrelation (vertical and transverse scales of fluctuation) and randomness (variation coefficient) on surface deformation during shield tunnel construction by using the combination of random field theory, finite difference method and Monte Carlo strategy. The change law of the shape and characteristics of surface deformation curve is discussed, and the surface deformation model is summarized and refined.

2 Random Analysis Method of Surface Deformation in Shield Construction

2.1 Parameter Spatial Variability

The geotechnical parameters have the dual characteristics of local randomness and overall structure. The random field theory regards the geotechnical parameters at any point as a random variable that approximately obeys a certain probability distribution. The spatial structure of the parameters is characterized by spatial concepts such as fluctuation range and autocorrelation structure.

2.2 Random Analysis Process of Surface Deformation

Fig. 1.
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Flow chart of random analysis of surface deformation.

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The random analysis method of surface deformation induced by shield tunneling based on random field theory is constructed. The analysis process is shown in Fig. 1: (1) The spatial variability characteristics of geotechnical parameters is statistically analysed, including the probability distribution characteristics of parameters (mean value, standard deviation and distribution type) and spatial correlation characteristics (related structure and scales of fluctuation). The finite difference software FLAC3D is used to construct the numerical model of shield tunnel, divide the grid and record the model size. (2) The parametric random field model is generated with the help of MATLAB platform. (3) Realize the mapping from the independently generated parameter random field model to the numerical calculation model. (4) With the help of Monte Carlo strategy, repeat steps (2)–(4) to realize multiple random analysis of surface deformation caused by shield tunnel construction. (5) With the help of probability statistics method, the surface deformation results obtained by Monte Carlo random calculation are analyzed.

3 Characteristics and Mode Analysis of Surface Deformation Curve

3.1 Numerical Calculation Model

Fig. 2.
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Schematic diagram of model

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Table 1. Physical and mechanical parameters of soil and segments in the model.
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Based on the shield tunnel project in weathered granite stratum in Xiamen, this paper simplifies the tunnel excavation problem into a two-dimensional plane strain model to carry out the random analysis of the surface deformation response of shield construction. Model size is 76 m × 34 m (width × Height), tunnel diameter is D = 6.2 m, axis buried depth is H = 15.4 m, and the maximum size of grid is about 0.75 m. Except that the surface is a free boundary, other boundaries are subject to normal constraints. The elastic-perfectly plastic body of M-C yield criterion is adopted for the soil, and the shell element is adopted for the simulation of lining structure. The values of physical and mechanical parameters of materials are shown in Table 1. The model is divided into upper and lower layers. When assigning the elastic modulus of subsoil, taking 3 times of the original modulus [7] to simply consider the loading and unloading characteristics of soil. The numerical calculation model is shown in Fig. 2.

This paper focuses on the influence of the spatial variability of soil elastic modulus on the surface deformation mode of shield tunnel construction, and other physical and mechanical parameters are constant. Considering the calculation scale of Monte Carlo random simulation, and the surface deformation mainly occurs in the stress release stage [7], the calculation result analysis is only carried out for the surface deformation in the stress release stage.

3.2 Deterministic Analysis

Fig. 3.
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Relationship between stress release coefficient and surface settlement deformation curve.

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Figure 3 shows the surface settlement curve under different stress release coefficients λ. According to the figure, it can be seen that the stress release coefficient increases, the surface settlement value increases accordingly, and the fitting results of peck formula are in good agreement with the numerical calculation results, which basically obeys the Gaussian distribution. It should be pointed out that the actual engineering situation and site environment are complex and there are many influencing factors, so it is generally difficult to accurately balance the corresponding stress release coefficient.

3.3 Random Analysis

Combined with random field theory, finite difference method and Monte Carlo strategy, the random analysis of surface deformation during shield tunnel construction is carried out. Without losing generality, the stress release coefficient is taken λ = 0.5 [7], the effects of vertical and transverse scales of fluctuation (θz, θx) and variation coefficient (COV) of soil elastic modulus on surface settlement curve model are systematically studied.

Referring to the suggestions on the value of scales of fluctuation of geotechnical parameters [8] (the transverse scales of fluctuation is generally 10.0–80.0 m and the vertical scales of fluctuation is 1.0–3.0 m), the basic value of soil elastic modulus scales of fluctuation is selected as θz = 0.3D = 1.86 m and θx = 6.0D = 37.2 m. On this basis, the random analysis condition is designed and divided into three types of random calculation condition groups, including MCS-z*-x (variable θz, there are 20 simulated working conditions, see Table 2, MCS-z-x* (variable θx, A total of 20 simulated working conditions) and MCS-E*-θ (variable COV, a total of 15 simulation conditions, see Table 3. The log normal distribution [9] is used to describe the uncertainty of soil elastic modulus. And the log modulus field satisfies the anisotropic exponential autocorrelation function [10], which can be expressed as

$$ \rho_{\ln E} (\tau_{x} ,\tau_{z} ) = \exp \left( { - \frac{{2\tau_{x} }}{{\theta_{x} }} - \frac{{2\tau_{z} }}{{\theta_{z} }}} \right) $$
(1)

Where \(\rho_{\ln E} (\tau_{x} ,\tau_{z} )\) is the autocorrelation coefficient of two points in the logarithmic modulus field, and \(0 \le \rho_{\ln E} (\tau_{x} ,\tau_{z} ) \le 1\), \(\tau_{x} ,\tau_{z}\) are the horizontal and vertical distances respectively, \(\theta_{x} ,\theta_{z}\) are transverse and vertical scales of fluctuation respectively.

Considering the calculation accuracy and efficiency, 1000 times are selected as the calculation times of random analysis under each working condition.

Table 2. Random analysis conditions of vertical scales of fluctuation.
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Table 3. Coefficient of variation random analysis condition.
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3.4 Analysis of Surface Deformation Curve Results

Surface deformation is an important index to reflect the impact of shield tunnel construction on the surrounding environment. Based on the random calculation results of surface deformation of three working conditions, the characteristics and types of settlement curve are discussed from the distribution of the location of maximum surface settlement, and the surface deformation mode is summarized and refined.

3.4.1 Shape Analysis of Surface Deformation Curve

Fig. 4.
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Surface deformation curve obtained by random and deterministic calculation.

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Figure 4 shows the ground settlement results of tunnel construction obtained from 1000 random calculations (gray curve in the figure) and deterministic calculations (modulus is the mean value used for random analysis) (black curve in the figure) under the working condition which the scales of fluctuation is θx = 6.0D, θz = 0.5D and coefficient of variation is COV = 0.3. For each realization of the parameter random field, the soil elastic modulus is spatially heterogeneous, and the calculation results are also different, which is shown as a cluster of discrete curves. Relevant studies [2, 4] have also shown that with the increase of spatial correlation and randomness of parameters, the dispersion degree of settlement curve distribution caused by tunnel construction increases accordingly, and the influence of parameter variation coefficient is more significant than scales of fluctuation.

3.4.2 Analysis of Surface Deformation Model

Fig. 5.
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Relationship between maximum settlement location and modulus distribution. (a) The maximum settlement position is left. (b) The maximum settlement position is right. (c) The maximum settlement is located on the axis.

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Comparing the settlement curve characteristics of deterministic analysis and stochastic analysis in Fig. 4, it can be seen that the surface settlement curve obtained by deterministic analysis is single peak, and the maximum value is located above the tunnel axis. Although the settlement curve obtained by random analysis still presents a single peak distribution, it is different from the deterministic results. The maximum settlement is no longer above a single tunnel axis. The specific location of the maximum settlement is closely related to the random distribution of soil modulus within the influence range of tunnel excavation [11]. When a relatively high (low) stiffness area appears on one side of the tunnel axis, the maximum value of the surface deformation curve obtained by random analysis will also deviate (deviate) from that side, which is more appropriate to the on-site monitoring [12], as shown in Fig. 5. Obviously, this influence will gradually weaken with the increase of the distance between the high (low) stiffness area and the excavation surface. When the high (low) stiffness area is outside the influence range of tunnel excavation, the influence on surface deformation can be ignored.

The location of the maximum surface settlement is counted, and the influence of the spatial variability of soil parameters on the surface deformation model is discussed.

  1. (1)

    Location of maximum surface settlement

    1. 1)

      Influence analysis of vertical scales of fluctuation

      Figure 6 shows the distribution of the location of the maximum surface settlement caused by tunnel excavation under the conditions of different vertical scales of fluctuation in MCS-z*-x working condition group (4 groups of different transverse scales of fluctuation). It can be seen that considering the influence of spatial variability of modulus, the location of maximum surface settlement caused by tunnel excavation is a probability distribution interval, in which Under the condition of θz = 0.5D and θx = 6.0D, the distribution range of maximum surface settlement is (−1.1625, 1.1625).

      As can be seen from Fig. 6(a)–(d), with the increase of vertical scales of fluctuation, the interval where the maximum surface settlement occurs expands, the dispersion degree also increases, and its low peak distribution characteristics become more obvious. Among which under the condition of θz = 0.5D and θx = 1.5D, the probability that the maximum settlement point is directly above the tunnel is only 29.7%. The main reason for this phenomenon is that the size of the element concentration area in the high (low) stiffness area in the parameter random model is affected by the scales of fluctuation, and the correlation of the horizontal parameters remains unchanged. The larger the vertical scales of fluctuation, the probability of a large range of high (low) stiffness area increases accordingly, and the probability of asymmetric distribution of parameters on both sides of the tunnel also increases.

      Fig. 6.
      figure 6

      Location statistics of maximum surface settlement under different vertical scales of fluctuation conditions. (a) θx = 1.5D. (b) θx = 3.0D. (c) θx = 6.0D. (d) θx = 12.0D.

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    2. 2)

      Influence analysis of transverse scales of fluctuation

      Figure 7 shows the distribution of the location of the maximum surface settlement caused by tunnel excavation under the conditions of different transverse scales of fluctuation in MCS-x*-z working condition group (4 groups with different vertical scales of fluctuation). It can be seen that under different transverse scales of fluctuation conditions, the distribution law of the location where the maximum surface settlement occurs is opposite to the vertical scales of fluctuation condition. With the increase of transverse scales of fluctuation, the peak distribution characteristics of the location where the maximum surface settlement occurs are prominent. Among which Under the condition of θx = 12.0D and θz = 0.2D, the probability that the maximum settlement point is directly above the tunnel is as high as 70.9%. This is mainly because in the anisotropic random field, with the increase of transverse scales of fluctuation, the correlation of horizontal parameters strengthens and gradually tends to the horizontal mean value, that is, the overall zonal distribution. The distribution of geotechnical parameters on both sides of the tunnel axis is more symmetrical and uniform, and the probability of relatively high (low) stiffness area on one side is small. Therefore, the location of maximum settlement value is mostly directly above the tunnel.

      Fig. 7.
      figure 7

      Location statistics of maximum surface settlement under different transverse scales of fluctuation conditions. (a) θz = 0.2D. (b) θz = 0.3D. (c) θz = 0.4D. (d) θz = 0.5D

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    3. 3)

      Influence analysis of coefficient of variation

      Figure 8 shows the distribution of the location of the maximum surface settlement caused by tunnel excavation under the conditions of different variation coefficients in MCS-E*-θ working condition group (two groups with different scales of fluctuation). It can be seen that with the increase of modulus variation coefficient, the dispersion degree of location distribution of maximum surface settlement also increases, where under the condition of COV = 0.5, θz = 0.3D and θx = 1.5D, the probability that the maximum settlement point is directly above the tunnel is only 23.5%. This is mainly because the larger the coefficient of variation in the parameter random field model, the probability of asymmetric distribution of parameters on both sides of the tunnel will be significantly improved.

      Fig. 8.
      figure 8

      Location statistics of maximum surface settlement under different coefficient of variation conditions. (a) θx = 1.5D. (b) θx = 6.0D.

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  2. (2)

    Settlement curve type

    According to the different positions of the maximum surface settlement, the surface settlement curve is divided into three types. I –III successively represent the single peak type which the wave peak is on the left, the single peak which type wave peak is above the tunnel axis, and the single peak type which wave peak is on the right. The distribution of various surface deformation modes under various random working conditions is studied.

    1. 1)

      Influence analysis of vertical scales of fluctuation

      Table 4 shows the variation of the type and number of surface deformation curves with the vertical scales of fluctuation. It can be roughly seen that in the MCS-z*-x random working condition group, with the increase of the vertical scales of fluctuation, the probability of the same surface model as the deterministic analysis decreases, that is, the random characteristics of the surface settlement form gradually increase. Under different transverse scales of fluctuation conditions, the vertical scales of fluctuation increases from 0.2D to 0.5D, and the probability of type II settlement mode decreases by 10%–20%. Under the condition of θx = 1.5D, the number of three settlement curves is roughly the same.

      Table 4. Statistics of surface settlement curve types under different vertical scales of fluctuation conditions.
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      Selecting a combination of scales of fluctuation (e.g. θx = 3.0D, θz = 0.3D), only one scales of fluctuation value is changed each time. Taking the number of class II curves as the research object, the influence of the change of vertical and transverse scales of fluctuation on the number of curves is analyzed. It is found that the vertical scales of fluctuation increases by only 0.1D, resulting in the change of the number of class II curves, which is roughly the same as that when the transverse scales of fluctuation increases by one time. This also shows that the vertical scales of fluctuation has a more significant impact on surface deformation.

    2. 2)

      Influence analysis of transverse scales of fluctuation

      Table 5. Statistics of surface settlement curve types under different transverse scales of fluctuation conditions.
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      Similarly, Table 5 shows the relationship between the type and number of surface deformation curves and the transverse scales of fluctuation. It can be seen that with the increase of transverse scales of fluctuation, the probability of the same surface model as the deterministic analysis increases significantly, that is, the random characteristics of surface settlement form gradually weaken. Under different vertical scales of fluctuation conditions, the transverse scales of fluctuation increases from 1.5D to 12.0D, the probability of type II settlement mode increases by 25%–30%, and the sensitivity of transverse scales of fluctuation to surface deformation mode is significantly weakened when the transverse scales of fluctuation increases to 9.0D. Under the condition of θx = 12.0D, the probability of type II settlement mode is more than 50%.

    3. 3)

      Influence analysis of coefficient of variation

      Table 6. Statistics of surface settlement curve types under different coefficient of variation conditions.
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    Table 6 shows the variation of the type and number of surface deformation curves with the modulus variation coefficient. With the increase of modulus variation coefficient, the random characteristics of surface settlement form gradually increase. Under different wave distance conditions, the coefficient of variation increases from 0.2 to 0.5, and the probability of type II settlement mode decreases by 20% –30%. When θx = 1.5D and θz = 0.3D, the number of three settlement curves is roughly the same under different coefficient of variation conditions. Combined with the analysis conclusion of scales of fluctuation condition, it can be seen that the diversity of surface deformation modes is affected by parameter correlation and randomness.

4 Conclusion

Aiming at the random response of surface deformation caused by shield tunnel construction, combined with random field theory, finite difference method and Monte Carlo strategy, this paper constructs a random analysis method of surface deformation of shield tunnel construction based on random field theory, and systematically studies the influence of autocorrelation and randomness of soil elastic modulus on surface deformation mode of tunnel construction. The main conclusions are as follows:

  1. (1)

    The location of the maximum surface settlement is closely related to the random distribution of parameters in the influence area above the tunnel. According to the location of the maximum surface settlement, three surface deformation modes are summarized.

  2. (2)

    The size of the element concentration area in the high (low) stiffness area and the probability of asymmetric distribution of parameters on both sides of the tunnel are closely related to the scales of fluctuation and variation coefficient.

  3. (3)

    With the increase of the vertical scales of fluctuation, the decrease of the transverse scales of fluctuation or the increase of the variation coefficient, the low peak distribution characteristics of the location of the maximum surface settlement caused by tunnel construction become more and more obvious. The influence of vertical scales of fluctuation is more significant than transverse scales of fluctuation, and the influence of parameter variation coefficient is significantly stronger than scales of fluctuation.

  4. (4)

    With the increase of the vertical scales of fluctuation, the decrease of the transverse scales of fluctuation or the increase of the coefficient of variation, the probability of the same surface model as the deterministic analysis decreases, that is, the randomness and chaos of the surface settlement curve gradually increase. When the vertical scales of fluctuation increases by 0.1D, the change of the number of type II curves is roughly the same as that when the transverse scales of fluctuation increases by one time.

  5. (5)

    The diversity of surface deformation modes is affected by parameter correlation and randomness, and there is an obvious superposition effect.

The existence of spatial variability of geotechnical parameters will have a significant impact on the deformation response characteristics of the ground surface during shield tunnel construction. Accurately estimating and characterizing the spatial variability of formation geotechnical parameters should be one of the main contents of routine design of shield tunnel engineering.