Introduction

Soldier piles are widely used in urban and industrial areas as a temporary or permanent retaining system for constructing the underground structures. The major advantages of this retaining wall system are its relatively low cost and fast construction. For these reasons, the soldier piles are a popular topic and have attracted the attention of researchers (e.g., Zhang 2009). To evaluate the performance of soldier-piled excavation, considering not only multiple failure modes but also soil properties uncertainties and the unsaturated state are crucial.

However, some studies have been presented in the literature for stability analysis of retaining wall in the unsaturated state (e.g., Vo and Russell 2014), but there is still a need for the implementation of this state in stability analysis of soldier-piled excavation. Sahoo and Ganesh (2018) investigated the distribution of active earth pressure against the face of retaining wall with unsaturated backfill. The results showed that the resultant active force exerted by unsaturated clay backfill was significantly affected by flow condition, while in the case of unsaturated sand backfill, active earth pressure remained unaffected. Shwan (2016) extended the limit analysis method to evaluate passive earth pressure in an unsaturated state. Vahedifard et al. (2015) developed a method for calculating earth pressures exerted by unsaturated backfill against retaining wall, which tackles the limitation connected with analytical methods in an unsaturated condition such as collapse and effective stress. Vo and Russell (2016) compared different retaining wall model tests with theoretical results in the unsaturated state.

Several studies showed the insufficiency of traditional deterministic excavation stability analysis in which stability is evaluated only based on the FS (e.g., Baecher and Christian 2005). Due to the inherent variability associated with soil properties and a limited number of boreholes, a reliability-based approach is present as a complementary measure to the FS to aid geotechnical experts to make acceptable designs (Johari and Mousavi 2019; Parhizkar et al. 2012). The relevant studies have addressed the probabilistic evaluation of the retaining system (e.g., Gao et al. 2019), but none of these involved a soldier-piled excavation. Wang (2013) presented a Monte Carlo simulation (MCS)–based design of a retaining system and found that factors adopted in Eurocode 7 do not satisfy its goal reliability. GuhaRay and Baidya (2015) combined the probability of failure (Pf) and sensitivity of a random variable of sheet pile wall and introduced a new factor, namely probabilistic risk factor. Due to a disability of probabilistic method to consider the spatial variability of soil properties, in the early 1990s, Griffiths and Fenton (1993) proposed a new probabilistic analysis approach called the random finite element method (RFEM). This method has been used for various geotechnical problems, but few studies involved the stability of excavation. Tang (2011) evaluated the excavation-induced lateral deflection and presented a design chart for estimating the Pf with respect to lateral deflection. Sert et al. (2016) assessed the role of spatial variability of sand properties on lateral displacement and bending moment of retaining wall using RFEM. Luo et al. (2018) conducted a probabilistic analysis of supported excavations using RFEM and revealed that spatial variation of soil properties has an essential role in several responses such as displacement and internal forces of the wall.

Most reliability analysis mainly focuses on the reliability index (β) corresponding to an individual failure mode, while geotechnical problems often involve numerous failure modes. As noted in the work of Luo et al. (2018), the failure modes to be considered in the design of a soldier-piled excavation may include global stability, lateral displacement, shear force, and bending moment. Since the system reliability index of soldier-piled excavation can be less than the reliability index of the individual failure mode, it is essential to use a system reliability analysis (Liu and Low 2017). System reliability analysis of the retaining system is rarely found in the literature. However, system reliability analysis of other geotechnical problems such as slope stability has received a great deal of attention recently (e.g., Johari and Lari 2017; Metya et al. 2017). Zevgolis and Daffas (2018) developed a model to investigate the system reliability of the soil nail wall. Zevgolis and Bourdeau (2010) conducted a system reliability analysis of retaining wall considering external stability as a series system. The result of this research indicated that the degree of correlation has an important effect on the overall Pf. Luo and Das (2016) evaluated the system reliability of serviceability, including excessive wall deflection and excessive ground surface settlement, for braced excavations, which cause damage to adjacent infrastructures.

The common feature of the mentioned stochastic studies was the consideration of spatial variability of soil properties in the retaining systems without considering unsaturated state, all geotechnical and structural failure modes, two-dimensional random field, and system reliability analysis. The main aim of the current study was to cover the mentioned limitations via system reliability analysis of unsaturated soldier-piled excavation using RFEM. To do this, a real soldier-piled excavation in the unsaturated state was analyzed deterministically using finite element–based program coded in MATLAB. Then, reliability analysis was performed using RFEM by the implementation of the random field in the program that was mentioned earlier to consider the inherent uncertainty of soil properties (i.e., cohesion, friction angle, and unit weight). Moreover, the effect of the unsaturated state was evaluated by using a physico-empirical model and considering water content (ωc) as a random variable. Then, the reliability indices of all individual failure modes (i.e., global, lateral displacement, shear force, and bending moment) were estimated. Finally, system reliability analysis was performed to offer a single index for quantifying the overall reliability of the system, instead of several partial reliabilities indices.

Methodology of analysis

In the present study, stochastic stability analysis of soldier-piled excavation in the unsaturated state is assessed. For this purpose, the random field is utilized in conjunction with the finite element method (FEM) to study the spatial variability of regionalized variables. In this way, the soil is assumed to be unsaturated, and the multiple failure modes of soldier-piled excavation are considered. Brief descriptions of the selected methodology are presented in the following subsections.

Finite element modeling

FEM is a powerful tool to model geotechnical problems such as excavation. One of the superiority of this method is the ability to predict the deformations of soldier-piled excavation. In the excavation project, the soil is removed from the ground. Since the ground is stressed before the removal of its part, the analysis must simulate the correct stress for remaining soil by applying force along the excavation surface, as shown in Fig. 1. Generally, the excavation forces (FBA) acting on a boundary depending on the stress state in the excavated soil (σA0) and the self-weight of it, regardless of the retaining system (Smith et al. 2013). Forces FBA can be calculated as follows:

$$ \left\{{F}_{\mathrm{BA}}\right\}=\underset{V_{\mathrm{A}}}{\int }{\left[B\right]}^T\left\{{\sigma}_{\mathrm{A}0}\right\}{dV}_{\mathrm{A}}+\gamma \underset{V_{\mathrm{A}}}{\int }{\left[N\right]}^T{dV}_{\mathrm{A}} $$
(1)

where [B] is the strain-displacement matrix, VA the excavated volume, γ the soil unit weight, and [N] the shape functions of elements.

Fig. 1
figure 1

The procedure for determining the excavation forces. (a) Initial stress state. (b) Equilibrium of the bodies (A and B). (c) Excavation forces FBA

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Failure limit states of soldier-piled excavation

Generally, for designing soldier-piled excavation, three main limit states: external stability, serviceability, and structural must be considered (Luo et al. 2018). Each of these limit states includes several failure modes with varying degrees of importance due to location, neighboring structures, and the depth of excavation. In order to demonstrate these limit states, the FS was used in relevant equations in this section for all the failure modes.

External stability

Generally, the external stability of a retaining system is considered as global and sliding stability. However, in soldier-piled excavation analysis, the sliding stability is neglected due to the base fixity. In the current study, the FS against global stability, which is represented with FSG, was evaluated using the finite element strength reduction analysis. The essence of the FEM with the shear strength reduction technique is the reduction of the soil strength parameters until the soil fails. The factored shear strength parameters c′f and φ′f are therefore given by:

$$ {c}_{\mathrm{f}}^{\prime }=\frac{c^{\prime }}{{\mathrm{FS}}_{\mathrm{G}}} $$
(2)
$$ {\varphi}_{\mathrm{f}}^{\prime }=\arctan \left(\frac{\tan \varphi^{\prime }}{{\mathrm{FS}}_{\mathrm{G}}}\right) $$
(3)

where c′ and φ′ represent the shear strength parameters of the soil.

Serviceability

The maximum lateral displacement of soldier-piled excavation is one of the most important components of the serviceability limit state and hence must be controlled strictly. For this purpose, the maximum lateral displacement should not be greater than the maximum allowable lateral displacement. FS against this failure mode (FSLD) is defined as (Burland et al. 1981):

$$ {\mathrm{FS}}_{\mathrm{LD}}=\frac{R_{\mathrm{LD}}}{\delta_{\mathrm{max}}} $$
(4)

where RLD is the maximum allowable lateral displacement and δmax is maximum lateral soldier-piled excavation displacement obtained from the elasto-plastic analysis. It is common to use the value of 0.5% of the excavation depth for maximum allowable displacement as a governing criterion for the safety of the works and neighboring structures (Zhang et al. 2015).

Structural

The structural limit state of soldier piles can be subdivided into shear force and bending moment stability. FS against shear force (FSSF) can be defined as (Burland et al. 1981):

$$ {\mathrm{FS}}_{\mathrm{S}\mathrm{F}}=\frac{R_{\mathrm{S}}}{S_{\mathrm{max}}} $$
(5)

In which RS is a shear force capacity of soldier piles, and Smax is the estimated maximum shear force of soldier piles. Also, the FS for bending moment (FSBM) can be expressed as (Burland et al. 1981):

$$ {\mathrm{FS}}_{\mathrm{BM}}=\frac{R_{\mathrm{M}}}{M_{\mathrm{max}}} $$
(6)

where RM is a bending moment capacity of soldier piles and Mmax is the obtained maximum moment of soldier piles.

Modeling of unsaturated condition

Identification of the behavior of unsaturated soils is pivotal to address the geotechnical problems, including stability analysis of the retaining system located above ground water table (GWT) (Tekinsoy et al. 2004). Shear strength, which is the most important parameter affecting the behavior of unsaturated soil, can be estimated directly or indirectly (Garven 2006). The direct approach is often more complicated, costly, and time-consuming compared to the indirect approach (Assouline et al. 1998; Fredlund et al. 2002). The indirect approaches can be classified into two categories, namely effective stress method and method based on two independent state variables (Johari and Gholampour 2018). Shear strength at failure (τf) in the effective stress method is given by:

$$ {\tau}_{\mathrm{f}}=c^{\prime }+\sigma^{\prime}\tan \varphi^{\prime } $$
(7)

where σ′, c′, and φ′ are effective stress, cohesion, and friction angle, respectively. The Bishop’s formulation is used in this paper as follows (Bishop 1959):

$$ \sigma^{\prime }=\left(\sigma -{u}_{\mathrm{a}}\right)+\chi \left({u}_{\mathrm{a}}-{u}_{\mathrm{a}}\right) $$
(8)

where σ is total stress, and ua and uw are the pore air and water pressure. (ua−uw) is matric suction and χ is an effective stress parameter. Among various existing expressions, the one proposed by Vanapalli et al. (1996) seems to be one of the most appealing for estimating the effective stress parameters.

$$ \chi =\frac{\theta_{\mathrm{w}}-{\theta}_{\mathrm{r}}}{\theta_{\mathrm{s}}-{\theta}_{\mathrm{r}}} $$
(9)

in which θw is the water content at the considered matric suction, θs is the saturated water content, and θr is the residual water content.

Estimation of soil water retention curve

As described in the previous section, matric suction provides a considerable contribution to shear strength. An efficient tool to measure matric suction is the soil water retention curve (SWRC), which demonstrates a soil’s ability to retain and release water as it is subjected to different suctions, with no necessity of direct method equipment. There are several methods, including direct and indirect methods, to obtain the SWRC of a particular soil, which is significantly affected by pore shape, from grain size distribution (GSD) for different depths, the particle-size distribution and the specific surface area (Johari and Golkarfard 2018). Several empirical methods have been proposed to overcome the excessive cost and time associated with a direct method. Among empirical methods, Arya and Paris’ (1981) method is a prominent one for predicting the SWRC from particle-size data. Based on the advantages and capabilities of this method, the modified form of it (Arya et al. 1999) was implemented in the current study. The model first translates a particle-size distribution into a pore-size distribution. Then, the cumulative pore volumes corresponding to progressively increasing pore radii are divided by the sample bulk volume to give the volumetric water contents, and the pore radii are converted to equivalent soil water pressures using the equation of capillarity. This model offered regression fitting parameters (i.e., a and b) for limited soil type from a different region, which may lead to less accurate prediction. The great influence of SWRC fitting parameters on stability (Nguyen et al. 2019) and the intention to use the proposed model in a case study made it desirable to estimate them for the city of the case study. To tackle the shortcomings associated with this model, the work of the senior author (Johari et al. 2011) was used in this study. The soil samples from 14 different locations in the city of Shiraz were tested and their SWRCs were determined using a pressure plate apparatus. To determine the fitting parameters of each sample, some of which are presented in the Appendix (Figs. 41 and 42), the model was fitted to the experimental curve. For each type of soil, the mean value of fitting parameters which are summarized in Table 13 in the Appendix was utilized to determine the SWRC in case study.

System reliability analysis

The variability of the soil properties, even within homogeneous layers, is the most dominating source of uncertainty in geotechnical engineering. Hence, considering the spatial variability of soil properties by using the appropriate method is crucial. The RFEM offers a useful tool to incorporate spatial variability of soil properties into stability analysis by combining a random field with the FEM (Griffiths and Fenton 2004).

Random field theory

The theory of random fields, which was achieved by Vanmarcke (1983), is a powerful and rigorous method for modeling the variability of soil properties. The set of random variables at all positions within the entire volume of the soil is mentioned as a random field according to a correlation between adjacent random variables. It is expected that the correlation value decreases by increasing the distance between the two positions. This distance in which the soil parameters are rationally well-correlated to its neighbors is called the autocorrelation length (l).

The spatial correlation of soil parameters can be characterized by their correlation function within the framework of random fields (Vanmarcke 1983). The correlation function of specified soil properties can be computed from known data of the properties at various positions (Baecher and Christian 2005). The correlation between two different random variables (e.g., x1 and x2) is measured by a correlation function defined as follows (Ahmed 2012):

$$ \rho \left({x}_1,{x}_2\right)=\frac{1}{\sigma_{x_1}+{\sigma}_{x_2}}E\left[\left({x}_1-{\mu}_{x_1}\right)\left({x}_2-{\mu}_{x_2}\right)\right] $$
(10)

in which x1 and x2 might be the values of two different properties or the values of the same property at two different locations. \( {\sigma}_{x_i} \) and μxi are, respectively, the standard deviation and the mean value of the variable xi (i = 1, 2). In the current research, the Markov correlation function was utilized to characterize the spatial autocorrelation of the soil properties (Nguyen and Likitlersuang 2019) as:

$$ \rho \left({\tau}_{\mathrm{x}},{\tau}_{\mathrm{y}}\right)=\exp \left\{-\frac{2{\tau}_{\mathrm{x}}}{l_{\mathrm{x}}}\right\}\exp \left\{-\frac{2{\tau}_{\mathrm{y}}}{l_{\mathrm{y}}}\right\} $$
(11)

where τx and τy are the differences between the horizontal and vertical directions of any two points in the random field, respectively, and lx and ly are the autocorrelation lengths in the horizontal and vertical directions, respectively.

Due to the significant influence of the autocorrelation lengths on stability of geotechnical problems in unsaturated state (Nguyen et al. 2017), it would be more appropriate to evaluate them from known data. The sample autocorrelation function (ACF) can be a simple and useful tool for this purpose. The ACF is the graph of the sample autocorrelation at lag k, rk, for lags k = 0, 1, 2, …, m, where m is the maximum number of lags allowed for obtaining reliable estimates. The rk is defined as follows (Jaksa 2013):

$$ {r}_{k=}\frac{\sum \limits_{i=1}^{N-k}\left({X}_i-\overline{X}\right)\left({X}_{i+k}-\overline{X}\right)}{\sum \limits_{i=1}^N{\left({X}_i-\overline{X}\right)}^2} $$
(12)

where Xi and Xi+k are the values of the variable at points i and i+k respectively, and \( \overline{X} \) is the mean value of the variable. Vanmarcke (1983) suggested that autocorrelation lengths can be determined by fitting one of the models to the sample ACF, as given in Table 1, where ∆z is the depth interval.

Table 1 Theoretical autocorrelation functions used to determine the autocorrelation lengths
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The numerical methods such as finite element have discrete nature; hence, a continuous-parameter random field must also be discretized into random variables. This process is commonly known as the discretization of a random field. Various methods have been presented for the discretization of a random field (e.g., Li and Der Kiureghian 1993). In the current research, the covariance matrix decomposition approach (Fenton and Griffiths 2003) was utilized for the discretization of the random field.

The system reliability approach

Physical systems which are consisted of several components can be categorized as series, parallel, and combined systems (Johari and Lari 2017). In the case of a series system, the failure of each component led to the failure of the whole system. However, the parallel system would not fail unless all failure modes occur. Since the failure of at least one component of soldier-piled excavation leads to the failure of the whole retaining system, it can be modeled as a four-component series system (i.e., global, lateral displacement, shear force and bending moment). Several approaches have been presented for combining the components of parallel systems (i.e., the intersection of component events), series systems (i.e., a union of the component event), and combined systems (Genz and Bretz 2002; Pandey and Sarkar 2002). To tackle the problems associated with the previous approaches, Kang and Song (2010) proposed an efficient method (SCM) in which the reliability of the components is first computed, and the components are subsequently combined into equivalent components two at a time until the full system reliability is obtained. Distinct superiority of the SCM over the others is minimizing the complexity of logical operation as it involves only two components in each process.

Consider compounding two components E1 and E2 coupled by union into a single equivalent event E1or2. For example, this compounding can appear in a series system, and can be compounded as follows:

$$ {\displaystyle \begin{array}{c}P\left({E}_1\cup {E}_2{\cup}_{\cdots}\cup {E}_{10}\right)=P\left({E}_{1 or2}\cup {E}_3{\cup}_{\cdots}\cup {E}_{10}\right)\\ {}P\left(\left({E}_1\cup {E}_2\cup {E}_3\right)\cap \left({E}_4\cup {E}_5\cup {E}_6\right)\right)=P\left(\left({E}_{1 or2}\cup {E}_3\right)\cap \left({E}_4\cup {E}_5\cup {E}_6\right)\right)\end{array}} $$
(13)

First, using De Morgan’s rule and the symmetry of the standard normal distribution, the reliability index of the compound event E1or2 is obtained by:

$$ {\beta}_{1 or2}=-{\Phi}^{-1}\left[P\left({E}_1\cup {E}_2\right)\right]=-{\Phi}^{-1}\left[P\left({\overline{\mathrm{E}}}_1\cap {\overline{E}}_2\right)\right]={\Phi}^{-1}\left[P\left({\overline{E}}_1\cap {\overline{E}}_2\right)\right]={\Phi}^{-1}\left[\Phi \left({\beta}_1\right)\Phi \left({\beta}_2\right)+\underset{0}{\overset{\rho_{1,2}}{\int }}{\varphi}_2\left({\beta}_1,{\beta}_2,\rho \right) d\rho \right] $$
(14)

where β1 and β2 are the reliability indexes of E1 and E2, respectively, and ρ12 is the correlation coefficient between the standard normal random variables Z1 and Z2, which respectively represent E1 and E2. Next, the aim is to find the equivalent correlation coefficient ρ(1or2),k that would provide the same estimate on the probability of the following event Ωu = [(Z1 ≤  − β1) ∪ (Z2 ≤  − β2)] ∩ (Zk ≤  − βk) after compounding, i.e.,

$$ \underset{\Omega_{\mathrm{u}}}{\int }{\varphi}_3\left({z}_1,{z}_2,{z}_3;{\rho}_{1,2},{\rho}_{1,k},{\rho}_{2,k}\right) dz={\Phi}_2\left({\beta}_{1 or2},-{\beta}_k;{\rho}_{\left(1 or2\right),k}\right) $$
(15)

where Ω denotes the domain of a system event defined in the space of n standard normal random variables.

Using the decomposition and approximation used in conditional probabilities, Eq. (15) is approximated as

$$ 1-{\Phi}_2\left({\beta}_{\left.1\right|k},{\beta}_{\left.2\right|k};{\rho}_{\left.1,2\right|k}\right)=\Phi \left(-{\beta}_{\left.\left(1 or2\right)\right|k}\right) $$
(16)

where Ф2(.) and Ф(.) respectively denote the joint cumulative distribution function (CDF) of the bivariate standard normal distributions and the cumulative distribution function of the standard normal distribution.

$$ {\beta}_{\left.\left(1 or2\right)\right|k}=\left({\beta}_{1 or2}-{\rho}_{\left(1 or2\right),k}A\right)/\sqrt{1-{\rho}_{\left(1 or2\right),k}^2}B $$
(17)

where A = φ(−βk)/Φ(−βk) and B = A(−βk + A) in which φ(.) denotes the probability density function (PDF) of the standard normal distribution.

The bivariate CDF in Eq. (16) can be computed by performing the single-fold numerical integration as follows:

$$ {\Phi}_2\left({\beta}_{\left.1\right|k},{\beta}_{\left.2\right|k};{\rho}_{\left.1,2\right|k}\right)=\Phi \left({\beta}_{\left.1\right|k}\right)\Phi \left({\beta}_{\left.2\right|k}\right)+{\int}_0^{\rho_{\left.1,2\right|k}}{\varphi}_2\left({\beta}_{\left.1\right|k},{\beta}_{\left.2\right|k};\rho \right) d\rho $$
(18)

where at each compounding, Eqs. (16) and (17) are solved numerically for ρ(1or2),k k = 3,…n in which n is the current total number of components in the system during a sequential compounding process with the constraint − 1 ≤ ρ(1or2),k ≤ 1.

Execution process of system reliability analysis of soldier-piled excavation

In prior sections, the processes for determining the shear strength parameters of unsaturated soil and finite element modeling and generating random fields were presented. The main emphasis in the current section is on the implementation of the proposed reliability analysis for soldier-piled excavation in unsaturated soils. The execution process is schematically illustrated in Fig. 2, and can be described as follows:

  1. (1)

    Extracting the required data from boreholes and discretization of the domain

  2. (2)

    Obtaining the SWRC from GSD for different depths

  3. (3)

    Generating random fields for considered parameters (i.e., water content, cohesion, friction angle, and unit weight of the soil excavation domain)

  4. (4)

    For each element:

  1. (a)

    Calculating the suction stress by ωc and SWRC related to the element’s depth

  2. (b)

    Estimating the total stress by considering soil unit weight as gravity loads

  3. (c)

    Determining the effective stress using the estimated suction stress and total stress

  4. (d)

    Estimating the shear strength of unsaturated soil

  1. (5)

    Performing finite element elasto-plastic and strength reduction analysis to determine the FS of the failure modes (i.e., FSG, FSLD, FSSF, and FSBM)

  2. (6)

    Repeating steps (3) to (5) as many as the number of simulations in order to obtain the reliability and statistics properties for all the failure modes

  3. (7)

    Conducting system reliability analysis to obtain the overall reliability of the system, instead of several partial reliabilities indexes

Fig. 2
figure 2

Flowchart of system reliability analysis of soldier-piled excavation in an unsaturated state

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Case study

A case study of the soldier-piled excavation in the unsaturated state is presented to bring out the performance of the suggested method and validation of MATLAB code. For this purpose, deterministic analysis was followed by a reliability evaluation to take in to account the variability of soil properties and unsaturated state. In the next step, system reliability analysis was conducted to offer a single index for quantifying the overall reliability of the system, instead of several partial reliabilities indices.

The site characterization and geotechnical soil properties

The intended site is situated in Shiraz, a city in Iran, as illustrated in Fig. 3. The key aim for choosing the site was that it mainly contains fine-grained soil in which unsaturated state can cause significant suction and consequently affect the stability of retaining systems. Furthermore, the uniform geometry of the intended excavation is appropriate for plane strain conditions. To specify the soil properties of the site, two boreholes (i.e., BH.1 and BH.2) with 20 m depth are drilled which their relative locations are illustrated in Fig. 4. The soil properties are obtained through the field and laboratory tests. Soil properties of BH.1 and BH.2 are summarized in Tables 2 and 3, respectively.

Fig. 3
figure 3

Satellite overview of site location

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Fig. 4
figure 4

Relative location of the boreholes

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Table 2 Geotechnical soil properties from BH.1
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Table 3 Geotechnical soil properties from BH.2
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Modeling and verification

A finite element–based program is coded in MATLAB to estimate FS, deflection, and internal forces of soldier piles. The program is for two-dimensional plane strain analysis of elastic perfectly plastic soils with a Mohr-Coulomb failure criterion utilizing eight-node quadrilateral elements with reduced integration (four Gauss points per element) in the gravity loads generation, the stiffness matrix generation, and the stress redistribution phases of the algorithm. The soil is initially assumed to be elastic, and the model generates normal and shear stresses at all Gauss points within the mesh. These stresses are then compared with the Mohr-Coulomb failure criterion. If the stresses at a particular Gauss point lie within the Mohr-Coulomb failure envelope, then that location is assumed to remain elastic. If the stresses lie on or outside the failure envelope, then that location is assumed to be yielding. Global shear failure occurs when a sufficient number of Gauss points have yielded to allow a global failure mechanism to develop, causing the displacements in the excavation to increase dramatically.

The accuracy and computational efficiency are critical factors in FEM modeling. Although the accuracy of the FEM model improves slightly with enlarging the dimensions of it, it leads to a significant increase in computation time and cost. The problem is exacerbated in the case of stochastic analysis in which the process is repeated several hundred times. Hence, a sensitivity analysis is needed to evaluate the influence of the model dimensions on responses of soldier-piled excavation. To do this, the problem was modeled with various dimensions, as tabulated in Table 4. The results of dimension sensitivity analysis are presented in Fig. 5. Also, the required time for one realization is illustrated in Fig. 6. It can be seen that the rate of variation of FSs is significantly different before and after case no. 4. On the other hand, enlarging the dimensions of the problem greatly increases the computation time. Hence, case no. 4 was considered as a model geometry since it was more efficient than others in terms of accuracy and computation time. Also, the geometry of the model was verified by the study of Brinkgreve et al. (2014), which revealed that the minimum distance between the retaining system and excavated-side boundary is 3 to 5 times the embedment depth of the soldier piles. Furthermore, the minimum distance between the retaining system and the unexcavated-side boundary was 2 to 3 times of the soldier piles length while the minimum distance between the bottom of the soldier piles and the bottom boundary was soldier piles length. The bottom boundary was restrained from both horizontal and vertical movements, while the left- and right-side boundaries were only restrained horizontally.

Table 4 Properties of the model in dimension sensitivity analysis
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Fig. 5
figure 5

Dimension sensitivity analysis

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Fig. 6
figure 6

Required time for one simulation

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The general conditions of the studied retaining system, such as its model geometry, surcharge load, and boreholes locations are illustrated in Fig. 7 through cross-section 1-1 of Fig. 4. The properties of soldier piles and soil that are used in the deterministic analysis are given in Tables 5 and 6, respectively, while the particle density was considered 2.65 g/cm3.

Fig. 7
figure 7

Soldier-piled excavation geometry and borehole location (section 1-1)

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Table 5 Properties of soldier piles applied in the modeling
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Table 6 The soil parameters applied in the deterministic analysis
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The suction which is applied in the analysis was calculated as a continuous one-dimensional profile. The suction profile was estimated from SWRCs which were obtained for different depth using a physico-empirical model (Arya et al. 1999). For this purpose, the PSDs were extracted from GSDs which were obtained by fitting the Fredlund et al. (2000) equation to the measured data. Then, the SWRCs were estimated through the mentioned approach in the “Methodology of analysis” section by fitting the Fredlund and Xing (1994) equation to the experimental data.

The GSDs, PSDs, and the related SWRCs for the soil samples of BH.1 and BH.2 are shown in Figs. 8, 9, 10, 11, 12, and 13, respectively. Using the estimated suctions, the stability analysis was conducted deterministically and the FSG of 1.3 and 2.2 was determined with and without considering unsaturated state, respectively. As is was expected and has been stated in the literature (e.g., Li et al. 2006), considering the unsaturated state in stability analysis led to improving FSs. Although the soldier-piled excavation was stable (i.e., FS > 1) before considering suction in deterministic analysis, it was in unsafe condition based on national code, which recommended the minimum value of 1.25 for FS with respect to all failure modes.

Fig. 8
figure 8

The GSD for BH.1 in different depths

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Fig. 9
figure 9

The GSD for BH.2 in different depths

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Fig. 10
figure 10

The PSD for BH.1 in different depths

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Fig. 11
figure 11

The PSD for BH.2 in different depths

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Fig. 12
figure 12

The SWRC for BH.1 in different depths

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Fig. 13
figure 13

The SWRC for BH.2 in different depths

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The finite element mesh with boundary conditions and the shear strain of the soldier-piled excavation are illustrated in Figs. 14 and 15, respectively. Also, the simulated profiles of lateral deflection, bending moment, and shear forces of soldier piles are shown in Fig. 16.

Fig. 14
figure 14

Finite element mesh and boundary conditions

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Fig. 15
figure 15

The shear strain of soldier-piled excavation

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Fig. 16
figure 16

Simulated excavation-induced responses of the soldier piles proposed model. (a) Lateral displacement. (b) Shear force. (c) Bending moment

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For the verification, the deterministic analysis results of the coded program in the saturated and unsaturated state were compared with those from the commercial finite difference software package FLAC. For this purpose, a soldier-piled excavation with the same geometry and boundary condition was modeled by two methods. As indicated in Figs. 17 and 18 and Table 7, the results showed a good agreement with the coded program.

Fig. 17
figure 17

Simulated excavation-induced lateral displacement of the soldier piles in the unsaturated state by FLAC

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Fig. 18
figure 18

Simulated excavation-induced bending moment of the soldier piles in the saturated state by FLAC

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Table 7 Comparison of the excavation-induced responses of the soldier piles in saturated state
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Reliability analysis

To conduct a reliability analysis of soldier-piled excavation, random fields were generated for unsaturated effective parameters of soil. To identify these parameters, in addition to ωc which play key roles in controlling the behavior of unsaturated soils, a sensitivity analysis was conducted. To do this, each parameter was increased by 10% of its value, while other input parameters were kept constant. Table 8 indicates the changes in the FS due to sensitivity analysis. It can be seen that unit weight is the most effective parameter on the FS, while Poisson’s ratio and modulus of elasticity have no significant effect on it. Also, the next effective parameter is cohesion and friction angle, respectively.

Table 8 Sensitivity analysis of input parameters
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Due to the availability of sufficient soil data in a vertical direction, the ly was estimated using the ACF, and single exponential curve model (i.e., model no. 2 in Table 1) described in the “Random field theory” section. The ACFs and the fitted model for the soil parameters from BH.1 and BH.2 are illustrated in Figs. 19 and 20, respectively. The related fitting parameter values for different soil properties are tabulated in Table 9. According to the results, the ly was considered 5.0 m as twice the mean value of parameter b obtained from boreholes, which was within the range of 0.1–7.2 m estimated by various studies (Cami et al. 2020). Furthermore, lx was considered as 25 m due to the fact that the horizontal correlation length is usually much larger, and its influence is much less important compared with the vertical correlation length (Phoon and Kulhawy 1999).

Fig. 19
figure 19

Sample and model ACF of BH.1

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Fig. 20
figure 20

Sample and model ACF of BH.2

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Table 9 Parameter b estimated from fitted model
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The stochastic parameters were modeled within 3σ of difference from the mean, using truncated normal probability distribution functions with the mean and standard deviation, which are presented in Table 10. The correlation coefficient between cohesion and friction angle was selected ρc,φ = − 0.5 based on previous studies (e.g., Cherubini 2000) which proposed several ranges for ρc,φ, from − 0.24 to − 0.7. A negative correlation means one variable may take on a large value while the other assumes a small one and vice versa. The positive correlation, which implies that both variables tend to assume either large values or small values simultaneously, between friction angle and unit weight suggested within the range of 0.1 to 0.7 (Wu 2013). In this study, this value was selected as ρφ,γ = 0.7.

Table 10 The mean and standard deviation of considered stochastic parameters
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The number of realization necessary for any statistical analysis is a function of the desired precision of the analysis, the greater the number of realizations, the more precise the results are. Increasing the number of realizations beyond a certain limit, however, can be regarded as a tedious process with very little improvement in the numerical accuracy of the analysis results. Previous researchers (e.g., Huang et al. 2017; Rahman and Nguyen 2012) used statistical parameters of the system response to obtain the required number of realization, such as mean, standard deviation, and the coefficient of variation (COV), which is defined as the ratio of the standard deviation to the mean. In this study, to estimate the effective number of realization for selected geometry, a sensitivity analysis was carried out using a COV of FS with respect to different failure modes. Figure 21 depicts the variations of calculated COV of FSs against the number of simulations. The adequate number of realizations was found to be 500, which beyond it, no significant change occurred in the value of COVs. The random field results for all parameters are depicted in Figs. 22, 23, 24, and 25.

Fig. 21
figure 21

Sensitivity analysis of number of realization in unsaturated state

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Fig. 22
figure 22

The domain random field of 1th realization for cohesion (kN/m2)

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Fig. 23
figure 23

The domain random field of 1th realization for friction angle (Deg.)

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Fig. 24
figure 24

The domain random field of 1th realization for the unit weight (kN/m3)

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Fig. 25
figure 25

The domain random field of 1th realization for water content (%)

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Effects of unsaturated state on FS and reliability indices of failure modes

To evaluate the influence of the unsaturated state on FS of all failure modes, the analysis was performed with and without taking into consideration the soil suction. In the case of the saturated state, the negative pore pressure above GWT was neglected and 500 random fields were generated for random parameters except for water content (i.e., cohesion, friction angle, and unit weight). However, in an unsaturated state, random fields were generated for all random parameters due to the important role of negative pore pressure in soil shear strength. As a result of the stochastic analysis, the uncertainties of lateral displacement, shear force, and bending moment profiles of the soldier piles for the saturated and unsaturated state are presented in Figs. 26, 27, and 28, respectively. Lognormal distributions were fitted to the variation of lateral displacement at the top of the soldier piles and at the positive and negative points of the shear force and the bending moment of them. As can be seen, by taking suction into account, the mean and standard deviation of lateral displacement decreased by 26.9% and 54.7%, respectively. Also, the mean and standard deviation of the maximum shear force decreased by about 31.6% and 64.6%, respectively. In line with the results, by considering suction, the mean and standard deviation of the maximum bending moment decreased by about 31.7% and 57.8%, respectively. These significant changes can lead to completely different outcomes in the reliability analysis.

Fig. 26
figure 26

Lateral displacement uncertainties of the soldier piles due to stochastic analysis. (a) Saturated state. (b) Unsaturated state

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Fig. 27
figure 27

Shear force uncertainties of the soldier piles due to stochastic analysis. (a) Saturated state. (b) Unsaturated state

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Fig. 28
figure 28

Bending moment uncertainties of the soldier piles due to stochastic analysis. (a) Saturated state. (b) Unsaturated state

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The PDFs of FS with respect to all failure modes (i.e., global, lateral displacement, shear force and bending moment) in saturate and unsaturated states are illustrated in Figs. 29, 30, 31, and 32. The good lognormal fits were obtained for PDFs despite the use of normal probability distribution for random variables. Considering unsaturated state improves PDF of FS with respect to all failure modes by shifting from risky zone to safe zone based on Engineers, U.A.C.o. (1997). These figures indicate that neglecting the unsaturated state leads to conservative design for the retaining system with the same properties. The related statistical parameters and reliability index of all failure modes are tabulated in Table 11.

Fig. 29
figure 29

Comparison of the PDF of FSBM for saturated and unsaturated states

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Fig. 30
figure 30

Comparison of the PDF of FSSF for saturated and unsaturated states

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Fig. 31
figure 31

Comparison of the PDF of FSLD for saturated and unsaturated states

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Fig. 32
figure 32

Comparison of the PDF of FSG for saturated and unsaturated states

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Table 11 Statistical parameters of FS for various failure modes in a saturated and unsaturated states
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The reliability index (β) is an alternative measure of safety, or reliability, which is uniquely related to the Pf. The value of β indicates the number of standard deviations between failure and the most likely value for safety factor which may be defined for lognormal distributions as follows:

$$ \beta =\frac{\ln \left[\frac{\mu }{\sqrt{1+{COV}^2}}\right]}{\sqrt{\ln \left(1+{COV}^2\right)}} $$
(19)

Due to the symmetry of the normal distribution, Pf can be expressed as follows:

$$ {p}_f=1-\Phi \left(-\beta \right) $$
(20)

in which Φ is the cumulative standard lognormal distribution function.

Based on the presented COV of the failure modes in Table 11, it can be seen that the uncertainty of the soil parameters has the most significant effect on the global safety factor of the excavation, while shear force failure mode is less affected than others.

To compare the failure probability of saturated and unsaturated conditions, the CDFs of FS with respect to all failure modes (i.e., global, lateral displacement, shear force, and bending moment) are plotted in Figs. 33, 34, 35, and 36. These curves can also provide a frame for the direct implementation of probabilistic and reliability analyses in the structural design process. The procedure of obtaining the required data for the purpose of structural design (i.e., lateral displacement, shear force, and bending moment) can be summarized as follows. First, determine the target Pf to achieve the desired performance level which can be found in the US Army Corps of Engineers (Engineers UACo 1997) and are tabulated in Table 12. Next, obtain the FS that takes on a selected value of Pf using CDF of each failure mode. Then, calculate the lateral displacement, shear force, and bending moment for the purpose of structural design using the following equations:

$$ {\delta}_d=\frac{R_{LD}}{FS_L} $$
(21)

where δd, Sd, and Md are design lateral displacement, shear force, and bending moment, respectively. FSL, FSS, and FSM are respectively FS with respect to lateral displacement, shear force, and bending moment failure obtained from related CDF.

Fig. 33
figure 33

Comparison of the CDF of FSBM for saturated and unsaturated states

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Fig. 34
figure 34

Comparison of the CDF of FSSF for saturated and unsaturated states

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Fig. 35
figure 35

Comparison of the CDF of FSLD for saturated and unsaturated states

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Fig. 36
figure 36

Comparison of the CDF of FSG for saturated and unsaturated states

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Table 12 Target reliability indices (Engineers UACo 1997)
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System reliability analysis

The key advantage of the system modeling approach is that it provides a single index for quantifying the overall reliability of the system, instead of several partial reliability indices. Several approaches have been presented for combining the partial components of a system. In this paper, SCM that is one of the most popular ones was utilized. Conducting the system reliability analysis through SCM involves the calculation of reliability indices and the correlation matrix of all failure modes.

$$ {S}_d=\frac{R_s}{FS_s} $$
(22)
$$ {M}_d=\frac{R_M}{FS_M} $$
(23)
$$ {\rho}_{\mathrm{sat}}=\left[\begin{array}{ccc}{\mathrm{FS}}_{\mathrm{BM}}& {\mathrm{FS}}_{\mathrm{SF}}& {\mathrm{FS}}_{\mathrm{LD}}\kern0.5em {\mathrm{FS}}_{\mathrm{G}}\kern0.5em \\ {}1& 0.867& \begin{array}{ccc}0.731& 0.799& {\mathrm{FS}}_{\mathrm{BM}}\end{array}\\ {}\begin{array}{c}\\ {}\\ {}\operatorname{sym}\end{array}& \begin{array}{c}1\\ {}\\ {}\end{array}& \begin{array}{c}\begin{array}{ccc}0.590& 0.849& {\mathrm{FS}}_{\mathrm{SF}}\end{array}\\ {}\begin{array}{ccc}1& 0.464& {\mathrm{FS}}_{\mathrm{LD}}\end{array}\\ {}\begin{array}{ccc}& 1& {\mathrm{FS}}_{\mathrm{G}}\end{array}\end{array}\end{array}\right] $$
(24)
$$ {\rho}_{\mathrm{unsat}}=\left[\begin{array}{ccc}{\mathrm{FS}}_{\mathrm{BM}}& {\mathrm{FS}}_{\mathrm{SF}}& {\mathrm{FS}}_{\mathrm{LD}}\kern0.5em {\mathrm{FS}}_{\mathrm{G}}\kern0.5em \\ {}1& 0.767& \begin{array}{ccc}0.709& 0.713& {\mathrm{FS}}_{\mathrm{BM}}\end{array}\\ {}\begin{array}{c}\\ {}\\ {}\operatorname{sym}\end{array}& \begin{array}{c}1\\ {}\\ {}\end{array}& \begin{array}{c}\begin{array}{ccc}0.684& 0.832& {\mathrm{FS}}_{\mathrm{SF}}\end{array}\\ {}\begin{array}{ccc}1& 0.425& {\mathrm{FS}}_{\mathrm{LD}}\end{array}\\ {}\begin{array}{ccc}& 1& {\mathrm{FS}}_{\mathrm{G}}\end{array}\end{array}\end{array}\right] $$
(25)

To estimate the correlation matrix, which indicates how the failure modes depend on each other, the correlation between each failure mode with other failure modes must be determined. For this purpose, the correlation between FSs of two failure modes was used to estimate the correlation coefficient (Johari et al. 2020). This procedure was utilized for all components and the results were collected to the correlation matrix, Eqs. (24) and (25) for the saturated and unsaturated state, respectively. Two typical samples for determining the correlation between different failure modes in saturated and unsaturated states are shown in Figs. 37 and 38, respectively. Moreover, the procedure of determining the overall reliability of the soldier-piled excavation system is illustrated in Figs. 39 and 40, for the saturated and unsaturated state, respectively. It can be seen that by taking the unsaturated state into account, the overall reliability index improved from 0.33 to 2.95, or in other words, the performance level shifted from hazardous to above (Engineers UACo 1997).

Fig. 37
figure 37

Determination of correlation between FSSF and FSLD in the saturated state

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Fig. 38
figure 38

Determination of correlation between FSBM and FSG in the unsaturated state

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Fig. 39
figure 39

The procedure for determining the soldier-piled excavation system reliability index in the saturated state

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Fig. 40
figure 40

The procedure for determining the soldier-piled excavation system reliability index in the unsaturated state

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Conclusions

Most of retaining systems problems such as soldier-pilled excavation generally consist of multiple failure modes and are located in unsaturated areas. Furthermore, complex geological deposition, post-deposition processes, and limited site investigation data can result in the spatial variability of soil properties.

However, addressing these issues simultaneously has not been paid enough attention to, and only some aspect of the object is investigated in the literature. In the current study, a system reliability analysis of a soldier-piled unsaturated excavation was examined. For this purpose, a real site with two boreholes of 20 m depth was considered and the soil parameters were estimated via field and laboratory tests. To simulate the unsaturated soil condition, the SWRC was determined for different depths using a physico-empirical method. Knowing the ωc and SWRC for each depth, the suction was obtained and implemented in analysis. The analysis was carried out deterministically using the elasto-plastic finite element–based program coded in MATLAB. Then, the stochastic analysis was performed using RFEM by the implementation of a random field with normal distribution to take into account the inherent uncertainty associated with soil parameters (i.e., cohesion, friction angle, and unit weight). In this way, in the unsaturated state, ωc was considered as a stochastic parameter and consequently, suction for each element was calculated using related SWRC. The reliability indices of all individual failure modes (i.e., global, lateral displacement, shear force, and bending moment) were extracted. Finally, the soldier-piled excavation was modeled as a four-component series system (i.e., global, lateral displacement, shear force, and bending moment) and the SCM was used as an efficient numerical procedure to obtain system reliability index from the reliability indices of individual failure modes. The most important observations can be summarized as follows.

From PDF of FS with respect to all failure modes, it was illustrated that considering the unsaturated state will increase the mean value of FS with respect to lateral displacement, shear force, bending moment, and global stability by 36.2, 45.3, 44.1, and 43.4%, respectively. Also, it was found that implementing suction into model decreases the standard deviations of FS with respect to lateral displacement, shear force, bending moment, and global stability by 35.1, 29.1, 12.5, and 29.3%, respectively, which can be counted as a goal of reliability analysis.

Comparing the COV of failure modes revealed that the impact of the soil properties’ uncertainty on global stability is high in comparison to others, while shear force failure mode is less affected than others.

The results highlighted the great significance of the suction in improving reliability indices of failure modes, which changes the stability condition performance level.

The correlation matrix, which indicates how the failure modes depend on each other, showed that the failure mode with respect to shear force and bending moment are strongly correlated with each other in both saturated and unsaturated state.

Results of system reliability analysis indicated that neglecting system reliability analysis lead to overestimated reliability indices. Besides, the results revealed that global failure is the primary mechanism compared to other failure modes, and therefore it determines the probability of the overall failure to a large extent.

However, the proposed method evaluates the stochastic behavior of soldier-piled excavation in an unsaturated state, but rainfall effects such as variation of pore water pressure, soil stiffness, and hydraulic parameters were not considered. Hence, further researches are required to study the application of the proposed method in soldier-piled excavation subject to rainfall effects.

CRediT authorship contribution statement

Ali Johari: Conceptualization, methodology, writing—review and editing, supervision, formal analysis, validation.

Alireza Kalantari: Software, methodology, writing—original draft, investigation, validation.