The elastic modulus, Poisson ratio, cohesion, and friction angle of the cohesive soil layer in the proximity area are the main parameters for the settlement assessment induced by undercrossing tunnel excavation [
29,
30]. These mechanical parameters can be measured using basic geotechnical tests. In this study, the underground penetration depth of the subway is about 10.5 m, and the soil layer is mainly composed of mixed fill (0~0.5 m), silty clay (0.5~5.0 m), and clay (5.0~10.5 m). The mixed fill has low self-stability and poor engineering properties but thin thickness, and the settlement deformation after engineering compaction is basically affected. The silty clay has low strength and poor engineering geological properties. The strength of the clay is normal, and the engineering geological properties are normal. The shallow buried depth of the site diving and micro-confined water level has a certain influence on the formation deformation. The deformation between the upper foundation of the subway underpass area and the top of the subway tunnel is mainly controlled by cohesive soil. Therefore, in this paper, cohesive soil samples were taken at ten locations, basic soil tests were carried out, and test data were obtained.
Figure 1 is the test data of elastic modulus, Poisson ratio, cohesion, and friction angle of the cohesive soil layer in the proximity area. These four mechanical parameters were tested using a conventional triaxial testing apparatus, strain gauge, and hydraulic shear testing apparatus. It can be seen that the mechanical parameters of cohesive soil are discretely distributed. There are two main reasons for this. Firstly, the size of the soil sample taken for testing may not fully represent the entire volume or characteristics of the soil mass. Small samples might not capture the variations present in larger volumes, leading to discrepancies in the measured parameters. Secondly, soil properties often exhibit spatial heterogeneity, meaning that they can vary significantly over short distances. The soil at one location may have different characteristics than the soil just a few meters away. This spatial variability can result in disparate measurements when sampling at different points. Therefore, to calculate the settlement of the cohesive soil layer induced by undercrossing tunnel excavation, the limited sample data and spatial heterogeneity needs further clarification.
2.1. Generation Method of Limited Sample Data
Geological and soil conditions can be highly complex and heterogeneous. Limited data may result from challenges in accurately characterizing the subsurface layers, especially in regions with diverse lithology and soil types. Conducting comprehensive geotechnical investigations involves significant costs. Budget limitations may constrain the extent and depth of field explorations, leading to a reduced number of test points and limited data coverage [
31]. Copula theory finds application in geotechnical engineering, specifically in modeling the complex dependencies among various soil and rock parameters. In subsurface investigations, understanding the joint distribution of factors like soil strength, permeability, and consolidation is crucial for an accurate risk assessment and design. Copulas allow engineers to capture and model the intricate relationships between geotechnical variables, providing insight into their joint behavior without making assumptions about individual distributions. This is particularly valuable in scenarios where traditional methods may oversimplify or overlook the complexities of soil behavior [
32]. In slope stability analysis, for example, copulas can help characterize the joint probability of factors contributing to failure, such as soil cohesion, friction angle, and groundwater conditions. Similarly, in foundation design, copula theory aids in assessing the combined influence of parameters like bearing capacity and settlement.
Multivariate copula theory is a statistical framework used to model and analyze the joint distribution of multiple random variables, emphasizing the dependence structure while allowing for flexibility in capturing complex relationships. In the context of multivariate copulas, the emphasis is on understanding how variables co-move or co-vary without being restricted by specific marginal distributions. The theory assumes the independence of marginal distributions, enabling the separation of the marginal behaviors from the dependence structure. This separation allows practitioners to model various types of dependencies, including positive, negative, and tail dependencies, making multivariate copulas particularly useful in fields like finance, environmental science, and reliability analysis [
33,
34]. Multivariate copulas can be classified into different families, such as Archimedean and Elliptical copulas, each offering unique ways to model dependencies. According to copula theory, if
F1(
x1),
F2(
x2), ···,
Fn(
xn) are the marginal distribution functions of the desired n-dimensional parameters, then there must be a copula function which can be expressed as the n-dimensional joint distribution function (JDF)
F(
x1,
x2, ···,
xn):
The corresponding probability density function (PDF) can be written as:
where
fn(
xn) is the PDF of the variable
xn;
D(·) is the PDF of
C; and
θ is the copula parameter.
Spearman rank correlation coefficient defines the correlation between variables through the consistency of the rank between two variables, and its definition can be expressed as follows: if (
x11,
x21) and (
x12,
x22) are two sets of observations of random variables (
X1,
X2), if (
x11,
x21)(
x12,
x22) > 0, the two sets of observations are consistent, while the opposite is inconsistent. Consider (
X1,
X2) two independent identical distribution vector (
X′
1,
X′
2) and (
X″
1,
X″
2); if
P [(
X1 −
X′
1) (
X2 −
X″
2) > 0] says they are a consistent probability and
P [(
X1 −
X′
1) (
X2 −
X″
2) < 0] says they are not a consistent probability, then the Spearman rank correlation coefficient
γ is defined as the difference between the probability of consistency and the probability of inconsistency between a random vector and any independent equally distributed vector in this direction:
Since the Spearman rank correlation coefficient represents the rank consistency of any independent same distribution vector with the sample population or distribution population, the correlation of the Spearman rank correlation coefficient is closer to the population than that of a Kendall rank correlation coefficient. At the same time, the relationship between the rank correlation coefficient γ of Spearman and the parameter θ of the copula function is as follows:
The γ can be obtained by the Pearson correlation coefficient of the probability value of the edge distribution of the parameter. The computational relationship can be expressed as:
According to Equations (1)–(5), the JDF and PDF of the sample data for the cohesive soil layer induced by undercrossing tunnel excavation can be constructed.
Bootstrap is a resampling technique used for statistical inference, providing a robust method to estimate the sampling distribution of a statistic. It involves creating multiple datasets from sampling with replacement from the original data. This process generates pseudo-populations that mimic the underlying distribution of the observed data [
35,
36]. The key principle is to approximate the sampling distribution of a statistic by repeatedly drawing samples from the observed data. This allows for the calculation of standard errors, confidence intervals, and other statistical measures without relying on parametric assumptions. In essence, bootstrapping leverages the observed data to simulate a multitude of hypothetical datasets, enabling a more comprehensive understanding of the variability inherent in the sample. It is particularly valuable when the underlying distribution is unknown or complex, providing a versatile tool for statistical analysis and hypothesis testing. The bootstrap sample simulation method mainly includes the following three steps:
Step 1: Sampling with Replacement. Begin by randomly selecting ‘n’ observations from the original dataset, allowing for replacement. In this step, each observation has the potential to be picked multiple times or not at all. The goal is to create a resampled dataset of the same size as the original.
Step 2: Statistic Calculation: Compute the desired statistic (e.g., mean, median, standard deviation) for the newly created resampled dataset. This statistic serves as an estimate for the parameter of interest.
Step 3: Repeat and Aggregate: Repeat the sampling with replacement and statistic calculation process a large number of times. This repetition generates a collection of bootstrap samples and their corresponding statistics. The aggregated results provide insights into the distribution of the statistic of interest.
These three steps are fundamental to the bootstrap method, allowing researchers and statisticians to simulate the variability of a statistic. The method is particularly useful when analytical methods may be challenging or when the underlying population distribution is not well known. The resulting bootstrap distribution provides valuable information for constructing confidence intervals and understanding the uncertainty associated with the estimated statistic.
Based on the above three steps, the subsample with the same distribution of the limited data for the soil layer in the adjacent construction area can be constructed.
2.2. Characterization Process of Spatial Heterogeneity
Soil heterogeneity refers to the variation or diversity in soil properties within a given area or volume. These properties may include factors such as texture, composition, structure, moisture content, and nutrient levels [
37]. In other words, soil heterogeneity indicates that different parts of the soil exhibit different characteristics. This variability can occur at various scales, ranging from small micro-environments within a single soil profile to larger spatial scales across a landscape. Factors contributing to soil heterogeneity include geological processes, weathering, biological activity, and human influence such as land use and management practices. Geotechnical engineering considers spatial heterogeneity when assessing the stability and load-bearing capacity of soil structures. Variations in clay properties can lead to differential settlement, affecting the performance of foundations, roads, and other infrastructure projects [
38,
39]. Random field methods in geotechnical engineering are employed to quantify the spatial variability. In this approach, soil characteristics like shear strength, permeability, and settlement are considered as random fields, acknowledging their spatial distribution across a site. The process involves modeling these parameters as stochastic functions, capturing their variability over space. To describe spatial variability, statistical tools such as variogram analysis and covariance functions are utilized. Variograms illustrate the spatial correlation structure of soil properties, depicting how they change with distance. Random field methods incorporate these variograms to simulate and predict the spatial distribution of soil parameters. The outcome is a probabilistic representation of the subsurface, allowing engineers to assess uncertainties and design structures that account for the inherent variability in soil behavior. By integrating random field methods, geotechnical engineers gain a comprehensive understanding of the complex and heterogeneous nature of soils. This facilitates more accurate risk assessments, reliability analyses, and ultimately leads to safer and more robust designs in geotechnical projects.
The random field model proposed by Vanmarcke considers the soil layer as statistically uniform and the spatial distribution of soil properties as random fields. The essence of this model is to simulate soil profiles with continuous stationary random fields, depict the autocorrelation of geotechnical materials with autocorrelation functions (or autocorrelation distances), and establish the variance calculation method for the transition from the point characteristics obtained from test data to the spatial average characteristics. The statistical properties of random field can be expressed by its finite dimensional joint distribution function or probability density. However, it is often difficult to determine the distribution function or probability density in practical problems. The key parameters mainly include mean function, variance function, correlation function, covariance function, and correlation coefficient. In this paper, the elastic modulus, Poisson ratio, cohesion, and friction angle of the cohesive soil layer in the proximity area is simulated as four independent random fields (
e1,
e2,
e3,
e4). The mean and variance function of the any parameter random field can be written as:
where
mi is the mean of the parametric random field;
σi is the standard deviation of the parametric random field.
The autocorrelation function of any parameter random field can be expressed as:
The cross-correlation function of the two parameter random field can be expressed as:
The covariance function of the two parameter random field can be expressed as:
The correlation coefficient of the two parameter random field can be expressed as:
According to Equations (7)–(10), the spatial heterogeneity of the cohesive soil layer in the proximity area can be quantified.
2.3. Calculation Detail of Settlement
To study the stress field and settlement of the cohesive soil layer induced by undercrossing tunnel excavation, the first step is to study the constitutive model and numerical algorithm of the cohesive soil layer in the adjacent construction area. Due to the complexity of the cohesive soil layer in the adjacent construction area, a study of its constitutive model is still in the exploratory stage. Most of the constitutive models that have been constructed are in the experimental and theoretical analysis stage, and generally can only reflect one or two of the shear shrinkage, dilatancy, hardening, and softening characteristics [
40,
41]. The study object may have shear shrinkage, dilatancy, hardening, and softening local areas at the same time. Therefore, the double-yield surface constitutive model is adopted. The double-yield surface constitutive model for cohesive soils is an advanced geotechnical framework that comprehensively characterizes the intricate behavior of cohesive soils. This model is crucial for simulating the response of soils to various loading conditions. It incorporates two distinct yield surfaces to accurately represent normal and shear loading responses. The primary yield surface is associated with normal loading, while the secondary yield surface corresponds to shear loading. This model integrates the Mohr–Coulomb yield criterion with additional parameters, encompassing aspects like dilation, strain hardening, and anisotropy. It offers a nuanced depiction of soil deformation, shear strength, and stress–strain relationships. Engineers utilize this model to optimize the design of foundations, slopes, and retaining structures, ensuring structural stability in geotechnical projects.
Based on the modified Cambridge model, the yield function of the cohesive soil layer induced by undercrossing tunnel excavation can be expressed as:
where
p is the mean normal stress;
q is the generalized shear stress.
M and
pr are two model parameters, their values are
and
, respectively.
The iso-directional consolidation test results of cohesive soil are plotted in the coordinate system
εv-ln
p, and the iso-directional consolidation curve can be written as:
After that, the current yield surface function of the cohesive soil layer induced by undercrossing tunnel excavation can be expressed as:
where
p0 is the spherical stress at the current yield surface corresponding to the initial volume strain.
The relationship between the change in material volume strain and the spherical stress during the whole stress path from a shear to critical state under different confining pressure conditions can be obtained by a triaxial compression test. It can be written as:
After that, the refer yield surface function of the cohesive soil layer induced by undercrossing tunnel excavation can be expressed as:
where
p0 is the spherical stress at the current yield surface corresponding to the initial volume strain.
According to the classical elastic–plastic theory, the total settlement of the cohesive soil layer induced by undercrossing tunnel excavation can be divided into an elastic strain increment and plastic strain increment. The stress–strain relationship can be written as:
where [
D]
e is the elastic constitutive matrix; [
D]
ep is the elastic–plastic constitutive matrix; [
D]
p is the plastic constitutive matrix.
A is the function of hardening parameter;
is the plastic potential function, and
.
Substituting Equations (13) and (15) into (16), the detailed expression of the stress–strain relationship can be obtained.
Finite Element Method (FEM) plays a crucial role in analyzing foundation deformation. It involves discretizing the complex foundation–soil system into smaller elements to simulate real-world behavior. First, the model gathers geotechnical data to establish material properties and loading conditions [
42,
43]. It creates a model with finite elements representing the foundation and surrounding soil and applies boundary conditions reflecting actual constraints, including external loads and soil–structure interactions. Using FEM to solve equations describing the behavior of each element, the model provides a comprehensive analysis of deformation patterns. FEM aids in understanding the stress distribution, settlement, and potential failure zones. As well, time-dependent analyses consider factors like consolidation and creep, offering insights into long-term behavior. Then, the model is validated against field measurements for accuracy. Sensitivity analyses explore the impact of varying parameters on deformation predictions. This iterative process refines the model, ensuring a reliable tool for studying and mitigating foundation deformations using the powerful Finite Element Method. For the settlement of the cohesive soil layer induced by undercrossing tunnel excavation, the FE formulae can be written as:
where [
K] is the stiffness matrix; {Δ
R} is the increment of equivalent nodal forces vector; [
B] is the element strain matrix; [
D]
g is the transitional matrix.
The FEM for elasto-plastic analysis incorporates material nonlinearity to simulate structural behavior under varying loads. It models plastic deformation, capturing the transition from elastic to plastic response. This method employs a stiffness matrix modification, yielding accurate predictions of plastic strains, stress redistribution, and structural performance under cyclic loading. Therefore, the transitional matrix can be written as:
where
is the equivalent yield stress increment.
is the load step stress increment.
According to Equations (16)–(20), the settlement of the cohesive soil layer induced by undercrossing tunnel excavation can be calculated. The width, height, crown radius, and invert depth of the undercrossing tunnel is 3.8 m, 4.1 m, 2.4 m, and 1.3 m, respectively.
Figure 2 shows the statistical characteristics of uncertain settlement characteristics of the cohesive soil layer induced by undercrossing tunnel excavation. It can be seen that the settlement of the cohesive soil layer are discretely distributed. This result is reasonable because of the limited sample data and spatial heterogeneity of the cohesive soil layer.