1 Introduction

With the population of major cities increasing, more underground engineering works are under construction or will be constructed in the near future. Braced excavation has been used widely. During the advance of an excavation, the retaining wall deflection (RWD) and surface settlement (SS) are the key reference to evaluate the safety conditions of underground engineering works and prevent the excavation from failure. In general, the maximum RWD and maximum SS caused by excavation are determined using serviceability limit state (SLS) design [1] and need to satisfy the limiting values specified by the local regulatory agency. Table 1demonstrates an example of criteria for excavation-induced responses in Shanghai, China, according to the ‘Specification for excavation in Shanghai metro construction’ [2]. Thus, it is necessary to predict the maximum RWD and SS during the design of a braced excavation.

Table 1 Criteria for the limiting values of excavation-induced responses in Shanghai, China [2]
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It is acknowledged that the values of soil parameters exert a significant influence on the soil/wall deformation caused by excavations. Thus, it is important to describe the soil parameters accurately. Numerous attempts [3,4,5] have been made to investigate the responses induced by excavations, but limitations exist in these methods in that soils are usually considered as isotropic and homogeneous materials. In reality, uncertainty prevails in geotechnical engineering. Morgenstern [6] divided this uncertainty into three categories: parameter uncertainty, model uncertainty, and human factor uncertainty. Among them, research into geotechnical parameters shows that the parameter uncertainty mainly results from the spatial variability. In this regard, Lumb [7] proposed the concept of ‘spatial variability’ of soil parameters and demonstrated that the spatial variability is attributed to the difference in material composition during the sedimentation process and the effects of various uncertain external forces later in the period. Vanmarcke [8] treated soil parameters as a random field and established a random field model encompassing spatial variability. Using the random field model, many studies have been conducted and reported foci include, e.g., foundation settlement [9], ground movements caused by tunneling [10], and slope stability [11, 12]. Recently, the random field model is also employed in the field of braced excavations. Luo et al. [13] assessed the influences of soil spatial variability on excavation-induced responses and identified the possibility of geotechnical and structural failure in engineering design. Ching et al. [14] explored the phenomenon of a worst-case scale of fluctuation (SOF) in basal heave analysis for excavation in spatially variable clays. Gholampour and Johari [15] developed a beneficial method for reliability-based analysis of braced excavation considering the uncertainties involved in the soil–structure interaction. Sert et al. [16] implemented random finite element modeling (RFEM) to estimate the wall deflection and bending moment of the retaining wall with consideration of the spatial variations of the effective friction angle φ′. Lo and Leung [17] proposed an approach to obtain improved predictions for a braced excavation on the basis of the Bayesian updating of subsurface spatial variability. According to the aforementioned studies on soil parameters spatial variability, some beneficial works have been conducted in relation to excavations: However, it is noteworthy that most existing studies considering soil parameter spatial variability focus on the isotropic spatial variability, which is inconsistent with site conditions. In reality, anisotropic spatial variability prevails in site. In the framework of anisotropy random fields, the horizontal SOF of soil parameter variability is different from their vertical SOF [18,19,20]. It is necessary to assess the influences of anisotropic spatial variability on the excavation-induced deformation responses.

In reality, the anisotropy of soft clay has been considered in geotechnical engineering [21,22,23,24,25,26]. Many attempts have been also made for excavation analyses considering the clay anisotropy: n, ground settlement [27], wall deflection [28], earth pressure [29], and basal heave stability [30,31,32]. It is noted that most of these literature adopted the NGI-ADP model [25], which is an anisotropy shear strength for clay using nonlinear stress path-dependent hardening relationship. In this regard, the random field in the framework of anisotropy structure is rarely reported in the investigations of excavation-induced responses. In addition, with respect to different mechanisms of RWD and SS caused by braced excavation, it remains to be described and explained. Meanwhile, the effects of the anisotropy soil on the probability of failure for RWD and SS should also be investigated in the study of braced excavation-induced responses.

In this paper, the effects of soil parameter spatial variability on the deformation assessments of braced excavation are ascertained, with consideration of anisotropic correlation structures. In the framework of Monte Carlo simulation (MCS), the study establishes anisotropy random field models to evaluate the influences of coefficients of variability (COVs) and scales of fluctuation (SOFs) on excavation-induced deformations. Probabilistic analyses are conducted on the basis of stochastic calculations to assess the failure of SS and RWD. The study can provide a reference in the field of reliability-based design braced excavations, with consideration of soil and structural failure.

2 Numerical Method for Modeling Braced Excavations

2.1 Plastic Hardening-Small-Strain Model

According to Puzrin et al. [33] and Burland et al. [34], the strain of soil in engineering works such as tunnels and braced excavations is mainly concentrated within a narrow range of values. Figure 1 provides the curves of soil stiffness decaying nonlinearly with increasing strain [35]. On a logarithmic scale, stiffness reduction curves exhibit a characteristic S-shape; the small-strain behavior shall be considered in the study of soil and structure responses induced by excavation. In this regard, the small-strain characteristics have been considered in many studies of braced excavations [36,37,38]. Thus, the plastic hardening-small-strain model (‘PH-SS model’) is adopted to simulate the constitutive relationship of clay in this study. The PH-SS model is based on the hardening soil model, which can approximate the observed stress–strain behavior in clays, using Eq. (1):

$$\varepsilon _{1} {\text{ = }}\frac{1}{{E_{i} }} \cdot \frac{{\text{q}}}{{1 - {q \mathord{\left/ {\vphantom {q {q_{{\text{a}}} }}} \right. \kern-\nulldelimiterspace} {q_{{\text{a}}} }}}}$$
(1)

where qa denotes the asymptotic value of the deviatoric stress and Ei is the initial soil Young’s modulus at a very low-strain (< 10−6). To account for the small-strain characteristic, the initial or very small-strain shear modulus G0 and the shear strain level γ0.7 are additionally taken into account in FLAC3D software [39]. The two parameters can be employed to address the small-strain stiffness via Eq. (2):

$$\frac{{\text{G}}}{{{\text{G}}_{{\text{0}}} }}{\text{ = }}\frac{{\text{1}}}{{\left( {{\text{1 + 0}}{\text{.385}} \cdot \left| {{\gamma \mathord{\left/ {\vphantom {\gamma {\gamma _{{0.7}} }}} \right. \kern-\nulldelimiterspace} {\gamma _{{0.7}} }}} \right|} \right)^{{\text{2}}} }}$$
(2)
Fig. 1
figure 1

Characteristic stiffness-strain behavior of soil with typical strain ranges for laboratory tests and structures [35]

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The soil parameters affect the excavation-induced deformation of soil. Hence, significant importance shall be attached to determining parameters of the PH-SS model. Previous research [40] was carried out to present a method of determination of parameters for clay layers based on laboratory tests. As an alternative, the empirical equations provided in the reference of FLAC3D Version 6.0 [39] are used in this study, as shown by Eq. (3):

$$9 \cdot E_{{oed}}^{{ref}} = 9 \cdot E_{{50}}^{{ref}} = 3 \cdot E_{{ur}}^{{ref}} = G_{0}^{{ref}}$$
(3)

For simplicity, the paper conducts the studies in a single clay layer while ignoring the influence of soil layering. The clay parameters are listed in Table 2.

Table 2 PH-SS model parameters: soft clay
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2.2 Numerical Model

Inspired by an example in the literature [39], a two-dimensional numerical model for the braced excavation is developed to study the SS and RWD. In reality, the numerical model is pseudo-two-dimensional with a size of 35 m (x axis) × 2 m (y axis) × 20 m (z axis). Figure 2 illustrates the geometry of the numerical model for braced excavations under plane-strain conditions. Due to the symmetry of the braced excavation cross section, only half of the excavation is used for modeling. The final depth of excavation is 6 m from the surface (H = 6 m) and its half-width is 7 m (B = 7 m). It is noted that the geometry of the model (35 m × 20 m) meets the requirements for minimizing the boundary effect according to the Saint-Venant principle. The diaphragm walls extends to 12 m depth and are braced at the top by horizontal struts at a 2-m interval. With reference to the technical specifications for a braced excavation, the vehicles and related personnel around the braced excavation are simplified to a uniform applied pressure of 20 kPa.

Fig. 2
figure 2

Braced excavation model and its size

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The PH-SS model is utilized to simulate the elastoplastic behavior of soil. Additionally, the soil is assumed to be completely without precipitation throughout the area to be excavated. In this regard, the area to be excavated is regarded as being in an undrained state.

The retaining system for a braced excavation typically consists of retaining walls and strut components. In this study, the liner element is used to simulate retaining wall, and the beam element is employed to represent structural components. Only the first support for braced excavation is considered in the numerical model. It is noted that the equivalent thickness of the wall is 0.51 m with its equivalent Young’s modulus being 24 GPa. In addition, the equivalent modulus of internal support is set to 30 GPa. Tables 3 and 4 list the parameters of the retaining wall and internal support, respectively.

Table 3 Retaining wall parameters
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Table 4 Internal support parameters
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The average value of the shear strength for interface element between soil and wall is used here. Due to the spatial variability of soil properties, the stiffness of the adjacent soils varies with spatial position. In the code in FLAC3D [39], the normal stiffness Kn and the shear stiffness Ks can demonstrate the deformation performance of the interface element. If both stiffnesses are set too low, the interface element will be deformed too much and will penetrate the adjacent soils too much, rendering the simulation results unreasonable. If the values are high, it will cause difficulties in convergence. In this regard, the suggested values of interface stiffness are given as follows:

$$K_{n} = K_{s} = 10 \times \frac{{K + \frac{4}{3}G}}{{\Delta z}}$$
(4)

where K + 4G/3 represents the compression stiffness, K and G denote the average values of the soil bulk stiffness and shear stiffness, respectively. Δz is the element size on the low-stiffness side among the adjacent soil elements.

In this study, the excavations are conducted as follows:

  • Step 1. Generate the initial stresses in a convergent state;

  • Step 2. Activate the liner element;

  • Step 3. Excavate the soil of the braced excavation to 2 m below the surface;

  • Step 4. Activate the beam element;

  • Step 5. Continue to excavate the soil to 6 m below the surface.

Figure 3 shows the results of RWD and SS induced by braced excavation in a deterministic framework. For the RWD, it appears convex. The maximum value of RWD is − 26.25 mm, about 0.44% of the excavation depth. It is noted that the maximum RWD occurs around the base. Meanwhile, the maximum SS is located 4.5 m away from the wall. The maximum SS is − 15.29 mm, about 0.26% of the excavation depth. Based on the calculated results, the ratio between the maximum RWD and SS is 1.72. Additionally, the settlement value extends to the side away from the retaining wall and gradually tends to be stable, while the settlement of soil close to wall is relatively large due to the weight of the retaining wall.

Fig. 3
figure 3

Deformation induced by braced excavation in a deterministic framework

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3 Random Finite Difference Method

3.1 Generation of Random Field in Clays

The inherent spatial soil variability is one of the main sources of uncertainty in geotechnical problems because of depositional and post-depositional processes. With an attempt to address the spatial variability of soil properties, Vanmarcke [8] proposed a method of establishing random fields. One of the effective approaches to generating a random field is taking the prescribed auto-correlation functions (ACFs) into account using the covariance matrix decomposition method (CMDM) [41]. The method is employed for its simple implementation and high accuracy for a small simulated domain, making it suitable for excavation-induced deformation, herein. In this regard, the CMDM is used in this paper.

In the implementation of CMDM, the mean, coefficient of variation (COV), and SOF of the soil parameters shall be firstly specified for characterizing a random field. The stationary random field is generated to simulate a homogeneous soil layer in this paper. Meanwhile, the lognormal Gauss random field is utilized to describe the soil parameter spatial variability for its nonnegativity. To determine the spatial correlation of geotechnical parameters in both vertical and horizontal directions, an anisotropic random field identified by exponential auto-correlation function (ACF) is used, associating with the horizontal and vertical SOFs. Studies [18, 20] found that the parameters of natural soils are spatially anisotropic due to their geological deposition processes, illustrating that the horizontal SOF is generally 10–80 m, and the vertical SOF is 1–3 m. The anisotropic coefficient ξ is used to describe the transversely isotropic correlation framework.

A uniform clay layer is discretized into m × n quadrilateral elements. Figure 4 depicts the discretization for random fields, where m and n denote the number of elements in each direction, respectively, and (xi, zj) denotes the coordinates of the centroid of each element, i = 1, 2, …, m, j = 1, 2, …, n. The single exponential ACF is used to elucidate the correlation between different points in clay, as follows:

$$\rho \left( {\tau _{x} ,\tau _{z} } \right) = \exp \left[ { - 2 \times \left( {\frac{{\left| {\tau _{x} } \right|}}{{\theta _{x} }} + \frac{{\left| {\tau _{z} } \right|}}{{\theta _{z} }}} \right)} \right]$$
(5)

where τx and τz represent the absolute distance between any two spatial points in horizontal and vertical directions, respectively, and θx and θz denote the horizontal and vertical SOFs. Then, the auto-correlation coefficient can be calculated using Eq. (5), and the auto-correlation matrix Cn×n is written as follows:

$$C_{{n \times n}} = \left[ {\begin{array}{*{20}c} 1 & {\rho \left( {1,2} \right)} & \cdots & {\rho \left( {1,n} \right)} \\ {\rho \left( {2,1} \right)} & 1 & \cdots & {\rho \left( {2,n} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {\rho \left( {n,1} \right)} & {\rho \left( {n,2} \right)} & \cdots & 1 \\ \end{array} } \right]$$
(6)

where ρ represents the auto-correlation coefficient between any two centroid points.

Fig. 4
figure 4

Discretization for random fields

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Since Cn×n is a positive definite symmetric matrix, the Cholesky decomposition technique is used to decompose Cn×n into the product of a lower triangular matrix L and its corresponding transpose:

$$C_{{n \times n}} = L \cdot L^{T}$$
(7)

where LT denotes the transpose of the matrix L. Then, a realization of the auto-correlated standard Gaussian random field can be deduced according to:

$$Z = L \cdot Y$$
(8)

where Y is a vector randomly generated. Using the regenerated Y, the Gaussian random field can be derived.

To incorporate the random field model into numerical model, the finite difference code FLAC3D is used herein. With the built-in FISH language, FLAC3D can identify the positions of the discretized elements according to the proximity principle between numerical elements and random field coordinates, thereby realizing the mapping of an independently generated random field model to the numerical model.

In this paper, \(E_{{{\text{oed}}}}^{{{\text{ref}}}}\) is considered as a spatially variable parameter. On the basis of Eq. (3), other stiffness parameters are derived to indicate the corresponding stiffness random field.

3.2 Calculation of the Probability of Failure

Since random fields are used to describe the spatial variability of soil parameters, the responses of excavation-induced deformation can become stochastic. In this paper, MCS is adopted to generate Ns realizations of random fields for stochastic calculation and to perform a subsequent reliability analysis of structure and soil responses. To cope with such stochastic problems, the probability of failure (Pf) is employed to express the possibility of excavation-induced deformation. The method of calculating Pf can be expressed as follows:

$$P_{f} = P\left[ {Z < 0} \right] = \frac{1}{N}\sum\limits_{{i = 1}}^{N} {I\left[ {Z < 0} \right]}$$
(9)
$$Z = S_{\rm sto} - S_{\rm lim}$$
(10)

where Z is the serviceability limit state function (LSF), in which Ssto is the response of stochastic calculation and Slim refers to the limiting value of the corresponding response, N denotes the number of MCS, and I[–] represents the indicator function. When Z < 0, I[–] is 1, otherwise zero.

3.3 Implementation Procedure of RFDM for Reliability Analysis

Here, random field theory and numerical analysis are combined to investigate the influences of clay property spatial variability on the responses of soil and structure deformation induced by excavation. Based on the aforementioned introduction of the proposed methods, i.e., RFDM, more details of the implementation procedure are outlined as below:

  • Step 1: Determine the values of the clay parameters, uncertain clay properties, their salient statistical parameters, and the numerical model geometry.

  • Step 2: Discretize the numerical model, and generate a random field model in the same size of the corresponding numerical model.

  • Step 3: Perform Ns MCS to obtain different stochastic results, including the values of RWD and SS.

  • Step 4: Discuss the influences of SOFs and COVs of soil stiffness on the RWD and SS, then investigate the probability of failure of excavation-induced deformations.

Figure 5 shows the flowchart for excavation-induced deformation assessment using the RFDM.

Fig. 5
figure 5

Flowchart through of MCS for excavation-induced responses

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4 Stochastic Analysis for the Braced Excavation

In this section, stochastic responses for SS and RWD induced by excavations are illustrated. A series of anisotropic random fields (θx ≠ θz) are generated to identify the influences of both the COVs and the SOFs on the excavation-induced responses.

Based on the literature [18, 20], the horizontal and vertical SOFs of soil stiffness lie in the range of 10–80 m and 1–3 m, respectively. In this regard, this subsection evaluates the responses caused by excavation for the baseline case in the conditions of θx = 4.8H = 28.8 m, θz = 0.3H = 1.8 m, and COV = 0.3. To assess the effects of COVs and SOFs on the responses, a series of cases are investigated under different combinations of COVs and SOFs (Table 5).

Table 5 Calculated cases of stochastic analyses
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Before conducting MCS, the number of simulations is determined. Figure 6 shows the means and COVs of maximum deformations (i.e., SS and RWD, the same below) with the numbers of simulations. It can be found that the means and COVs of both maximum SS and maximum RWD tend to be stable when Ns reaches 1000; thus, Ns = 1000 is adopted here.

Fig. 6
figure 6

Mean and COV of maximum deformation

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4.1 Influences of SOFs on Excavation-Induced Deformation Responses

In this subsection, SOFs are paid enough attention in the field of deformations influenced by spatial variability of soil parameters. It is noted that both horizontal and vertical SOFs arise in engineering practice; thus, both SOFs are considered here. For excavation-induced responses, a series of cases, named MCS-z1 to MCS-z6, are analyzed to identify the influences of vertical SOFs while the cases (MCS-x1 to MCS-x6) are analyzed to study the effects of horizontal SOFs.

Figure 7 shows the stochastic results corresponding to cases MCS-z1, MCS-z4, and MCS-z6. For better comparison, the deterministic results are also shown with the black line while the stochastic results are denoted by the gray line. As shown in Fig. 7, the stochastic results exhibit a trend of aggregation with an increase in the values of ξ (corresponding to the decrease of vertical SOFs); however, the SOFs exert no influence on the shape of the deformation curve. The stochastic calculated results fluctuate around the solutions obtained by deterministic calculation. Overall, the stochastic results in majority are larger than those obtained in the deterministic scenario, which is mainly due to the dominant effects of low stiffness and the asymmetry of the logarithmic random distribution of stiffness. It is noted that the stochastic results in the analysis of cases MCS-z1 to MCS-z6 are similar to those obtained when analyzing cases MCS-x1 to MCS-x6. Meanwhile, the stochastic results are more influenced by vertical SOFs than by horizontal SOFs.

Fig. 7
figure 7

Retaining wall deflection and surface settlement under different SOFs

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Based on the Monte Carlo method, 1000 simulations have been conducted for all cases. It can be observed from Fig. 7 that the maximum deformations obtained by any stochastic calculation are different. In this regard, statistical analysis is conducted to identify the effects of SOFs on the excavation-induced maximum deformations. Figure 8 shows the influences of SOFs on the mean values and COVs of maximum deformations. Mean values of both the SS and the RWD are less affected by the SOFs of the soil stiffness, which indicates the maximum deformations are generally the same in the ‘average sense’ of the influences of SOFs. Further to investigate the ‘average sense’ of the maximum deformation, the entire 1000 simulations are analyzed here. It is assumed that both the soil stiffness and maximum deformation follow the lognormal distribution. For the lognormal distribution (Y = lnX − N(μln, σln2)), its characteristic values and those of the normal distribution can be mutually inter-converted (Table 6).

Fig. 8
figure 8

Effects of SOF on the means and COVs of maximum deformations

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Table 6 Characteristic values of the lognormal distribution
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For soil stiffness E, ln E follows a normal distribution with mean and standard deviation given by:

$$\begin{array}{*{20}l} {\sigma _{{\ln }}^{2} = \ln (1 + \sigma ^{2} /\mu ^{2} )} \hfill \\ {\mu _{{\ln }} = \ln (\mu ) - \frac{1}{2}\sigma _{{\ln }}^{2} } \hfill \\ \end{array}$$
(11)

According to the literature [9], since the soil stiffness follows a lognormal distribution, the probability that the stiffness is less than the average value of stiffness is greater than 50% in random field model of the soil stiffness. It is recommended that engineers use the geometric mean stiffness as the equivalent characteristic stiffness. It can be seen from Table 6 that the average soil stiffness is greater than its geometric mean; thus, the results obtained by using the average value of soil stiffness are lower than the results using the geometric mean. The expression of geometric mean shows it is independent of the SOF, but is related to the standard deviation (or COV), which further explains the conclusion that the mean of the maximum deformations is independent of the SOF.

For the COVs of maximum deformations in Fig. 8b, they show a tendency to increase as the SOFs (horizontal and vertical) increase. The influences of vertical SOFs on the COVs of maximum deformations are found to be greater than those of horizontal SOFs. When the vertical SOFs lie in the range of 0.6H to 2.4H, the COV of SS shows a relatively smooth trend with the anisotropy coefficient ξ while that of RWD does not show any abnormalities, as defined by the correlation of vertical SOF. When the vertical SOFs are close to the size of the excavation, the probability that high-stiffness regions and low-stiffness regions occur in the excavation area increases, making the excavation-induced response more broadly scattered. It is noted that the influences of vertical SOFs on SS and RWD are differently, perhaps as a result of the difference in stiffness between the soil and the wall.

Figure 9 shows the location of maximum deformation considering different SOFs. It is observed that the location of maximum SS mainly lies in the range of 0.8H to 0.9H with those of RWD around the base. Meanwhile, with increasing anisotropy coefficient ξ (the decrease of vertical SOFs), the scatter degree of maximum SS and its location first decreases, then increases. In the case of SOF corresponding to the size of the excavation, the scatter of SS is the greatest, consistent with the correlation of vertical SOF. For the RWD, its location becomes more concentrated with increasing anisotropy coefficient ξ, which is different from the influence on SS. This may be due to the difference in stiffness between the soil and the wall. Based on the locus of maximum RWD (Fig. 9b), two modes of RWD are presented in this subsection, as shown in Fig. 10.

Fig. 9
figure 9

Location of maximum deformation

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Fig. 10
figure 10

Two modes of RWD with consideration of its location

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Mode 1: The location of maximum RWD is above the base, as shown in Fig. 10a. Due to the inhomogeneity of strong regions and weak regions, different loci of maximum RWD may occur. The weak region occurs above the base and the base below tends to be stronger, making it difficult to transfer displacement to the base below.

Mode 2: The location of maximum RWD lies below the base, as shown in Fig. 10b. The strong region of unloading stiffness lies above the base while the weak region is below the base. In this regard, more deformation is assigned to the area below the wall.

Figure 11 shows the variations of the ratios between maximum SS (δvm/H) and maximum RWD (δhm/H) for different SOFs, corresponding to MCS-z1, MCS-z4, and MCS-z6. It shall be shown that the maximum SS (δvm/H) and the maximum RWD (δhm/H) are dimensionless. This study uses a range angle α to describe the degree of scatter of the ratios between δvm/H and δhm/H. In all the cases, the ratios between δvm/H and δhm/H are distributed within a particular range. As an example, α is 13.74°, 10.90°, and 9.95° at ξ = 2, 16, and 64, respectively. With increasing ξ, the vertical SOFs decrease, the range angle α decreases, and the average ratios between the two deformations (δvm/H and δhm/H) tend to increase. This may be explained by the fact that more soil displacement will be assigned to surface settlement instead of horizontal displacement for such a braced excavation in variable clays. Compared to the deterministic result (1.72), the ratios between δvm/H and δhm/H calculated in a stochastic framework vary widely, with a bigger value in the ‘average sense’. This may result from the inhomogeneity of strong regions and weak regions and it indicates that the ratios between δvm/H and δhm/H will be underestimated if the spatial variability of soil properties is not considered.

Fig. 11
figure 11

Relationship between maximum SS and maximum RWD

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As illustrated, the deformed curves may be stochastic due to the inhomogeneity of soil stiffness. In this regard, four modes corresponding to different combinations of SS and RWD are proposed by comparing data with the deterministic results. It is noted that the most scattered results may be obtained in the case when SOF corresponds to the size of the excavation. Thus, scenario MCS-z2 is taken as an example to illustrate the modes of deformation of SS and RWD. Figure 12 shows four modes of SS and RWD by comparing the data with deterministic results. Mode 1 indicates that the SS and RWD are both more deformed than their deterministic results. This can be explained that the weak region of soil stiffness is widely distributed in the area around the excavation, increasing the values of SS and RWD. Mode 2 represents a larger SS and a smaller RWD while the opposite results arise in Mode 3, which results from the inconsistent distribution of weak regions and strong regions (or intermediate regions) on the two sides of the retaining wall. Mode 4 denotes the scenario wherein smaller values of SS and RWD are obtained in the stochastic calculation when strong regions are located around the excavated areas. In general, the inhomogeneity of strong regions and weak regions leads to different modes for SS and RWD.

Fig. 12
figure 12

Four modes of SS and RWD compared with the deterministic results

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4.2 Influences of COVs on Excavation-Induced Deformation Responses

As can be imagined from engineering experience, the COV of soil stiffness will affect the characterization of soil spatial variability, which subsequently influences the excavation-induced responses. A series of cases, as shown in Table 5 (from MCS-v1 to MCS-v5), are analyzed to study the corresponding influence.

Figure 13 shows the calculated results when COV = 0.1, 0.3, and 0.5, corresponding to MCS-v1, MCS-v3, and MCS-v5, respectively. It is noted that three cases of COV = 0.1, 0.3, and 0.5 indicate three levels of soil stiffness variability (low, moderate, and high). As the COV increases, the degree of scatter of curves increases accordingly, whether they are plots of SS or RWD. However, the shape of the curve does not vary with the variation of COV, which is consistent with the result obtained in the study of ξ. These conclusions might result from the soil spatial correlation, indicating the distributions of low-stiffness areas and high-stiffness areas. The correlation of soil stiffness shows a decreasing trend with an increase in COVs.

Fig. 13
figure 13

RWD and SS under different COVs (ξ = 16, θx = 4.8H, θz = 0.3H)

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Figure 14 illustrates the influences of COVs on the mean values and COVs of maximum deformations. The results associated with the mean values of maximum deformation are opposite thereto: the surface settlement increases slightly while the results for the retaining wall show a slight decreasing trend with increasing COV. This is in line with the aforementioned conclusion. From the expression of geometric mean given in Table 6, it is noted that this value is independent of the SOF of the stiffness, only the standard deviation (or COV) thereof. For the COVs of maximum deformation in Fig. 14b, both results increase in a quasi-linear manner with an increase in soil stiffness COV.

Fig. 14
figure 14

Effects of the COVs of stiffness on the COVs and mean values of maximum deformation

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Figure 15 shows the location of maximum deformation in different cases. The location of maximum SS occurs in the range of 0.6–0.8H while that of maximum RWD is distributed around the base. With the increase of COVs, the location of maximum SS broadens. The same conclusion is manifest for the maximum values of RWD.

Fig. 15
figure 15

Location of maximum deformation

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Figure 16 describes the relationship between the maximum SS and the maximum RWD in different cases with the variations of COVs. As shown in the aforementioned study, some similar conclusions can be drawn with regard to the influences of COVs on these ratios. An increasing trend in angle α, corresponding to the increase of its degree of scatter, can be found as the COV increases. For a given SOF, the average ratios between δvm/H and δhm/H are found to increase as the COV increases. It is noted that the influence of ξ (or SOFs) is more moderate, compared to the effects of COVs. In addition, similar to the study on the influences of SOFs, the ratios between δvm/H and δhm/H will also be underestimated if the spatial variability of soil properties is ignored.

Fig. 16
figure 16

Relationship between maximum SS and maximum RWD

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5 Probability Analysis for Deformations Induced by Excavation

In the stochastic calculations, the deformation results for the surface and retaining wall obtained from each MCS are different. Thus, it is necessary to study the characteristics of SS and RWD caused by excavation using probabilistic analysis.

The baseline case (COV = 0.3, θx = 4.8H = 28.8 m, θz = 0.3H = 1.8 m) is selected for reliability analysis. Meanwhile, based on the baseline case, three groups of cases (MCS-z, MCS-x, and MCS-v) are analyzed to study the influences of vertical SOFs, horizontal SOFs, and COVs on the excavation-induced stochastic responses.

Figure 17 shows the frequency histogram of maximum values of the deformations and their cumulative distribution functions in the baseline case. By observing the results, the average value of maximum SS is − 14.92 mm while the mean of the maximum RWD is − 26.56 mm. It is noted that the average maximum SS is slightly less than the corresponding deterministic result (− 15.29 mm) while the average maximum RWD is slightly larger than that in the deterministic calculation (− 26.25 mm). This may be explained by the inhomogeneous distribution of the soil strength, especially in zones of weakness [9, 42]. With the influence of the weak region, soil is more assigned to the direction of the retaining wall in its displacement field, increasing the value of RWD.

Fig. 17
figure 17

Histogram of maximum deformations induced by excavation

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It is acknowledged that risks of excessive deformation may occur during the construction of braced excavation. Thus, it is necessary to determine safe limiting values of excavation-induced deformation. This study performs probability analyses of the maximum deformations based on 1000 runs of the MCSs. As provided in Eqs. (9) and (10), Pf and LSF are used to address the problem. Figure 18 compares the probability of failure estimated at a series of limiting deformations (i.e., SS and RWD) at various combinations of COVs and SOFs. For Fig. 18a, b, this section just shows the solutions resulting from variations in vertical SOFs. Overall, the probability of failure by excessive deformation can be overestimated or underestimated when ignoring the spatial variability of soil properties. Additionally, it is found that Pf decreases with the limiting SS or RWD at different combinations of COVs and SOFs (or ξ). Nevertheless, COVs and SOFs can dominate the rate of decrease of Pf. As the SOFs increase (ξ decreases), the maximum deformations vary more widely, and the probability of exceeding limiting values increases accordingly. In this regard, a similar conclusion can be obtained with an increase in COVs.

Fig. 18
figure 18

Influences of spatial variability on the probability of exceeding the specified limiting deformations at various combinations of SOF (or ξ) and COV

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Figure 18 provides a beneficial reference for an assessment of maximum SS or maximum RWD before the construction of a braced excavation.

6 Conclusion

This study presents the effects of spatial variability of soil stiffness on the braced excavation in clays using the RFDM. Attention is mainly paid to SOFs (including horizontal and vertical variations thereof) and COVs when illustrating the spatial variability. The spatial variability of soil properties is modeled in the framework of random field theory using a series of combinations of SOFs and COVs. On the basis of the results presented in this study, several conclusions are drawn and summarized as follows:

  1. 1.

    Combining the small-strain characteristics of the soil and the spatial variability of soil properties, an algorithm is developed to facilitate the FDM-based probabilistic assessment in the framework of MCS.

  2. 2.

    The effects of weak stiffness region: a series of modes for RWD and SS are proposed in the paper. In addition, the average maximum SS is slightly less than the corresponding deterministic result while the average maximum RWD is slightly larger than that in the deterministic calculation. These may be explained by the inhomogeneous distribution of the soil stiffness, especially in zones of weakness.

  3. 3.

    The effects of anisotropic random field: the effects of vertical SOFs on the excavation-induced responses are greater than those of horizontal SOFs. In the paper, the correlation of vertical SOF is proposed to explain that the most scattered result occurs when vertical SOF is close to the excavation size, which does not occur when investigating the influence of horizontal SOF.

In the present research, some useful conclusions have been drawn in the investigation of the excavation-induced deformations in spatially variable clays; however, only the first support of such a braced excavation is considered and the analysis of the base has not been involved, thus allowing scope for future research.