Influence of non-stationarity in friction angle on the performance of the braced excavation system

Abstract

This study aims to quantitatively evaluate the deformational response of braced excavations in the presence of existing infrastructure while considering the spatial variability of the internal friction angle of soil. The spatial variability is characterized by a non-stationary random field of linearly increasing mean and constant coefficient of variation with depth. Probabilistic analysis with stationary random fields is first performed to provide a reference for the subsequent analysis with non-stationary random fields. The influence of friction angle gradient and vertical scale of fluctuation on excavation-induced deformation responses, such as maximum lateral wall deflection and maximum ground surface settlement, is studied. The findings indicate that neglecting the depth-dependent variation of the friction angle results in an overestimation of lateral earth pressure, which in turn leads to higher values of maximum lateral wall deflection and maximum ground surface settlement. It is also observed that the friction angle gradient significantly affects the location of maximum lateral wall deflection in braced excavation. Additionally, the failure probability of individual components and the overall failure of the braced excavation system are evaluated, considering multiple ultimate and serviceability criteria. A new parameter, building wall torsional tilt, which indicates the degree of differential settlement, is adopted as a serviceability criterion for buildings.

Introduction

Excavation projects are commonly undertaken for constructing foundations of multi-storied structures and tunnels in urban transportation projects. The success and failure of excavation projects should not only be related to the stability and safety of excavation support structures but also the stability of adjacent structures, fostering responsible infrastructure development. A braced excavation system typically consists of a retaining wall, known as a diaphragm wall, and struts provided at various levels of excavation to reduce earth pressure on the retaining wall. This, in turn, reduces lateral wall deflection (LWD) and ground surface settlement (GSS) in the vicinity of the excavation. For the safe construction of braced excavations, it is essential to satisfy both geotechnical and structural requirements. Geotechnical requirements include: (1) the ultimate limit state (ULS) and (2) the serviceability limit state (SLS). The ULS requirement is ensured by designing the braced excavation to achieve a minimum factor of safety against failure. The type of factor of safety required depends on the type of soil being excavated. In the case of sands, a minimum factor of safety is needed against piping failure, while for clays, a minimum factor of safety is required against basal heave¹.

For geotechnical SLS, two requirements must be satisfied: (1) wall deflections and (2) Ground settlement in the vicinity of the excavation. In numerous excavation projects, governing bodies or regulatory authorities define acceptable limits for the factor of safety, wall deflections, and ground settlements to prevent excavation failures and protect nearby infrastructure². For example, in the construction of the Shanghai Metro, design requirements were proposed for braced excavations based on the importance of the project and the critical structures located near the excavations^3,4,5. Structural ULS requirements include ensuring the safety of the embedded retaining wall against failure caused by excessive bending moments and shear forces, as well as ensuring safety against excessive axial forces in the struts⁵.

To study this complex soil-structure interaction problem of braced excavation, various researchers adopted empirical approaches^6,7,8, analytical methods^9,10,11, field observation^12,13, numerical methods such as finite element method^{14,15,16,17,18,19,20,21,22},and finite difference method^23,24. Among the methods, numerical methods are widely adopted. Numerical methods can explicitly model realistic conditions, including multilayered soil, groundwater conditions, complex geometries, nonlinear soil behaviour, and construction phases¹. Input parameters for numerical analysis are typically derived from a limited set of field tests, which might not accurately reflect the actual site conditions. To address this, probabilistic methods are widely used to assess geotechnical engineering responses, considering the uncertainties in soil properties. Various sources of uncertainty in geotechnical engineering include inherent variability, measurement uncertainty, and transformation uncertainty. Inherent variability refers to the uncertainty resulting from spatial variation in soil properties and is considered the most significant type of uncertainty in geotechnical engineering. Vanmarcke²⁵ proposed the random field theory to model the inherent variability of soil, and presently it is widely used for probabilistic assessment of geotechnical engineering applications. Numerical methods based random finite element method and random finite difference method are widely used for quantitative assessment of deformational behaviour of braced excavations, considering the inherent variability of soil^{26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41}.

Although various studies have adopted random field theory for the probabilistic assessment of braced excavation performance, the soil parameters are modelled as stationary random fields. However, in the field, most soil parameters exhibit depth-dependent linear trend. This linear increase in soil properties is attributed to the effective overburden pressure and confining stress. In this view, modelling the soil parameters as stationary random fields is inconsistent with the realistic field conditions. Various researchers have employed non-stationary random fields of undrained shear strength for probabilistic assessment in geotechnical engineering applications, such as slope stability and foundation analysis^42,43,44,45. The trend function of undrained shear strength is widely reported in the literature. However, the trend function of friction angle is not well documented, perhaps due to the difficulty in obtaining undisturbed sand samples. Li et al.⁴⁴ analysed cone penetration data, derived a trend function for the friction angle using an empirical equation, and applied it to the infinite slope problem. They found that the increase in friction angle with depth does exist in the field. Thus, in this study, probabilistic analysis of braced excavation is carried out by considering the friction angle as a non-stationary random field.

It is also important to note that although various studies have dealt with the failure probability of braced excavation systems by considering multiple ultimate and serviceability limit states, such as structural failure modes (bending moments, shear forces, and strut axial forces) and geotechnical failure modes (ground surface settlement, and wall deflection). To the best of the author’s knowledge, only a limited number of the studies have incorporated differential settlement as serviceability criteria (which affects adjacent building stability) for obtaining the system failure probability of braced excavations. This study incorporated the differential settlement criterion for failure probability assessment of braced excavation, as differential settlement has greater consequences for adjacent buildings compared to maximum ground surface settlement.

This paper entails modelling the complex soil structure interaction problem of braced excavation using the finite difference software FLAC 2D 9.0. The effect of stationary and non-stationary random fields of frictional angle on the braced excavation deformation performances, such as maximum lateral wall deflection (MLWD) and maximum ground surface settlement (MGSS), are studied. A parametric study is also conducted to study the influence of the vertical scale of fluctuation (VSF) on the MLWD and MGSS. Additionally, the failure probabilities of individual components and the overall system are evaluated considering multiple serviceability failure modes such as maximum ground surface settlement (MGSS), maximum lateral wall deflection (MLWD), and building wall torsional tilt (BWTT), and ultimate failure modes such as maximum wall shear force (WSF), maximum wall bending moment (WBM), and strut axial force (SAF). BWTT is a new parameter adopted in this study to address the issue of differential settlement serviceability criteria for buildings.

Methodology

Finite difference modelling of braced excavation

The finite difference method is widely used in geotechnical engineering for solving complex soil-structure interaction problems. The braced excavation problem is analysed using finite difference software FLAC 2D 9.0⁴⁶. A case study from Luo et al.⁵ is implemented in this study and depicted in Fig. 1. Plane-strain condition is assumed, and only half the width of the excavation is modelled in this study to reduce computation demand. Sert et al.⁴⁷ suggested that the minimum distance between the unexcavated side boundary and the retaining wall should be 2–3 times the embedded wall depth, and the minimum distance between the bottom boundary to the wall bottom should be the depth of the wall. Based on the above-mentioned criteria, a numerical model with a size of 47 m in the horizontal direction and 20 m in the vertical direction is adopted, as shown in Fig. 1. Quadrilateral zones are adopted in this study, and based on a mesh convergence study, a zone size of 0.5 m is adopted. The surcharge load adjacent to the braced excavation consists of uniformly distributed loads of 10 kPa and 60 kPa. The bottom boundary is restrained, and roller boundaries are applied to the vertical sides.

The groundwater table is situated 3.5 m beneath the ground, while the excavation extends to a depth of 6 m and a width of 7 m. The supporting system consists of an embedded retaining wall extending to a depth of 10 m and a strut positioned 1.5 m beneath the ground surface. The Mohr-Coulomb (MC) constitutive model is used to simulate soil behaviour. The retaining wall is modelled using a structural liner element, while the strut is modelled using a structural beam element. A linear elastic constitutive model is applied to simulate the behaviour of both the retaining wall and strut. Table 1. presents the soil and structural element properties utilized in this study.

The interaction between the embedded diaphragm wall and the soil is represented by a linear spring-slider system, where the interface shear strength is determined by the Mohr-Coulomb failure criterion. The relative displacement between the soil and the retaining wall is governed by the normal stiffness (K_n) and shear stiffness (K_s) of the interface. A typical approach is to set the normal and shear stiffness properties of the interface to approximately ten times greater than the equivalent stiffness of the adjacent zone with the highest stiffness⁴⁶. This practice ensures that the deformability at the interface minimally affects both the overall model compliance and computational efficiency. The apparent stiffness of a zone in the normal direction is determined using the following Eq⁴⁶.

$$\:{K}_{s}={K}_{n}=\:10\:\times\:max\left[\frac{K+\left(\raisebox{1ex}{$4$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.\right)G}{{\varDelta\:Z}_{min}}\right]$$

(1)

Where G and K represent the shear and bulk modulus, respectively.

$\:{\varDelta\:Z}_{min}$ refers to the minimum width of an adjacent zone in the normal direction.

Max [] denotes the maximum value across all zones adjacent to the interface.

Based on the above expression, the values of normal and shear stiffness are given in Table 1.. The nodes at the soil-wall interface are assigned a low shear strength, approximately two-thirds of the friction angle, to simulate a relatively smooth soil-wall interface. The excavation process adopted in this study is explained below:

• Initialize stresses in the soil mass due to gravity and surcharge loads.

• The retaining wall is activated and simulated using the structural liner, with soil-wall interface parameters assigned to both sides of the wall.

• Excavate the soil to a depth of 2 m beneath the ground surface.

• Model the strut 1.5 m beneath the ground surface using a beam element.

• Excavate the soil to a depth of 3.5 m beneath the ground surface.

• Dewater to a depth of 7 m.

• Excavate the soil to a depth of 6 m.

Table 1. Material properties adopted⁵.

Validation of the numerical model

For validation, Hardening Soil (HS) and Mohr-Coulomb (MC) constitutive models are used. Constitutive model parameters for the hardening soil model are the same as Luo et al.⁵ The parameters of the Mohr-Coulomb constitutive model are provided in Table 1., with the exception of the friction angle and Young’s modulus, which are taken from the previous study. Table 2 provides the MLWD values and their locations obtained in this study, and values from Luo et al.⁵

Table 2 Comparison of MLWD and their locations.

From Table 2, the MLWD and its location obtained using the HS constitutive model are 23.73 mm and 5.51 m, respectively, while those with the MC constitutive model are 24.36 mm and 6 m, respectively. The difference between the MLWD values obtained from Luo et al.⁵ and this study for HS and MC constitutive models are 1.12% and 1.5%, respectively. The difference between the location of MLWD obtained from Luo et al.⁵ and this study for HS and MC constitutive models are 1.25% and 7%, respectively. MLWD is within the range of 1.5% for both constitutive models adopted. However, the difference in the location of MLWD is greater when the MC constitutive model is adopted. This is due to the following reasons:

• Parameters related to the groundwater table, embedded retaining wall, and strut are not mentioned in the previous study; therefore, these values are assumed.

• The Mohr-Coulomb constitutive model is a linearly elastic, perfectly plastic model that assumes constant stiffness and ignores non-linear behaviour.

Due to these limitations, an approximately 7% difference is observed in the predicted location of the maximum lateral wall deflection (MLWD) when the Mohr–Coulomb (MC) model is compared with the Hardening Soil (HS) model. Nevertheless, the MC model is adopted because it requires roughly half the computational time of the HS model, which is advantageous for Monte-Carlo simulations. Although advanced constitutive models can represent soil nonlinearity more rigorously, Juang et al.⁴⁸ showed that models with a larger number of parameters amplify input uncertainty more strongly, as quantified by higher amplification factors and larger response variability in wall deformation in the braced excavation. In contrast, the MC model exhibited lower amplification factors and reduced sensitivity to parameter uncertainty, leading to more robust probabilistic predictions of response variability and reliability. Furthermore, Huang et al.⁴⁹ demonstrated through field comparisons that both the mean response and variability obtained using the MC model are consistent with observed tunnel excavation data. Therefore, while the MC model may introduce minor discrepancies in the exact location of MLWD, these differences are not expected to affect the estimated failure probability trends investigated in this study.

Spatial variability of soil properties

Even in homogeneous soils, soil properties can vary from one point to another. This phenomenon is known as the spatial variability of soil. Spatial variability of soil is attributed to deposition and post-deposition processes, which include variation in mineral content, overburden pressure, and confining pressure. It is considered a major source of uncertainty in geotechnical engineering and is included in the reliability assessment of geotechnical engineering applications. Vanmarcke²⁵ introduced the random field theory to address spatial variability. Random field theory is widely used to model spatial variability for probabilistic assessments in geotechnical engineering applications, including slope stability analysis, foundations, tunnelling, and braced excavation.

Fig. 1

(Adapted from Luo et al.⁵)

Braced excavation model.

Random field theory entails modelling the soil properties as random variables characterized by a probabilistic distribution, and the properties of adjacent points are correlated to each other with the help of the autocorrelation function and scale of fluctuation (SOF). Due to insufficient site investigation data, obtaining the autocorrelation function of soil properties is challenging²⁸. Various theoretical autocorrelation functions are widely used in geotechnical engineering, including the single exponential, squared exponential, Gaussian, cosine exponential, and second-order Markov exponential functions. In this study, the Gaussian autocorrelation function is adopted because the Gaussian autocorrelation function is differentiable near the origin and has a smooth, isotropic surface in space. This property facilitates a more effective and realistic characterization of the spatial correlation of soil properties⁵⁰. This choice is further supported by the fact that excavation-induced responses, such as wall deflection and ground settlement, generally vary gradually in space due to the soil continuity and the stiffness of the retaining systems, a behaviour that is more appropriately captured by the strong short-range continuity implied by the Gaussian model. The Gaussian autocorrelation model is expressed as follows.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\rho\:\left(\varDelta\:x,\:\varDelta\:y\right)=\text{exp}\left(-\left(\:{\left(\frac{{\varDelta\:x}^{\:}}{{{{\delta\:}_{x}}_{\:}^{\:}}_{\:}}\right)}^{2}+\:\:{\left(\frac{{\varDelta\:y}^{\:}}{{{{\delta\:}_{y}}_{\:}^{\:}}_{\:}}\right)}^{2}\right)\right)$$

(2)

Where $\:\rho\:\left(\varDelta\:x,\:\varDelta\:y\right)$ is the autocorrelation function, $\:\varDelta\:x=\:\left|{x}_{i\:}-{x}_{j}\right|,\varDelta\:y=\left|{y}_{i\:}-{y}_{j}\right|$ are the absolute separation distances in the x and y directions, respectively. $\:{\delta\:}_{x}$ and $\:{\delta\:}_{y}$ are the scale of fluctuation (SOF) in the x and y directions, respectively. SOF quantifies how soil characteristics/properties vary with distance.

Generation of stationary and non-stationary random fields of friction angle

A random field is considered weakly stationary if it has a constant mean and its covariance depends solely on the separation distance rather than the absolute location. The random field is considered strong stationary if the complete probability distribution depends solely on the separation distance rather than the absolute location⁵¹. A random field that does not follow the conditions of stationarity is said to be a non-stationary random field. In geotechnical engineering, soil properties often exhibit non-stationarity in their mean values. However, the variance and covariance structures are generally assumed to be stationary because estimating these parameters requires a significant amount of data. Consequently, while the mean is modelled as non-stationary to account for the smear effect, the variance and covariance structures of the random fields are treated as stationary⁵². Li et al.⁴⁴ found that the increase in the friction angle with depth exists in the field and trend function of the friction angle varies from 0.24^o/m to 2.38^o/m. Due to the limited availability of data related to the trend function of the friction angle in sand, trend gradients of 0.2, 0.4, and 0.6°/m are adopted in this study, representing loose to medium-dense sand conditions where the increase in CPT resistance and interpreted friction angle with depth is relatively gradual. Other parameters, such as covariance and variance, are assumed to be stationary. From this point onward, non-stationarity in the mean of the friction angle is regarded as non-stationary.

To avoid negative values, the friction angle is considered log-normally distributed. The mean and standard deviation of the lognormal distributed friction angle are obtained using the following expression⁵⁰.

$$\:{\mu\:}_{ln\varnothing\:}=\text{ln}\left(\mu\:\right)-0.5\text{ln}(1+{cov}_{\:}^{2})$$

(3)

$$\:{\sigma\:}_{ln\varnothing\:}=\sqrt{\text{l}\text{n}(1+{cov}_{\:}^{2}})$$

(4)

Where $\:{\mu\:}_{ln\varnothing\:}$ and $\:{\sigma\:}_{ln\varnothing\:}$ are the mean and standard deviation of the lognormal distribution. $\:\mu\:$ and $\:cov$ represents the mean and coefficient of variation (COV) of the normal distribution.

In this study, a linearly increasing friction angle can be expressed as⁴⁴

$$\:\varnothing\:\:\left(z\right)=\:{\varnothing\:}_{0}+\:{k}_{\varnothing\:\:}z$$

(5)

Where $\:\varnothing\:\:\left(z\right)$ is the friction angle mean at depth z, $\:{k}_{\varnothing\:\:}$ is the mean gradient of the friction angle, $\:{\varnothing\:}_{0}$ is friction angle mean at the ground surface, and z indicates the depth. The non-stationary random field for the friction angle is generated using the following expression⁵⁰.

$$\:{H}_{\varnothing\:}=\text{exp}\left({\mu\:}_{ln\varnothing\:}+{k}_{\varnothing\:\:}z+{H}_{w}{\sigma\:}_{ln\varnothing\:}\right)$$

(6)

Where $\:{H}_{\varnothing\:}\:$is a non-stationary random field, $\:{H}_{w}$ is a zero-mean stationary random field. A zero-mean stationary random field is obtained using the following expression⁵³.

$$\:{H}_{w}\:=L\:\xi\:\:\:$$

(7)

Where $\:\xi\:\:$is a column matrix of standard normal independent random variables, and L is the lower triangular matrix of the Cholesky decomposition of the autocorrelation matrix and is obtained as follows⁵³.

$$\:\rho\:\left(\varDelta\:x,\:\varDelta\:y\right)=L\:{L}^{T}$$

(8)

A stationary random field can be generated by considering the gradient of the friction angle as zero in Eq. (6). Based on the above-mentioned procedure, stationary and non-stationary random fields of friction angle are generated. Phoon & Kulhawy⁵⁴ mentioned that for most soils, the mean friction angle varies from 20^o to 40^o. For this range of mean values, the COV of the friction angle is found to vary between 5% to 15%⁵⁴. Thus, in this study, the friction angle mean and COV considered are 28^o and 5%, respectively. Figure 2 summarizes the stationary and non-stationary approaches.

Fig. 2

Shows the generated 1D random fields to summarize the stationary and non-stationary approaches.

Number of Monte Carlo simulation (MCS) runs

The required number of Monte Carlo simulation runs for the probabilistic analysis is determined, with simulations conducted using a mean friction angle of 28°, a coefficient of variation (COV) of 5%, and an isotropic correlation length of 1 m. A total of 3,000 realizations are performed, and the results are presented in Fig. 3. For each realization, MLWD and MGSS are determined. Subsequently, the first and second moments are calculated, and the cumulative mean and standard deviation are plotted for each realization. From Fig. 3(A), it is observed that the mean and standard deviation of MLWD converge at 600 simulations. Similarly, Fig. 3(B) illustrates that the mean and standard deviation of MGSS also converge at 600 simulations. So, in this study, 600 simulations are chosen for probabilistic analysis. It takes approximately 1 min for FLAC 2D to run a single random finite difference simulation on an Intel(R) Xeon(R) Gold 6226R CPU @ 2.90 GHz with 256 GB RAM.

Procedure of the random finite difference method (RFDM)

In this study, FLAC 2D is used for probabilistic analysis. Random fields of friction angle are generated in MATLAB and then imported into FLAC 2D. A total of 600 Monte Carlo simulations are conducted, and the resulting output responses are obtained. The RFDM procedure is illustrated in Fig. 4 and is explained in detail below.

(1) Numerical model for braced excavation: A numerical model is created for braced excavation, which consists of a single-layered sandy soil, an embedded retaining wall, and a strut. The soil and structural members are assigned the model parameters as given in Table 1., except for the friction angle and dilation. The adopted mesh consists of quadrilateral zones measuring 0.5 m on each side.

(2) Random field generation: Stationary and non-stationary random fields of friction angle are generated using the Cholesky matrix decomposition method with the help of MATLAB. The generated random fields are stored as a single line in txt files, as required by FLAC2D.

(3) Importing random fields and MCS: Random field values stored in a text file are imported into FLAC and assigned to the centroid of each zone using the Python console. Based on the friction angle in each zone, the dilation angle is calculated and assigned to the corresponding zones to account for the effect of dilative behaviour in sands. Using the Python console, 600 MCS are performed, and the output responses are stored in Excel files.

(4) Determination of the first moment, second moment, and PDF of output responses: Excel files are then analysed in MATLAB to obtain the first moment, second moment, and probability density function of output responses. The Kolmogorov-Smirnov test is conducted to determine the best-fit distributions.

(5) Parametric study: Effect of frictional angle gradient and VSF on the first and second moments of MLWD and MGSS are studied.

Fig. 3

Convergence curves for (A) MLWD and (B) MGSS.

(6) Component and system failure probability: Component failure probabilities are obtained by considering multiple allowable limits for MLWD, WSF, WBM, MGSS, BWTT, and SAF. The system failure probability is determined using a series system of failures, indicating that the failure of any individual component causes the entire system to fail.

The maximum values of LWD and GSS from each realization are used to examine the effects of the gradient and VSF of the friction angle. Different parameter combinations used in the probabilistic analysis are presented in Table 3.

Results

In this section, effects of stationary and non-stationary random fields on the magnitude and location of braced excavation induced deformation responses such as MLWD and MGSS are studied. The influence of vertical scale of fluctuation on the magnitude of MLWD and MGSS are also studied. Figure 5 shows a typical random field generated for cases of stationary and non-stationary gradients of $\:{0.2}^{^\circ\:}/\text{m}$, $\:{0.4}^{^\circ\:}/\text{m}$, and $\:{0.6}^{^\circ\:}/\text{m}$, respectively, for an isotropic scale of fluctuation of 10 m and 5% COV. As can be seen from the figure, in the case of a stationary random field, the mean friction angle is independent of depth. Whereas in the case of a non-stationary random field, as the gradient of the mean friction angle increases from $\:{0.2}^{^\circ\:}/\text{m}$, $\:{0.4}^{^\circ\:}/\text{m},\:$ and $\:{0.6}^{^\circ\:}/\text{m}$, the values of the friction angle increase with depth. This increasing trend of friction angle with depth is consistent with the field observations of friction angle in fine to medium-grained sand as reported by Singh & Chung⁵⁵.

Table 3 Parameters used in probabilistic analysis.

Fig. 4

Procedure of RFDM.

Figure 6 presents typical realizations of LWD and GSS obtained from the random finite difference method coupled with MCS for stationary case with isotropic scale of fluctuation of 1 m and a 5% coefficient of variation (COV). MLWD from the deterministic analysis is 28.5 mm, while in the probabilistic case, it ranges from 24.48 mm to 33 mm. Similarly, the MGSS is 21.5 mm in the deterministic case compared to a range of 15.57 mm to 26.76 mm from probabilistic analysis. It can be inferred that while probabilistic analysis accounts for all possible outcomes, deterministic analysis may either underestimate or overestimate output responses such as LWD and GSS. Hence, neglecting the spatial variability of friction angle can result in inaccurate predictions of MLWD and MGSS.

Fig. 5

Generated random fields (A) Stationary, (B) Non-stationary with a gradient of 0.2$\:^\circ\:$/m, (C) Non-stationary with a gradient of 0.4$\:^\circ\:$/m and (D) Non-stationary with a gradient of 0.6$\:^\circ\:$/m.

Influence of friction angle gradient on MLWD

The effects of gradient of friction angle which is a non-stationary characteristic on MLWD are investigated in this section at various levels of spatial variability and at COV of 5%. Using RFDM, 600 realisations of LWD are obtained for both stationary and non-stationary random fields. Figure 7A shows the influence of gradient on the mean of MLWD for varying VSF while HSF is fixed at 10 m. It is evident that the non-stationary characteristic of friction angle results in lower mean values compared to both the stationary random field and deterministic case. More explicitly, when the gradient increases from 0 to 0.6°/m, the mean of MLWD decreases from 29 mm to 18 mm. Mean of MLWD for deterministic and stationary cases are 28.5 mm and 29 mm respectively.

Fig. 6

Comparison of deterministic and probabilistic analysis for (A) LWD and (B) GSS.

The influence of the friction-angle gradient on MLWD at different HSFs, while maintaining a VSF of 10 m, is shown in Fig. 7c. A trend similar to that observed for varying VSF is obtained. It is noted that neglecting the depth-dependent variation of the friction angle, i.e., adopting a stationary RFDM, leads to an overestimation of MLWD. For a given HSF and VSF, an increase in the mean friction-angle gradient results in higher effective shear strength over most of the excavation depth, which reduces the mobilized active earth pressures acting on the wall. Consequently, the lateral earth pressure decreases, leading to a reduction in the mean MLWD.

Figure 7b shows the effect of the friction-angle gradient on the coefficient of variation (COV) of MLWD for different VSF values at a constant HSF of 10 m. No clear or consistent trend between the friction-angle gradient and the COV of MLWD can be identified. A similar response is observed in Fig. 7d, where the COV of MLWD is examined for different HSF values while maintaining a constant VSF of 10 m. These results suggest that, although the friction-angle gradient has a noticeable influence on the mean MLWD, its effect on the variability of MLWD is relatively small. This can be explained by the stiffness-controlled behaviour of wall deflection, in which structural compatibility and load redistribution limit the influence of spatial soil variability. As a result, when the input COV of the friction angle is held constant, the COV of MLWD remains largely insensitive to changes in the friction-angle gradient.

Influence of friction angle gradient on the MGSS

Stationary and non-stationary RFDM are employed to examine their effects on MGSS for different VSF levels at a fixed HSF of 10 m. In addition, the influence of varying HSF is investigated while maintaining a VSF of 10 m. Figure 8a illustrates the effect of the friction-angle gradient on the mean MGSS at different VSF levels. The non-stationary random field of friction angle results in lower mean MGSS compared to the deterministic and stationary cases. For instance, the mean MGSS values for the deterministic and stationary cases are 21.5 mm and 22 mm, respectively, whereas the non-stationary cases yield mean MGSS values of approximately 17.5 mm, 14 mm, and 10.3 mm for friction-angle gradients of 0.2, 0.4, and 0.6, respectively. A similar trend is observed in Fig. 8c, where the mean MGSS decreases with increasing friction angle gradient. This reduction is attributed to the increase in effective shear strength over the settlement influence zone caused by the depth-dependent variation of the friction angle, which enhances soil resistance to deformation and reduces vertical strain accumulation. Consequently, the mean MGSS decreases as the friction angle gradient increases.

Figure 8b illustrates the effect of the friction-angle gradient on the COV of MGSS for a constant horizontal scale of fluctuation of 10 m. In contrast to MLWD, the COV of MGSS increases with increasing friction-angle gradient. This behaviour is attributed to the fact that MGSS is governed by far-field soil deformation mechanisms, such as stress redistribution and arching behind the excavation. As the friction-angle gradient increases, the mean settlement decreases due to higher overall soil strength. However, the settlement response remains highly sensitive to localized weak zones, resulting in a relatively high dispersion. Consequently, the COV of MGSS increases with increasing friction-angle gradient. A similar trend is observed in Fig. 8d, where the vertical scale of fluctuation is maintained at 10 m.

Fig. 7

Influence of friction angle mean gradient on (A) Mean of MLWD for 10 m HSF, (B) COV of MLWD for 10 m HSF, (C) Mean of MLWD for 10 m VSF (D) COV of MLWD for 10 m VSF.

Figure 9 A and B show the histograms of MLWD and MGSS for friction angle gradients along with the fitted probability density functions. The Kolmogorov-Smirnov and Anderson-Darling tests are conducted to find the best suitable distribution for MLWD and MGSS. It is found that the lognormal distribution fits best for MLWD^{30,36,56,57,58} and MGSS^36,56,57,58. The figures also demonstrate that as the gradient of friction angle increases, the histograms shift to the left which indicates that the output responses, such as MLWD and MGSS decreases.

Influence of friction angle gradient on the location of MLWD and MGSS

The effects of stationary and non-stationary random fields on the locations of MLWD and MGSS are examined in Fig. 10. Understanding these effects is crucial for evaluating the structural stability of diaphragm walls and adjacent buildings. In the stationary case, the mean location of MLWD is higher compared to non-stationary cases (Fig. 10A). For non-stationary cases, an increase in the friction angle gradient results in a slight upward shift of the MLWD location. This shift occurs because the friction angle increases with depth, strengthening the lower soil layers and causing maximum wall deflection to occur in relatively weaker upper layers.

Figure 10B illustrates that the friction angle gradient has no effect on the mean horizontal position of the MGSS. The MGSS location is measured from the left boundary of the model domain, and across all stationary and non-stationary analyses, it remains fixed at approximately 35.5 m. This position corresponds exactly to the right-hand edge of the adjacent building located beside the excavation. This can also be observed from Fig. 6B, where the typical realizations of the GSS profiles for the stationary case are provided. The reason for this invariance is that the MGSS position is primarily governed by the structural and geometric configuration, such as the relative locations of the excavation, the retaining wall, and the adjacent building. These factors dictate the deformation mechanism and the shape of the settlement trough. In contrast, variations in the friction angle gradient influence the magnitude of ground deformation but do not shift the settlement trough horizontally. As a result, the MGSS continues to occur at the same location regardless of whether the soil profile is stationary or non-stationary.

Effect of VSF on the mean and COV of MLWD and MGSS

Figure 11 shows the influence of VSF on the mean and COV of MLWD and MGSS for a non-stationary case with friction angle gradient of 0.2°/m and COV of 5%. In Fig. 11A it is seen that the mean of MLWD and MGSS increases when VSF increases from 1 to 20 m. This trend levels off when VSF is greater than 20 m. Figure 11A also includes the results from the deterministic study.

Fig. 8

Effect of friction angle gradient on (A) Mean of MGSS for 10 m HSF, (B) COV of MGSS for 10 m HSF, (C) Mean of MGSS for 10 m VSF, and (D) COV of MGSS for 10 m VSF.

Similarly, Fig. 11b shows that the COV of MLWD and MGSS increases as the vertical scale of fluctuation (VSF) increases from 1 to 20 m. Beyond 20 m, which corresponds to the depth of the numerical model, this trend levels off. When the VSF is small, soil properties vary rapidly with depth, and the effects of these local variations tend to average out, leading to reduced variability in the structural response. In contrast, a larger VSF implies that the soil behaves more uniformly with depth, resulting in less local averaging and hence greater variability in the output responses. Once the VSF exceeds the model depth, further increases in VSF have little influence on the response variability. Other researchers have found similar results in their previous studies^1,47.

Fig. 9

Histogram plot for (A) MLWD and (B) MGSS.

Fig. 10

Influence of frictional angle gradient on the location of (A) MLWD and (B) MGSS.

Assessment of component failure probability

As outlined in the previous section, structural and geotechnical serviceability limit states are commonly used to assess the failure probability of individual components in the braced excavation system. Structural ultimate limit states involve WSF and WBM in the diaphragm wall, as well as SAF. Geotechnical serviceability limit states include parameters such as the MLWD and MGSS. However, the MGSS focuses solely on maximum ground settlement serviceability and does not explicitly account for differential settlement, which can be a critical factor in geotechnical designs, especially for the safety of structures located next to the braced excavation. To address this differential settlement serviceability criterion, this study adopts building wall torsion tilt (BWTT), a parameter proposed by Xu et al.⁵⁹ as a new serviceability criterion for buildings. This parameter introduces an additional aspect for assessing building performance, complementing existing serviceability criteria.

Fig. 11

Influence of VSF on (A) the mean of MLWD and MGSS, and (B) the COV of MLWD and MGSS.

BWTT is expressed as the ratio of the differential settlement to the length of the building wall. BWTT indicates the direction and degree of building inclination due to differential settlement.

BWTT is expressed as follows

$$\:\omega\:=\:\frac{\varDelta\:h}{l}\:\times\:100\%=\:\frac{{h}_{1}-\:{h}_{2}}{l}\:\times\:100\%$$

(9)

Where ω is the building wall torsion tilt, ∆h is the differential settlement of the building, l is the length of the wall (7 m), and h₁ and h₂ are the settlements of the building wall at the right and left ends of the wall, respectively. The calculation of BWTT is illustrated in Fig. 12. The figure also includes the deterministic GSS profile with the calculation of BWTT.

Table 4 Description of various protection levels used in Shanghai^3,4.

During the design and construction stage of braced excavation, the failure probability of a ultimate/serviceability limit state can be defined as the probability that the geotechnical or structural response exceeds the allowable limit. In this section, the failure probability is determined by considering component failure modes, such as the MGSS, BWTT, MLWD, WSF, WBM, and SAF separately.

The component failure probability is obtained as follows

$$\:{P}_{f}=\:\frac{{N}_{f}}{N}\:\times\:100\:\%$$

(10)

Where $\:{P}_{f}\:$ is the component failure probability, $\:{N}_{f}$ is the total count of cases in which the maximum output response exceeds the allowable limit, and N represents the total count of simulations.

To estimate the number of simulations required to accurately determine the failure probability of individual components and the braced excavation system, an analysis was performed using 5000 simulations. Figure 13 demonstrates that the failure probabilities for both individual components and the system converge after approximately 600 simulations, which is due to the 5% coefficient of variation (COV) of the friction angle used in the random field generation and stringent allowable limits adopted. As a result, 600 simulations are selected for this study to calculate the failure probabilities. Figure 14 provides the component failure probability curves.

Figure 14A illustrates the failure probability due to MLWD across a range of limiting MLWD values, from 5 mm to 55 mm. To understand the effect of stationary and non-stationary random fields of friction angle, the failure probability is obtained for stationary and non-stationary random fields, considering the isotropic scale of fluctuation of 10 m and COV of 5%. As observed from the figure, the stationary random field overestimates the failure probability when compared to non-stationary random fields. It is also important to note here that as the friction angle gradient increases, the failure probability reduces. For example, if the allowable LWD is 25 mm, the failure probability for the stationary case is around 84%, whereas, for non-stationary cases, it is 45%, 16%, and 3%, respectively. Thus, there is a substantial decrease in the failure probability when the non-stationary random field is adopted. The failure probability for other components, such as MGSS, BWTT, WSF, WBM, and SAF, follows a similar trend to that observed for MLWD. Figure 14B illustrates the influence of stationary and non-stationary random fields on the failure probability related to MGSS. For instance, with an allowable GSS of 20 mm, the failure probability in the stationary case is approximately 60%. In contrast, for non-stationary cases, the failure probabilities are about 27%, 7.5%, and 1.2%, respectively. Similarly, Fig. 14C shows the influence of stationary and non-stationary random fields on the failure probability for BWTT. For example, with a limiting allowable torsional tilt of 0.20%, the failure probability in the stationary case is around 38%, while for non-stationary cases, it is reduced to approximately 21%, 11%, and 3%, respectively.

Thus, considering the friction angle as a stationary random field overestimates the failure probability for all geotechnical and structural components when compared to a non-stationary random field. Additionally, as the gradient of the friction angle increases, there is a substantial decrease in failure probability, highlighting the importance of accounting for depth-dependent variability in component failure probability assessments.

Assessment of system failure probability

To obtain the system failure probability for braced excavation, geotechnical serviceability failure modes such as MLWD, MGSS, and BWTT, as well as structural ultimate failure modes such as WSF, WBM, and SAF, are considered. Since dewatering is carried out before excavation, there is no chance of piping failure, hence piping failure i.e. seepage induced failure is ignored in system failure probability calculations.

Fig. 12

Calculation of BWTT for the deterministic case.

Fig. 13

Number of Monte-Carlo simulations versus the failure probability of individual components and the braced excavation system.

Fig. 14

Influence of friction angle gradient on the probability of exceeding the allowable limits for (A) MLWD, (B) MGSS, (C) BWTT, (D) WSF, (E) WBM, and (F) SAF.

A series failure system is considered for obtaining the failure probability of a braced excavation system because one individual component failure in the braced excavation is considered as a system failure. To analyse the dependence/correlation among the components, the Pearson correlation coefficient is calculated among the components and result is presented in Fig. 15 (Stationary case). Similar correlation structure is found for non-stationary cases. As observed from the figure, the components are positively correlated with correlation coefficients higher than 0.8 in all the cases. This correlation structure indicates that failure of one component leads to the failure all other components as well as system failure.

Fig. 15

Correlation coefficient among the components.

The system failure probability of braced excavation is calculated using Monte Carlo simulation approach as explained below⁶⁰.

$$\:{P}_{f\:}=\:\frac{1}{N}\sum\:_{k=1}^{N}{I}_{k}=\:\frac{1}{N}\sum\:_{k=1}^{N}I({\exists\:}_{j}:{X}_{k,j\:}\ge\:\:{g}_{j})$$

(11)

Where $\:{P}_{f\:}$is system failure probability, $\:{\exists\:}_{j}$ is failure of any component, $\:k$ is number of Monte Carlo simulations, $\:j$ is number of components in the system (i.e. 6), $\:{X}_{k,j\:}$is the response obtained from the Monte Carlo simulation for a particular component and for a particular simulation and $\:{g}_{j}$ is the limiting value for the particular component $\:j.$ Kolmogorov–Smirnov (K–S) test is performed on the output responses. Based on the KS test, the components such as MGSS, MLWD, WSF, WBM and SAF are lognormally distributed. Whereas the BWTT is normally distributed. To generate the random numbers based on the obtained correlation structure, the Gaussian copula method is adopted and is explained below.

To generate n independent and identically distributed (i.i.d) samples $\:{X}^{\left(1\right)},\dots\:.,{X}^{\left(n\right)}$ from a k-dimensional distribution with a Gaussian copula (with correlation matrix $\:R\in\:\:{\mathbb{R}}^{k\times\:K}$) and specified marginal distribution, where the first marginal is $\:\mathcal{N}\left({\mu\:}_{N},\:{\sigma\:}_{N}^{2}\right)$ and the remaining k-1 marginals are Lognormal($\:{\mu\:}_{l},\:{\sigma\:}_{l}^{2})$⁶¹.

• 1. Generate an $\:n\times\:k$ matrix $\:Z$ whose rows are i.i.d. draws from the multivariate normal distribution $\:{\mathcal{N}}_{k}(0,\:R)$ :

$$Z_{{i~}} \sim ~{\mathcal{N}}_{k} \left( {0,~R} \right),i{\text{ }} = 1{\text{ }}, \ldots ,{\text{ }}n~$$

(12)

• 2. Apply the standard normal CDF $\:{\Phi\:}:\mathbb{R}\to\:\left(\text{0,1}\right)$ elementwise to obtain the $\:n\times\:k$ matrix $\:U:$.

$$\:{U}_{ij}={\Phi\:}\:\left({Z}_{ij}\right),\:i=1,\dots\:.,n,\:\:\:j=1,\dots\:,k.$$

(13)

The rows of $\:U$ are then i.i.d. draws from the uniform distribution on $\:{\left[\text{0,1}\right]}^{k}$ with Gaussian copula $\:{C}_{R}$.

• 3. Transform the rows of $\:U$ via the inverse CDFs of the target marginals to obtain the $\:n\times\:k$ sample matrix $\:X:$.

$$X_{{i1}} = \mu _{N} + \sigma _{N}^{~} \Phi ^{{ - 1}} \left( {U_{{i1}} } \right),~i = 1, \ldots ,n$$

(14)

for first normal marginal

$$~~X_{{ij}} = \exp (\mu _{{j - 1}} + \sigma _{{j - 1}}^{~} \Phi ^{{ - 1}} \left( {U_{{ij}} } \right))i{\text{ }} = {\text{ }}1, \ldots n,~j{\text{ }} = {\text{ }}2, \ldots ,k$$

(15)

for remaining lognormal marginals.

Where $\:{{\Phi\:}}^{-1}$ is the standard normal CDF function and $\:exp$ is the inverse CDF of Lognormal.

The rows $\:{X}_{i}={\left({X}_{i1,\dots\:,}{X}_{ik}\right)}^{T}$ for $\:i=1,\dots\:,n$ are the desired samples. The sample correlation matrix of the column of Z approximates R for large n.

Table 5 presents the limiting values adopted for evaluating the system failure probability considering multiple failure modes. The limiting values for MLWD and MGSS are defined based on the protection levels of the Shanghai Metro and are summarized in Table 4. The limiting values for BWTT are determined using the maximum allowable angular distortion criteria specified in the Chinese standard GB 50,007–2011⁶² and Eurocode-7⁶³, which recommend limits of 0.002 L and 1/500, respectively. These criteria correspond to a maximum allowable BWTT of 0.2%, which is adopted as the limiting value for level 3. For levels 1 and 2, more stringent BWTT limits are required relative to the level 3. Accordingly, limiting values of 0.10% and 0.05% are adopted for Levels 2 and 1, respectively. Xu et al.⁵⁹ reported a case study in which building collapse occurred when BWTT reached 0.10%. Although this observation is based on a single case study, it is used in the present work as a conservative intermediate threshold, rather than as the governing criterion for the most critical protection level. For level 1 more stringent BWTT limit of 0.05% is adopted. The limiting values for WSF, WBM, and SAF are assumed.

Figures 16, 17 and 18, and 19 show the individual and system failure probabilities obtained for stationary and non-stationary friction angle gradients. The limiting values adopted for calculating the failure probabilities are summarized in Table 5. The results indicate that, for level 1, the failure probability of both individual components and the overall system is equal to 1 for all stationary and non-stationary cases. This behaviour is primarily due to the very stringent serviceability criteria adopted for level 1. As the transition is made from level 1 to level 2 and then to level 3, there is a significant reduction in the failure probability of both individual components and the system. This is attributed to the higher allowable limits set at level 3 compared to level 2, and similarly, level 2 has higher limits than level 1. It is important to highlight that, for level 1 and level 2, the obtained failure probabilities are nearly 1 which is due to the very strict allowable limits prescribed in the Shanghai Metro standards and other allowable limits adopted from the codes. The protection levels adopted from the Shanghai Metro guidelines and other international standards are intentionally conservative, aiming to protect nearby buildings and the braced excavation system. When these deterministic limits are directly treated as a probabilistic failure threshold, even modest variability in soil parameters lead to frequent exceedance of the allowable limits. Consequently, the calculated system failure probability can approach unity, despite the system still being acceptable from a conventional deterministic design standpoint.Whereas in the level 3 case, individual components and system failure probability are very low. For example, for a stationary case, the system failure probability is around 47%. Whereas for a non-stationary case, it is around 26%, 13%, and 4%, respectively. The gradient of the friction angle is seen to have a considerable effect on the system failure probability.

Table 5 Limiting values adopted in this study for the system probability.

For protection level 3, BWTT significantly affects the system failure probability across all cases of stationary and non-stationary random fields while the effects of other failure modes are negligible. This means in absence of differential settlements as serviceability criteria, the failure probability of braced excavation system can be greatly underestimated leading to potential catastrophic failures of excavation support structure and/or the adjacent built structures.

Fig. 16

System failure probability for protection levels considering the stationary random field.

Fig. 17

System failure probability for various protection levels considering the non-stationary random field with a friction angle gradient of $\:0.2^\circ\:/\text{m}.$.

Fig. 18

System failure probability for various protection levels considering the non-stationary random field with a friction angle gradient of $\:0.4^\circ\:/\text{m}.$.

Fig. 19

System failure probability for various protection levels considering the non-stationary random field with a friction angle gradient of $\:0.6^\circ\:/\text{m}.$.

Conclusions

The study considers non-stationarity in the friction angle to quantitatively evaluate the performance and failure probability of the braced excavation system. The effects of a stationary random field of friction angle are also considered. The influence of parameters such as the friction angle gradient and the vertical scale of fluctuation on the mean and COV of output responses, such as maximum lateral wall deflection and maximum ground surface settlement, is studied. Finally, the failure probability for individual components and the system is obtained. The following conclusions are drawn.

• Ignoring the inherent spatial variability of soil leads to erroneous estimation of output responses of braced excavation, such as maximum lateral wall deflection and maximum ground surface settlement.

• Considering the friction angle as stationary random field results in higher values of the output responses compared to those from non-stationary random field. In the non-stationary case, as the gradient of friction angle increases, the mean of the output response decreases because of the increased shear resistance. It is observed that neglecting the depth-dependent variation of friction angle or use of stationary RFDM could lead to inaccurate estimation of excavation system behaviour.

• Location of the maximum lateral wall deflection is dependent on the gradient of friction angle. This is an important finding for the efficient design and assessment of the structural stability of diaphragm walls and excavation support system.

• Mean and COV of maximum lateral wall deflection and maximum ground surface settlement were found to be increasing when the vertical scale of fluctuation is between 1 m and 20 m. This trend levels off when the vertical scale of fluctuation is greater than the vertical domain, i.e., 20 m. This is due to the local averaging effect.

• For all the cases considered here such as stationary and non-stationary, the components are found to be highly dependent/correlated with each other. The Pearson correlation coefficient is found to be higher than 0.8 in all the cases, which indicates that one component failure leads to failure of all the components and braced excavation system.

• Consideration of friction angle as a stationary random field leads to higher component failure probability as well as system failure probability when compared with a non-stationary random field. As the gradient of friction angle increases, there is a substantial decrease in failure probability, highlighting the importance of depth-dependent variability in component failure probability assessments.

• The protection levels significantly influence both the individual failure probability of components and the overall system failure probability. The lowest failure probability is observed at Level 3, followed by Level 2, and then Level 1. In Level 1, for both stationary and non-stationary random fields, the failure probabilities of individual components and the system are approximately equal to 1, due to the stringent criteria adopted at this level. BWTT significantly affects the system failure probability across all cases of stationary and non-stationary random fields while the effects of other failure modes are negligible for protection level 3. In this view, ignoring the assessment of differential settlements could result in underestimation of risks to adjacent structures as well as the braced excavation system.