Reliability-based settlement analysis of embankments over soft soils reinforced with T-shaped deep cement mixing piles

Abstract

This paper presents a reliability-based settlement analysis of T-shaped deep cement mixing (TDM) pile-supported embankments over soft soils. The uncertainties of the mechanical properties of the in-situ soil, pile, and embankment, and the effect of the pile shape are considered simultaneously. The analyses are performed using Monte Carlo Simulations in combination with an adaptive Kriging (using adaptive sampling algorithm). Individual and system failure probabilities, in terms of the differential and maximum settlements (serviceability limit state (SLS) requirements), are considered. The reliability results for the embankments supported by TDM piles, with various shapes, are compared and discussed together with the results for conventional deep cement mixing pile-supported embankments with equivalent pile volumes. The influences of the inherent variabilities in the material properties (mean and coefficient of variation values) on the reliability of the piled embankments, are also investigated. This study shows that large TDM piles, particularly those with a shape factor of greater than 3, can enhance the reliability of the embankment in terms of SLS requirements, and even avoid unacceptable reliability levels caused by variability in the material properties.

Graphical abstract

Keywords

T-shaped deep cement mixing piles / piled embankments / settlement / reliability analysis / soil uncertainties

1 Introduction

1.1 Soil stabilization with T-shaped deep cement mixing piles

Soil stabilization with a cementitious mixture (referred to as deep cement mixing, DCM pile) is frequently used for soft soil treatment before constructing transport infrastructures, particularly for roadway embankments [1–6]. Generally, DCM piles are used as a reinforcing volume with a constant circular diameter for an entire length of soft soil layers. They accomplish to convey the load by way of the structures (e.g., embankments) to stronger soil layers at deeper levels. Previous studies on embankment construction [1–3,5,7–9] showed that DCM piles are successful to reduce the settlement of soft soil foundations. However, due to the difference in stiffness between piles and unimproved surrounding soils, unfavorable differential settlements between piles and adjacent soil were experienced (e.g., Refs. [10,11]). This differential settlement is classified as an important behavior of piled embankment over soft soil that can cause a high risk to the transport infrastructures during the operational period [12–14]). Several alternatives have been introduced to address this problem, such as geosynthetic reinforcement (e.g., Refs. [1,2,4,7,13,15]) and load transfer platforms (e.g., Refs. [16,17]). It was found that the use of these methods can substantially reduce the total and differential settlements of soft foundation soil under embankment loading.

In the last twenty years, the T-shaped deep cement mixing (TDM) piles is an innovative technique that has been used to support embankments on soft soils [18,19]. Unlike DCM piles whose shapes are slender cylinders with a constant diameter throughout their length, TDM piles have a small diameter at their lower part (pile body) and a larger pile diameter at the upper part (pile head). The double-mixing technique (foldable mixing blades) is employed to construct TDM piles to reduce the installation time as compared to the conventional device [18,20]. In addition, owing to the larger area improvement at shallow depth provided by the TDM pile, the TDM piles can be arranged at larger pile spacings than the DCM piles for obtaining similar embankment performance [18]. Hence, the number of piles and cement volume required in the case of TDM pile-supported embankment, are lower than those for the case of DCM pile-supported embankment [18]. Therefore, lower construction cost is experienced in the case of TDM pile-supported embankments. Furthermore, employing a TDM pile for supporting embankment eliminates the requirement of geosynthetic reinforcement (load transfer platform) as placed above the top of DCM pile–embankment systems. Consequently, the settlement at the base of the TDM pile-supported embankment without any geosynthetic reinforcement, could be less than that of the DCM pile-supported embankment [18]. Importantly, the use of TDM piles for supporting embankment could provide a lower construction budget than the other methods (minus the augmentative cost of geosynthetic reinforcements). As extensively reviewed and as proven by several previous studies [18,19,21–28], TDM piles have been used to enhance the load-bearing capacity and reduce ground settlement, lateral displacement, especially reduce the differential settlement between the piles and surrounding soil, and reduce construction cost. However, for embankments in service, excessive settlement (total and differential settlements) is still the most undesirable characteristic, which can lead to unexpected loss of service and damage to the embankment. As found in practice [3,18,21,29], the inherent natural soil variability, and the complex mixing process of soil–cement mixing (SCM) for piles (DCM and TDM piles), there remains high uncertainty in the natural soil properties, and in the pile mechanical properties. The strength and modulus values of cored soil-cement specimens sampled from the same project site can vary significantly (e.g., Refs. [18,21]). The performance of structures placed on an improved ground can thus be unsafe, or overly conservative, if only a deterministic analysis is considered. Therefore, reliability assessments of SCM pile-supported embankments, are of significant importance and should be performed.

1.2 Previous studies of soil–cement mixing materials on reliability framework

Studies considering the spatial variability of the mechanical properties of SCM materials have been reported in the literature (e.g., Refs. [30–35]). However, most of these previous studies have investigated only the effect of the variability of mechanical properties, mainly the unconfined compressive strength ( q_{\text{u}}), and the modulus (E), in tested samples or piles observed in the field. These studies have shown that the properties of SCM materials have large variations. Few studies have investigated the effect of the variability of SCM materials on the stability of field embankments. Al-Naqshabandy and Larsson [36] highlighted the influence of spatial variability in the soft soil and DCM piles, on the safety factor of the embankment. Navin and Filz [37] showed that the safety factor of DCM pile-supported embankments obtained through a deterministic analysis is not sufficient for the design of these systems. A reliability analysis is thus imperative at the design stage. Relatively few investigations, on reliability-based deformation analysis of embankments supported by DCM piles, have been reported. Only the most recent studies by Wijerathna and Liyanapathirana [38,39] included, a reliability assessment of the lateral movement combined with the settlement, using different reliability methods. They demonstrated that reliability assessments can help provide adequate safety levels in terms of deformation performance. Moreover, the random variable method (RVM) is suitable for analyzing the reliability of piled embankment systems at the preliminary design stage, as demonstrated in the study of Guo et al. [40]. However, only the influence of the uncertainty, in the DCM pile properties, was investigated. Moreover, the differential settlement, which is of great concern for road or railway embankments, was not investigated in these studies.

For TDM piles, which represent a relatively new technique, no attempt has been made to investigate the deformation of TDM piled embankments in a reliability framework, even though this TDM method is effective in reducing excessive differential settlement as compared to DCM piles [23,25–28]. Furthermore, based upon volume control (representing the material cost), the deformation and performance of the embankments also depend on the TDM pile shape (i.e., the sizes of the cap and body) [24–27]. Thus, it is of interest to consider the pile shape during reliability-based deformation analyses to highlight the advantages of using TDM piles.

1.3 Reliability approaches for geotechnical structures

Considering the reliability analysis in the field of geotechnical engineering in which the main task is to estimate the failure probability ( P_{\text{f}}) of a structure, the sample-based method, Monte Carlo Simulation (MCS) is widely used. However, this method requires many evaluations, which leads to a very high computational effort and is time-consuming. In particular, the cases with a complicated deterministic model or a small P_{\text{f}} value ( ) [41], if a high accuracy (i.e., a small coefficient of variation, coefficient of variation (COV) of P_{\text{f}}) is required [42], then significant effort and time are to be consumed. Alternatively, various approximation techniques such as the First-Order Second Moment (FOSM) Method, First-Order Reliability Method (FORM), and Second-Order Reliability Method (SORM), are considered for providing an estimation of P_{\text{f}} with a smaller cost. These techniques are commonly employed when the limit state function in the reliability analysis is quite uncomplicated and can be determined explicitly. Nevertheless, they cannot provide a highly accurate P_{\text{f}} prediction for the case of complex and nonlinear limit state functions [43]. To overcome this issue, these methods are further developed in combination with the Response Surface Method (RSM) [44]. Unfortunately, the combination of approximation techniques as described above, and RSM, are not recommended when the problem has numerous random variables [45]. To increase the efficiency of MCS, two advanced sampling methods, namely, Importance Sampling, IS [46] and Subset Simulation, SS [47], were developed to reduce the variance of the MCS estimator. These methods required a small number of deterministic calculations. The reliability analysis with IS method is not suitable for random fields in some cases, while the SS method can be used with both random fields and random variables [48]. However, these methods still require substantial computational time for estimating P_{\text{f}} using computationally expensive mechanical models [42]. Recently, several meta-modeling techniques have been developed for the reliability analysis of engineering problems, such as the Artificial Neural Networks (e.g., Ref. [49]), the Kriging (e.g., Refs. [42,50–53]), and the Support Vector Machine (e.g., Ref. [54]). Among these methods, the Kriging meta-modeling technique, enhanced by an adaptive sampling algorithm, in combination with MCS, namely the Adaptive Kriging Monte Carlo Simulation (AK-MCS) proposed by Echard et al. [50], was foremost recommended [42,51,53]. The AK-MCS method can efficiently estimate P_{\text{f}} by constructing a meta-model based on a relatively small number of deterministic simulations [41]. The AK-MCS method demonstrates high efficiency and can accurately provide P_{\text{f}} with a smaller number of iterations of the computational model than the MCS method [55]. Recently, this method has been employed with good performance for reliability analyses of strip footings [42,51], offshore monopile foundations [53], tunnels [55], and earth dams [41]. Therefore, AK-MCS is the reliability method selected to investigate the P_{\text{f}} of the pile–embankment system on soft soils presented in this study.

1.4 Objective of this study

The objective of this work is to apply the reliability framework into the piled embankment systems supported by the TDM piles. This is a relatively new kind of DCM pile for a comprehensive investigation, on serviceability behavior, and especially for vertical displacement. The study employs the finite element analysis, in the framework of two-dimensional axisymmetric numerical modeling, to investigate the effect of material uncertainties on the behavior of differential, and maximum settlements, which is important to the serviceability of piled embankment systems. Reliability-based analysis, based on RVM conducted in this study, focuses on the long-term behavior of settlement that corresponds to a consolidation degree of at least 90%, and simultaneously considers the uncertainties in the mechanical properties of the soil, embankment fill, and the pile, as well as the effect of the pile shape. The uncertainties of strength parameters for soil, and pile, are correlated to the deformation parameters through empirical equations. The influences of the variabilities of these materials on the differential, and maximum settlement, of TDM piled embankment systems, are discussed and highlighted. Reflecting the influence of the TDM pile shape, various cap sizes are considered under an equivalent pile volume. Both individual, and system failure modes, are considered. A direct coupling analysis between the mechanical model (finite element modeling), and a reliability algorithm (MATLAB software), is considered to avoid manual operations during the probabilistic analysis. A meta-model is used to perform the MCS. This is constructed using the Kriging theory combined with an adaptive experimental design algorithm. The performance of TDM piled embankments is evaluated based on the P_{\text{f}}, and reliability analysis of DCM piled embankments, is also performed for comparison. The effects of the COV and mean ( \mu) values on the system are investigated. Note that the failure probabilities in terms of lateral responses of piled embankment systems, such as lateral movements, and stability embankments, are not taken into consideration.

2 Incorporated assumptions and reliability analysis method

2.1 Method for accounting soil uncertainty

In reliability analysis, the methods used to account for the uncertainty in the soil properties can be broadly classified into two types: the RVM, and the random field method (RFM). It has been established that the RFM can more effectively represent the uncertainty of soil properties than the RVM (e.g., [56]), particularly when the spatial variability must be considered. However, a high computational effort is required, and the discretization effect for different RFM generations, should be investigated. For a preliminary design stage, considering a reliability assessment of DCM piled embankments, the RVM is sufficient [39], and no spatial variability is considered. The advantages of the RVM are that it is simpler than the RFM and allows results to be obtained more rapidly [39,41,56,57]. For this study and for simplicity, the uncertainty of parameters of the soft soil, SCM material, and embankment fill, are all considered as random variables in the TDM and DCM piled embankment system.

2.2 Random variables

The deformation parameters of soils and SCM materials can be correlated with their strength and their uncertainties are usually represented by the strength parameter. For soils, the soil strength ( s_{\text{u, soil}}) becomes the primary variable using these correlations. The details of these correlations are presented in Subsection 3.3. In the same fashion, the value of the pile modulus ( E^{\prime }), was determined to be proportional to the value of the unconfined compressive strength of the pile ( q_{\text{u, pile}}) as E^{\prime }=100q_{\text{u, pile}}, based on the empirical correlation reported in the literature (e.g., Refs. [1,4,5,24,29,39,58,59]). According to several previous studies, c_{\text{u}} of the pile can be derived using c_{\text{u}}=0.5q_{\text{u, pile}} (e.g., Refs. [3,5,39]). For the embankment fill, the stiffness is relatively high owing to field compaction and small contribution of the deformation of the fill layer to settlement of the embankment in the service stage. The embankment fill is present to apply the load on the improved soil. The embankment fill unit weight ( {\gamma }_{\text{emb}}) parameter, is thus chosen for analysis.

Based on previous studies [35,36,38,39,60], these three parameters, s_{\text{u, soil}}, q_{\text{u, pile}}, and {\gamma }_{\text{emb}}, are frequently used in reliability analyses of DCM piled embankments. These three properties are assumed to be uncorrelated and are considered simultaneously in the reliability analyses. All of the random variables are assumed to follow a log-normal probability law. This distribution type allows to provide some mathematical conveniences (e.g., simple conversion to the normal distribution, and two-parameter probability density function) and avoid generating negative values of the random variables for physical material properties [36,38,39]. It also offers good data-fit with the measurements of some soil properties as indicated in several previous studies [41,61]. The log-normal distributions are thus frequently adopted in the field of geotechnical reliability analyses, particularly for the cases with limited information. Several other distribution types can also provide a good representative for the soil variability, and can allow generating the random values within a positive range such as truncated normal, and gamma distributions. However, the log-normal distribution type is still a common alternative solution used in reliability analyses of geotechnical works for a preliminary prediction, or illustrative research [40,41,55]. Hence, log-normal distributions are employed to represent the variability of soil properties considered in this study for the sake of simplicity. The details of the three random variables used in the current study are listed in Tab.1.

2.3 Performance function for the reliability analysis

A performance function, G\left(x\right), is used to define a criterion for assessing an unexpected performance of the piled embankments considered in this study. This function is used to determine the limit state surface separating the failure and safety domains, where x represents a vector of the input random variables. The limit state surface is usually defined mathematically as G\left(x\right)=0, where G\left(x\right)⩽0 represents the failure domain, and G\left(x\right)>0 represents the safety domain. Regarding the settlement reliability of piled embankments (i.e., the serviceability limit state (SLS) requirements), two performance functions have been adopted in the current study. The first limit state function, corresponding to the differential settlement, is defined as follows:

(1)G_1\left(x\right)=\mathrm{\Delta }{s}^{\text{a}}-\mathrm{\Delta }s.

The second limit state function, related to the maximum settlement, can be expressed as follows:

(2)G_2\left(x\right)=s_{max}^{\text{a}}-s_{max},

where \mathrm{\Delta }{s}^{\text{a}} and s_{max}^{\text{a}} are the allowable differential, and maximum settlements, respectively, which are further detailed in Section 4. The differential settlement is defined as the difference in settlements between pile head (point C, see Fig.1) and surrounding soil (point D, see Fig.1) while the maximum settlement is defined as the average settlement realized on the top slab (line AB, see Fig.1).

The failure probability ( P_{\text{f}}), for each failure mode considered in this study, is expressed by:

(3)P_{{\text{f}}_i}=P\left[G_i\left(x\right)⩽0\right].

The two limit state functions described in Eqs. (1) and (2) represent two failure modes (i.e., differential settlement failure mode, P_{\text{f}}^{\text{diff}} and maximum settlement failure mode, P_{\text{f}}^{max}) of piled embankment for this study, by taking into account SLS requirements. In order to evaluate the P_{\text{f}} of the series system which is considered if it fails by any of the two possible modes as described above, the system probability of failure ( P_{\text{f}}^{\text{sys}}) as expressed in Eq. (4) [57], is then considered.

(4)P_{\text{f}}^{\text{sys}}=P\left[\underset{i=1}{\overset{n_{\text{LS}}}{\cup }}\left(G_i\left(x\right)⩽0\right)\right],

where n_{\text{LS}} is the number of the limit states ( n_{\text{LS}} = 2 for this current study according to the Eqs. (1) and (2)).

Utilizing MCS with N_{\text{MCS}} runs of the computational model, the failure probability, P_{\text{f}}, can be calculated as follows:

(5)P_{\text{f}}=\frac{1}{N_{\text{MCS}}}\times \sum _{i=1}^{N_{\text{MCS}}}{I}_{\text{MCS}},

where I_{\text{MCS}} is an indicator of failure; I_{\text{MCS}} is equal to 1 if the system fails ( ) and I_{\text{MCS}} = 0 otherwise. N_{\text{MCS}} is the total number of MCS samples; N_{\text{MCS}} should be large enough to obtain an accurate P_{\text{f}} with a small value of the coefficient of variation for P_{\text{f}} ( COV-P_{\text{f}}). COV-P_{\text{f}} can be estimated as follows:

(6)COV-P_{\text{f}}=\sqrt{\frac{1-P_{\text{f}}}{N_{\text{MCS}}\times P_{\text{f}}}}\times 100\mathrm{\%}.

2.4 Adaptive Kriging Monte Carlo Simulation (AK-MCS)

To estimate P_{\text{f}}, AK-MCS is used for the reliability analysis. This method is an active learning reliability method comprised of the combination of a Kriging meta-model and MCS, as proposed by Echard et al. [50]. The AK-MCS method is based on the Kriging theory, which guarantees the construction of a meta-model with a high accuracy in the vicinity of the limit state surface. Using this method, it is possible to estimate the failure probability, P_{\text{f}}, by generating a small number of realizations of the meta-model (i.e., analytical function) instead of the large number required for computational models (i.e., finite element models). In this method, a small number of samples (designs of experiments, DoE) are used to construct the Kriging meta-model in the initial stage (e.g., a dozen samples have been used [50]). This meta-model is then updated by adding a new sample in each iteration following the conditions of a powerful learning function. This procedure is repeated until an imposed stopping condition is achieved. To choose the next candidate for the new sample with the highest probability, the U-function expressed in Eq. (7) is used as the powerful learning function:

(7)U\left(x\right)=\frac{\left|{\mu }_{\text{G}}\left(x\right)\right|}{{\sigma }_{\text{G}}\left(x\right)},

where {\mu }_{\text{G}}\left(x\right) and {\sigma }_{\text{G}}\left(x\right), are the mean and standard deviation values of the Kriging predictions, respectively. Completion of the adjustment process for the Kriging model is employed utilizing the stopping criterion proposed by Schöbi et al. [62], which only considers the uncertainty of the P_{\text{f}} estimation:

(8)\frac{P_{\text{f}}^{+}-P_{\text{f}}^{-}}{P_{\text{f}}^0}⩽{\epsilon }_{{\text{P}}_{\text{f}}}

where P_{\text{f}}^0 is the mean estimation of the Kriging model, {\mu }_{\text{G}}\left(x\right), used to identify I_{\text{MCS}} ( P_{\text{f}}^0=\mathbb{P}\left({\mu }_{\text{G}}(x)⩽0\right)); P_{\text{f}}^{+} is the upper bound failure probability ( P_{\text{f}}^{+}=\mathbb{P}({\mu }_{\text{G}}\left(x\right)+k{\sigma }_{\text{G}}\left(x\right)⩽ 0)); and P_{\text{f}}^{-} is the lower bound failure probability ( P_{\text{f}}^{-}=\mathbb{P}(\left({\mu }_{\text{G}}\left(x\right)-k{\sigma }_{\text{G}}\left(x\right))⩽0\right)). A k value of 1.96 is selected based on a previous study [62]. In this study, the AK-MCS ends when the error estimation of the failure probability ( {\epsilon }_{{\text{P}}_{\text{f}}}), is less than 5%. This criterion value seems acceptable, as suggested and detailed by Schöbi et al. [62] and Guo and Dias [41].

2.5 Computational framework

In this study, the calculation of failure probabilities is facilitated by the UQLab software package [63]. The UQLab software is applied within MATLAB and can be connected to the PLAXIS finite element software. A personal computer equipped with an Intel Core i7 running at 4.0 GHz and having 16 GB of RAM is used for the computations.

3 Deterministic analysis of piled embankments

3.1 Reference piled embankment cases

Embankments supported by DCM or TDM piles are used as reference cases to construct the reliability analyses. A geological profile composed of a homogeneous soft clay layer with a thickness of 10 m is considered. This layer is situated above a nondeformable substratum. A 1.5-m-high embankment fill (weathered clay) is placed on top of the pile–improved subsoil foundation. A 0.2-m-thick concrete slab is located on top of the embankment fill to apply the loading. The ground water table is set at the original ground surface. Two different pile types, DCM and TDM piles, are chosen in the present study under the condition of a controlled pile volume. The reference DCM pile with a diameter ( D_{\text{DCM}}) of 0.8 m and a pile length ( L_{\text{DCM}}) of 6 m is selected, based on the prototype case in a previous study [26]. Based on the past studies, a TDM pile with shape factor ( {\alpha }_{\text{s}}) of at least 3.0 is recommended to ensure the effectiveness of enlarging the pile caps, for reducing the differential settlement [25] and improving of the pile capacity [24] compared with DCM piles under the same volume. The {\alpha }_{\text{s}} parameter represents the ratio of the bearing area of the TDM pile to that of the DCM pile, over the ratio of the skin area of the TDM pile to that of the DCM pile, as given by the following [24]:

(9){\alpha }_{\text{s}}=\frac{D_{\text{TDM}}^2/\phantom{D_{\text{TDM}}^2{D}_{\text{DCM}}^2}{D}_{\text{DCM}}^2}{\left[\left(D_{\text{TDM}}-d_{\text{TDM}}\right)H+d_{\text{TDM}}{L}_{\text{TDM}}\right]/\phantom{\left[\left(D_{\text{TDM}}-d_{\text{TDM}}\right)H+d_{\text{TDM}}{L}_{\text{TDM}}\right]D_{\text{DCM}}{L}_{\text{DCM}}}{D}_{\text{DCM}}{L}_{\text{DCM}}},

where D_{\text{TDM}} is the pile head diameter of the TDM pile, d_{\text{TDM}} is the pile body diameter of the TDM pile, H is the thickness of the enlarged pile cap of the TDM pile, and L_{\text{TDM}} is the TDM pile length.

Consequently, a TDM pile with an {\alpha }_s of 3.0 is chosen for this study as an additional reference case. A d_{\text{TDM}} of 0.5 m is selected based on previous studies [18,21,24,25], and L_{\text{TDM}} is set as equal to L_{\text{DCM}}. Accordingly, D_{\text{TDM}} and H are equal to 1.31 and 1.6 m, respectively. The piles are arranged in a square grid pattern with a center-to-center spacing ( S) of 2.0 m, corresponding to an area improvement ratio ( a_{\text{r}}) of 12.6% for the DCM pile case and 33.7% for the TDM pile case. This a_{\text{r}} ratio falls within the range of 10%–50% commonly used in engineering [5,18,19]. Based on the literature review, the unconfined compressive strength of SCM piles is in the range of 200–2700 kPa [3,5,18,21,64]. The minimum SCM pile strength value of 200 kPa is selected for both cases (DCM and TDM piles). Fig.1(b) and 1(c) also depict points A–D, which are used to monitor the settlement behavior of the piled embankment systems. Points A and B are located at the concrete slab crest, whereas points C and D are positioned at the ground surface. Line AB is chosen to investigate the maximum settlement, while line CD is selected for monitoring the differential settlement between the pile and the surrounding soil.

3.2 Numerical modeling

Considering the symmetry condition, it is possible to investigate the settlement behavior of the pile–embankment system using a two-dimensional (2D) axisymmetric model (see Fig.1(b) and 1(c)) with an equivalent diameter ( D_{\text{e}}) of 1.128S, as presented in Fig.1(a)). The problem is modeled as a single pile in a network situated far from the embankment slope. This modeling approach can provide results with good accuracy compared to three-dimensional (3D) modeling while also requiring less computational time [26,65]. In this study, 2D axisymmetric numerical calculations were conducted using the PLAXIS 2D finite element modeling program [66] to analyze the piled embankment systems. Fifteen-node triangle elements were used for the mesh generation. Fig.2 shows an example of the FE mesh used for this study, on a 1.128-m-wide (equal to an equivalent radius of D_{\text{e}}/\phantom{D_{\text{e}}2}2), and 11.7-m-deep domain. The bottom boundary of the finite element (FE) mesh was fixed in all directions because the soft soil layer was placed on the rigid substratum, while the top boundary was left free. The lateral boundaries were constrained in the normal direction. Water could drain freely at the ground surface and the bottom boundary. A perfect bonding between the soft soil and SCM piles was adopted for this study, because the shear strength at the interface between the SCM piles and the surrounding soil is greater than the soil shear strength, as commonly used in previous studies (e.g., Refs. [5,8,25]). FE analyses were carried out to simulate the consolidation behavior after finishing embankment construction, using coupled mechanical and hydraulic modeling that permitted a dissipation of excess pore pressures in the saturated clays as a function of time [27,28]. A constant surcharge load of 25 kPa over the concrete slab was applied to consider the settlement behavior with an elapsed time of 1500 d, corresponding to consolidation degree of not less than 90%. The simulation details for the piled embankment system are summarized in Tab.2.

3.3 Constitutive models and model parameters

In this study, the constitutive models and model parameters for the soft soil, SCM pile, and embankment fill were adopted from a previous study by Phutthananon et al. [25]. These parameter sets were well calibrated based on the oedometer and triaxial testing results for soil samples obtained from the actual site of a DCM piled embankment. Moreover, the simulated results for the DCM piled embankment were also validated based on monitoring data, and good agreement was obtained. A more detailed description of the calibration and validation are provided in Ref. [25].

3.3.1 Soft soil

The Hardening Soil (HS) model [67] was chosen to model the soft soil behavior. According to previous studies, the HS model is effective for predicting the deformation of soft soils [24,25,29,68–71]. The set of soft soil material parameters for the HS model was adopted from Phutthananon et al. [25], as listed in Tab.3. The HS model parameters can be divided into two main groups: shear strength parameters, and deformation parameters. The shear strength parameters based on the Mohr–Coulomb shear criterion, include the effective cohesion ( c^{\prime }), effective friction angle ( {\varphi }^{\prime }), and dilatancy angle ( {\psi }^{\prime }). Five basic deformation parameters were used for the HS model: the reference secant stiffness in standard drained triaxial tests ( E_{50}^{\text{ref}}), reference tangential stiffness for a primary oedometer loading ( E_{\text{oed}}^{\text{ref}}), reference unloading/reloading stiffness ( E_{\text{ur}}^{\text{ref}}), Poisson's ratio for unloading/reloading ( {\nu }_{\text{ur}} = 0.2 was used in this study [66]), and power of the stress-level dependency of the stiffness ( m = 1 was suggested by Surarak et al. [71] for soft clay soils). A value of E_{50}^{\text{ref}} = 160s_{\text{u, soil}} was considered in this study. This relationship falls within the range of 60s_{\text{u, soil}}-330s_{\text{u, soil}} used in several previous studies [24,72,73]. E_{\text{oed}}^{\text{ref}} and E_{\text{ur}}^{\text{ref}} were estimated as E_{\text{oed}}^{\text{ref}} = E_{50}^{\text{ref}}, and E_{\text{ur}}^{\text{ref}} = 3E_{50}^{\text{ref}}, respectively, as generally used in other works [8,73–75]. Procedures similar to those recommended by Jamsawang et al. [68] were adopted to determine the input HS model parameters; the calibration results for those parameters can be found in Phutthananon et al. [25].

3.3.2 SCM pile, embankment fill, and concrete slab

The SCM pile and embankment fill were modeled using linear elastic–perfectly plastic behavior with the Mohr–Coulomb failure criterion (named the MC model). This model has been successfully used to simulate the behavior of SCM piles and the embankment fill in the literature [3–5,8,12,16,76–78]. In the present study, sets of material parameters for the SCM pile and embankment fill were adopted from the work of Phutthananon et al. [25]. Tab.4 lists the input parameters for the MC model used in this study. For the input parameters of the SCM pile, the available procedure in PLAXIS, i.e., Undrained (B), was chosen for the analyses by using the effective elastic modulus ( E^{\prime }), and undrained shear strength ( c_{\text{u}}) [5,25,66]. The concrete slab was considered to be linear elastic (LE) with a unit weight ( \gamma) of 25 kN/m³, Young’s modulus ( E) of 10 GPa, and Poisson’s ratio ( \nu) of 0.20. It should be noted that although the coefficients of permeability of SCM piles adopted in this study are relatively high, they have insignificant effect on the computed settlement results discussed herein because the consolidation analyses are performed, until the consolidation degree of not less than 90% is achieved.

4 Settlement of the reference embankment cases using deterministic analysis

In this section, the behaviors of the embankments improved by DCM and TDM piles described in Subsection 3.1 are analyzed by assuming deterministic properties for all of the materials listed in Tab.3 and Tab.4. Fig.3 presents the results for the vertical stresses acting on the pile head and surrounding soil for both reference cases. As expected for both cases, the vertical stress acting on the surrounding soil is lower than that acting on the pile head. This can be attributed to the arching effect caused by the different moduli between the pile and the soil, as described in previous publications [19,22]. Comparing the results for the DCM and TDM piles, it reveals that the induced vertical stresses on the TDM pile head are lower than those on the DCM pile head. This is caused by the larger cross-sectional area of the TDM pile head (higher value of a_{\text{r}}), as noted in previous studies [19,23,25].

The changes in the settlement distribution inside the embankment fill and the slab for both cases are depicted in Fig.4. Large settlements are found at the slab top, which then decay from the slab top to the ground surface. For both cases, settlement arches are formed above the pile head and are approximately dome-shaped. The boundaries of the settlement arches are very close to the pile edge. The arch height above the DCM pile is higher than that above the TDM pile. As can be seen, the minimum settlements occur in the vicinity of the pile heads. A comparison of the pile types shows that the piled embankment system with DCM piles provides less settlement at both the slab top and ground surface. The settlements at the top of the slab (AB line, see Fig.1) and ground level (CD line, see Fig.1) are plotted in Fig.5. Under the same loading and pile volume conditions, the settlements of the TDM piled embankment at both positions are greater than those of the DCM piled embankment. The maximum settlement ( s_{\text{max}}, the average settlement of the slab top) of the TDM piled embankment is 303.60 mm, and s_{max} = 228.92 mm for the DCM piled embankment. This trend is in good agreement with the computed results for embankments supported by DCM and TDM piles reported by Phutthananon et al. [25]. This can be explained by the fact that a greater portion of the embankment load is applied to the TDM piles owing to the larger pile head [18,19,23]. Once large settlements occur in the piles themselves, they can no longer inhibit the settlement of the surrounding soil, resulting in large soil settlements [25]. However, when considering the differential settlements ( \mathrm{\Delta }s) in these two cases, the TDM piled embankment can provide a markedly lower \mathrm{\Delta }s than the DCM piled embankment. A \mathrm{\Delta }s value of 12.95 mm is obtained for the TDM piled embankment, whereas the \mathrm{\Delta }s for the DCM piled embankment is 44.41 mm. Interestingly, the use of TDM pile-supported embankments can reduce \mathrm{\Delta }s by approximately 70% compared to the use of DCM piles. This result is in good agreement with the results of physical model tests reported by Yi et al. [19] and Phutthananon et al. [25] and numerical simulations reported by Yi et al. [23] and Phutthananon et al. [25]. Therefore, in this study, the values of \mathrm{\Delta }s = 12.95 mm and s_{max} = 303.60 mm obtained from the TDM piled embankment ( {\alpha }_{\text{s}} = 3) were defined as the allowable differential ( \mathrm{\Delta }{s}^{\text{a}} in Eq. (1)), and maximum settlements ( s_{max}^{\text{a}} in Eq. (2)), respectively, in the reliability analysis in the following section.

5 Reliability analysis results

5.1 Validation of the reliability results provided by AK-MCS

Although the AK-MCS method is robust and not overly time-consuming [42,51,53], it is necessary to evaluate the accuracy of the P_{\text{f}} estimation obtained using this method. It is well known that the MCS is a rigorous and robust approach for reliability analyses. To this end, the P_{\text{f}} estimated using direct MCS is adopted as the benchmark. Fig.6 illustrates the P_{\text{f}}^{\text{diff}} for the reference TDM piled embankment case computed using direct MCS with a various number of samples. The result obtained using AK-MCS is also included. It can be seen that the direct MCS results provide an estimated P_{\text{f}}^{\text{diff}} of 0.413 with a coefficient of variation of P_{\text{f}}, COV-P_{\text{f}}, of approximately 2.18%, after approximately 3000 realizations. The estimated P_{\text{f}}^{\text{diff}} provided by the AK-MCS method is 0.416, which is very close to the result obtained with the direct MCS. Moreover, the COV-P_{\text{f}} obtained with the AK-MCS, is very low compared with that of the direct MCS (0.37% << 2.18%). These results indicate that the constructed meta-model based on Kriging theory is efficient for estimating the settlement values of piled embankment systems, thus providing a good result accuracy of reliability analysis. Hence, by adopting the AK-MCS method in this study, the failure probability of piled embankment systems can be estimated with reasonable accuracy.

5.2 Influence of the pile shape

An important independent variable in the design of TDM piled embankments is the pile shape factor, {\alpha }_{\text{s}}, which is the significant parameter for controlling the ultimate load bearing capacity and settlement of TDM piled embankments, over soft soils [24,25]. To investigate the effect of this factor on the failure probability, while also considering the cost effectiveness, {\alpha }_{\text{s}} is varied based on the condition of a controlled pile volume of the DCM pile ( D_{\text{DCM}} = 0.8 m and L_{\text{DCM}} = 6.0 m). The TDM pile length ( L_{\text{TDM}}) and pile body diameter ( d_{\text{TDM}}) are kept constant and are 6.0 m, and 0.5 m, respectively. TDM pile head diameters are considered in the range of 1.0–1.5 m, which corresponds to thicknesses of the enlarged pile cap in the range of 1.17–3.12 m. The pile shapes adopted in this study were carefully selected considering the available dimensions used in practice [5,18,21–26]. Hence, the reliability analyses are conducted for values of {\alpha }_{\text{s}} = 1.0, 1.6, 2.2, 3.0, 3.5, and 4.0. The detailed configurations of these pile shapes are presented in Tab.5.

The results obtained with AK-MCS for the two limit state functions described above, are displayed in Fig.7. Three failure probabilities, P_{\text{f}}^{\text{diff}}, P_{\text{f}}^{max}, and P_{\text{f}}^{\text{sys}}, are plotted in this figure. Three random variables (i.e., s_{\text{u, soil}}, q_{\text{u, pile}}, and {\gamma }_{\text{emb}}) are considered simultaneously to investigate the failure probabilities by considering the mean ( \mu) values, and coefficient of variation of mean ( COV-\mu) values, as listed in Tab.1. As seen in Fig.7, P_{\text{f}}^{\text{diff}} does not change significantly when {\alpha }_{\text{s}} increases from 1.0 to 2.2 (small TDM piles with thicker pile caps). However, it is evident that P_{\text{f}}^{\text{diff}} decreases considerably with increasing {\alpha }_{\text{s}} of greater than 2.2 (larger TDM piles with thinner pile caps). Interestingly, the use of TDM piles with {\alpha }_{\text{s}} = 4.0 can decrease P_{\text{f}}^{\text{diff}} by approximately 99% relative to the DCM pile case ( {\alpha }_{\text{s}} = 1.0). This is in good agreement with the results provided by numerical analyses [23,25,26] and physical model tests [19,25] of several previous studies that demonstrated the increase in TDM pile head diameter can effectively result in a reduction of differential settlement. In contrast, an increase in {\alpha }_{\text{s}} leads to an increase in P_{\text{f}}^{max}. In the studied range, P_{\text{f}}^{max} for the case of TDM piles with {\alpha }_{\text{s}} = 4.0, ( P_{\text{f}}^{max} = 0.572) is approximately 1.26 times greater than that for {\alpha }_{\text{s}} = 1.0 (DCM pile, P_{\text{f}}^{max} = 0.455). Again, this result is attributed to the enlarged pile cap of the TDM pile, as described previously in Section 4. This tendency is in good agreement with the computational results for embankments supported by TDM and DCM piles obtained by Phutthananon et al. [25]. Inspecting the results for P_{\text{f}}^{\text{sys}} reveals that the use of TDM piles can reduce P_{\text{f}}^{\text{sys}} dramatically compared to the use of DCM piles, particularly when {\alpha }_{\text{s}} is in the range of 2.2 to 4.0. Using TDM piles with {\alpha }_{\text{s}} = 4.0 can drastically reduce the value of P_{\text{f}}^{\text{sys}} from 0.99 (for the case of DCM piles) to 0.58. Based on these results, it can be concluded that the decrease of P_{\text{f}}^{\text{sys}} is mainly governed by the variation in P_{\text{f}}^{\text{diff}}. It is also noted that the decrease in P_{\text{f}}^{\text{sys}} is not as great as that of P_{\text{f}}^{\text{diff}} owing to the inclusion of P_{\text{f}}^{max}, which has a relatively large value. However, the rate of increase in P_{\text{f}}^{max} is much smaller than the rate of decrease in P_{\text{f}}^{\text{diff}}. Consequently, TDM piles with large heads can provide very low values of P_{\text{f}}^{\text{sys}} compared to those for DCM piles. This result is due to the fact that DCM piles induce a large value of P_{\text{f}}^{\text{diff}} (up to 0.94 in this study). The results of this study present a promising alternative of using TDM piles to support embankments on soft soils with a high degree of safety. The use of this pile type can effectively avoid undesired performance of the embankment in terms of serviceability (i.e., settlement).

6 Parametric studies

Parametric analyses were performed to investigate the effect of each random variable on the computed system failure probabilities, P_{\text{f}}^{\text{sys}} of the embankments supported by TDM and DCM piles. The mean ( \mu), and coefficient of variation of mean ( COV-\mu) values, for each random variable were varied in the reliability analysis, as listed in Tab.6.

6.1 Impact of the uncertainties of soft soil properties

The parametric analyses of the SCM piled embankment in this subsection, aim to assess the impact of the variability in the \mu -s_{\text{u, soil}}, and COV-s_{\text{u, soil}} values (soft soil parameters), on the estimated reliability results. The values of \mu -s_{\text{u, soil}} for very soft soils, and soft soils are considered to be in the range of 5–10 kPa and 15–25 kPa, respectively [79,80]. The ranges of the COV-s_{\text{u, soil}} values are adopted from Phoon and Kulhawy [81].

The variation between the \mu -s_{\text{u, soil}} and the P_{\text{f}}^{\text{sys}} of the embankments supported by SCM piles for different {\alpha }_{\text{s}} values, is shown in Fig.8. For the case of small {\alpha }_{\text{s}} values, the P_{\text{f}}^{\text{sys}} value decreases slightly with increasing \mu -s_{\text{u, soil}}. For example, for {\alpha }_{\text{s}} = 1.0 (DCM pile), the value of P_{\text{f}}^{\text{sys}} declines from 1 to 0.935 with the increase in \mu -s_{\text{u, soil}} from 5 to 25 kPa. A similar trend is also observed for {\alpha }_{\text{s}} = 1.6 and 2.2 (small TDM piles with larger caps). When {\alpha }_{\text{s}} exceeds 2.2, the P_{\text{f}}^{\text{sys}} value decreases substantially with increasing \mu -s_{\text{u, soil}}. For instance, in the case of {\alpha }_{\text{s}} = 4.0 (TDM pile with a large but thin cap), the value of P_{\text{f}}^{\text{sys}} decreases from 1 to 0.074 as \mu -s_{\text{u, soil}} increases from 5 to 25 kPa. Interestingly, increasing \mu -s_{\text{u, soil}} from 10 to 25 kPa can dramatically reduce P_{\text{f}}^{\text{sys}} by approximately 92%. From these results, it can be concluded that the use of DCM and TDM piles to support embankments on very soft soils (i.e., s_{\text{u, soil}} = 5–10 kPa) does not significantly change the reliability level (i.e., P_{\text{f}}^{\text{sys}}). This result is attributed to the difference in stiffness between the soil and pile materials, as revealed by previous studies on DCM piles [7] and TDM piles [25]. For this reason, unacceptably high failure probabilities (large P_{\text{f}}^{\text{sys}}) are obtained for SCM piled embankments over very soft soils regardless of the pile shape. However, when using this improvement system on soft soils (i.e., s_{\text{u, soil}} = 15–25 kPa), it is evident that the use of TDM piles (especially for cases where {\alpha }_s exceeds 3.0) can effectively provide good reliability levels (small P_{\text{f}}^{\text{sys}}) compared to the use of DCM piles. Interestingly, the deterministic analysis [25] also reveals that the use of TDM piles with {\alpha }_{\text{s}} of greater than 3.0 can significantly reduce the differential settlement. As a result, the confidence level of embankments supported by TDM piles can be increased. This finding suggests that the use of TDM piles with large heads can significantly reduce the occurrence of undesired settlement.

Fig.9 presents P_{\text{f}}^{\text{sys}} as a function of the COV-s_{\text{u, soil}} values in association with {\alpha }_{\text{s}}. The results show that the influence of COV-s_{\text{u, soil}} on P_{\text{f}}^{\text{sys}} is almost insignificant. P_{\text{f}}^{\text{sys}} decreases slightly with increasing COV-s_{\text{u, soil}} with {\alpha }_{\text{s}} ranging from 1.0 to 2.2, i.e., the P_{\text{f}}^{\text{sys}} value at COV-s_{\text{u, soil}} = 0.44 decreases by approximately 1% compared that at COV-s_{\text{u, soil}}= 0.04. Beyond that ( {\alpha }_{\text{s}} = 3.0–3.5), the value of P_{\text{f}}^{\text{sys}} gradually increases with increasing COV-s_{\text{u, soil}}. For the case of {\alpha }_{\text{s}} = 4.0, it appears that the magnitude of P_{\text{f}}^{\text{sys}} becomes almost constant with varying COV-s_{\text{u, soil}}. Therefore, it can be concluded that COV-s_{\text{u, soil}} has a slight impact on the reliability of SCM piled embankments in terms of P_{\text{f}}^{\text{sys}}.

6.2 Impact of the uncertainties of SCM pile properties

The effect of \mu -q_{\text{u, pile}} on P_{\text{f}}^{\text{sys}} is investigated in this section. The values of \mu -q_{\text{u, pile}} range from 200–800 kPa, as is frequently used in practice [4,29,38,39,72].

The variation in the value of P_{\text{f}}^{\text{sys}} with changing \mu -q_{\text{u, pile}} is presented in Fig.10. It is evident that for all values of \mu -q_{\text{u, pile}}, P_{\text{f}}^{\text{sys}} decreases with increasing {\alpha }_{\text{s}}, particularly for the cases of TDM piles with large caps (high {\alpha }_s). The change in P_{\text{f}}^{\text{sys}} with varying \mu -q_{\text{u, pile}} for the same {\alpha }_{\text{s}} shows that \mu -q_{\text{u, pile}} has a negligible effect on the change in P_{\text{f}}^{\text{sys}} when {\alpha }_{\text{s}} is less than 2.2. In other words, the use of TDM piles with small pile caps cannot enhance the reliability or suppress the excessive settlement of the piled embankment within the range of \mu -q_{\text{u, pile}} values used in this study. When {\alpha }_{\text{s}} is equal to 3.0, P_{\text{f}}^{\text{sys}} increases with increasing \mu -q_{\text{u, pile}}. As can be seen, the value of P_{\text{f}}^{\text{sys}} can be increased from 0.863 to 0.983 with the increase in \mu -q_{\text{u, pile}} from 200 to 800 kPa. This increasing trend is similar to the results for the case of {\alpha }_{\text{s}} = 3.5. This result is not surprising as a high q_{\text{u, pile}} can restrain the large pile settlement, consequently producing a large differential settlement [25] and resulting in an increase in P_{\text{f}}^{\text{sys}}. For the data set with {\alpha }_s = 4.0, the magnitude of P_{\text{f}}^{\text{sys}} gradually decreases from 0.579 to 0.517 until \mu -q_{\text{u, pile}} reaches 400 kPa. Beyond that, P_{\text{f}}^{\text{sys}} continues to increase with \mu -q_{\text{u, pile}} up to 800 kPa ( P_{\text{f}}^{\text{sys}} = 0.690). Interestingly, the use of TDM piles with {\alpha }_s = 4.0 for \mu -q_{\text{u, pile}} = 800 kPa can reduce P_{\text{f}}^{\text{sys}} by approximately 30% compared with using DCM piles at the same \mu -q_{\text{u, pile}}. From these results, it can be concluded that both the pile strength ( q_{\text{u, pile}}) and enlarged pile cap shape ( {\alpha }_{\text{s}}) play an important role in the reliability of SCM piled embankments, especially for large {\alpha }_{\text{s}} values. To ensure the effective enhancement of the reliability level, pile shapes with {\alpha }_{\text{s}} values of greater than 3 are recommended for all of the q_{\text{u, pile}} values used in this study. Moreover, it is also possible to use TDM piles with large caps ( {\alpha }_{\text{s}} > 3) at high values of q_{\text{u}} to increase the reliability level of SCM piled embankments, or to avoid excessive settlement of the overlying structure. This solution is very promising for soft soil improvement when the embankment is required to support a super-structure (i.e., a high q_{\text{u, pile}} is needed).

Fig.11 illustrates the estimated P_{\text{f}}^{\text{sys}} with varying COV-q_{\text{u, pile}} for different values of {\alpha }_{\text{s}}. The values of COV-q_{\text{u, pile}} considered in this study are adopted from the previous study by Wijerathna and Liyanapathirana [39]. Under the same COV-q_{\text{u, pile}}, the P_{\text{f}}^{\text{sys}} values initially decrease gradually ( {\alpha }_s in the range of 1.0–2.2) and then rapidly approach a P_{\text{f}}^{\text{sys}} of less than 0.6 when {\alpha }_s reaches 4.0. For the change in P_{\text{f}}^{\text{sys}}, with identical values of {\alpha }_{\text{s}}, the magnitude of P_{\text{f}}^{\text{sys}} slightly decreases with increasing COV-q_{\text{u, pile}} for {\alpha }_{\text{s}} in the range of 1.0–3.0. The highest degree of decrease in P_{\text{f}}^{\text{sys}} is approximately equal to 4%, which occurs with the change in COV-q_{u\text{, pile}} value from 0.3 to 0.7. This result is obtained with {\alpha }_{\text{s}} = 1.6. As the {\alpha }_{\text{s}} value increases (i.e., {\alpha }_{\text{s}} = 3.5 and 4.0), the magnitude of P_{\text{f}}^{\text{sys}} exhibits the opposite trend. For instance, the magnitude of P_{\text{f}}^{\text{sys}} increases by approximately 6% with the change in COV-q_{\text{u, pile}} from 0.3 to 0.7. These results imply that the influence of COV-q_{\text{u, pile}} on the reliability of SCM piled embankments is insignificant. Moreover, it seems that the influence of COV-q_{\text{u, pile}} on the reliability level is less than that of \mu -q_{u\text{, pile}} presented above. Therefore, in the reliability analysis of SCM piled embankments in terms of the system failure modes (considering both the differential, and maximum settlements), the effect of COV-q_{\text{u, pile}} can be ignored.

6.3 Impact of the uncertainties of embankment fill properties

For a better understanding of the effect of the uncertainty of {\gamma }_{\text{emb}} on P_{\text{f}}^{\text{sys}}, the \mu -{\gamma }_{\text{emb}} and COV-{\gamma }_{\text{emb}} values are varied here in association with {\alpha }_{\text{s}}. The ranges of \mu -{\gamma }_{\text{emb}} and COV-{\gamma }_{\text{emb}} values reported by Phoon and Kulhawy [81] are applied in this study.

Fig.12 displays the variation between P_{\text{f}}^{\text{sys}} and \mu -{\gamma }_{\text{emb}} for different values of {\alpha }_{\text{s}}. The results with the same values of \mu -{\gamma }_{\text{emb}}, indicate that the value of P_{\text{f}}^{\text{sys}} remains almost constant with an increase in {\alpha }_s from 1.0 to 2.2 and then rapidly decreases with further increases in {\alpha }_{\text{s}}. For example, for \mu -{\gamma }_{\text{emb}} = 14 kPa, P_{\text{f}}^{\text{sys}} decreases from 1 to 0.398 with the increase in {\alpha }_{\text{s}} from 2.2 to 4.0. In this range, the P_{\text{f}}^{\text{sys}} value decreases by approximately 60%. This is because the use of TDM piles with large caps can significantly reduce the differential settlement [25]. Hence, lower P_{\text{f}}^{\text{sys}} values can be obtained with large {\alpha }_{\text{s}}. Investigating the variability of \mu -{\gamma }_{\text{emb}} reveals that \mu -{\gamma }_{\text{emb}} has a slight effect on P_{\text{f}}^{\text{sys}} for small {\alpha }_{\text{s}} (DCM piles and small TDM piles with thicker pile caps). However, \mu -{\gamma }_{\text{emb}} has an important impact on P_{\text{f}}^{\text{sys}} when {\alpha }_{\text{s}} is in the range of 3.0 to 4.0 (TDM piles with large but thin caps). In this range, a decrease in \mu -{\gamma }_{\text{emb}} leads to a substantial decrease in P_{\text{f}}^{\text{sys}}. For {\alpha }_{\text{s}} = 4.0, P_{\text{f}}^{\text{sys}} decreases drastically from 0.850 to 0.398 as \mu -{\gamma }_{\text{emb}} decreases from 20 to 14 kN/m³. This may be attributed to the decrease in the applied load above the pile–soil system as a result of the reduction in \mu -{\gamma }_{\text{emb}}. This phenomenon leads to reduction of the settlement [5,9]. From the results in this study, it is promising to recommend that TDM piles with large heads (high {\alpha }_{\text{s}}) can be used to provide a high reliability of piled embankments compared to that of DCM piles with the same pile volume.

Fig.13 presents the distribution of P_{\text{f}}^{\text{sys}} for three different values of COV-{\gamma }_{\text{emb}} and six different pile shapes. The results show that the value of P_{\text{f}}^{\text{sys}} decreases with increasing {\alpha }_{\text{s}}. It is clear that P_{\text{f}}^{\text{sys}} for {\alpha }_{\text{s}} = 4.0 is approximately 40% smaller than that for {\alpha }_{\text{s}} = 1.0. This reduction is observed for all of the COV-{\gamma }_{\text{emb}} values considered in this study. Investigating the effect of COV-{\gamma }_{\text{emb}} reveals that the change in COV-{\gamma }_{\text{emb}} seems to have an insignificant influence on P_{\text{f}}^{\text{sys}} when {\alpha }_{\text{s}} is in the range of 1.0 to 2.2. When {\alpha }_{\text{s}} is greater than 2.2, P_{\text{f}}^{\text{sys}} decreases slightly with increasing COV-{\gamma }_{\text{emb}}. For example, for the case of {\alpha }_{\text{s}} = 4.0, P_{\text{f}}^{\text{sys}} decreases from 0.586 to 0.553 (a reduction of approximately 6%) when COV-{\gamma }_{\text{emb}} increases from 0.03 to 0.20. Therefore, it seems that the reliability levels of the SCM piled embankments in term of P_{\text{f}}^{\text{sys}} are not significantly affected by COV-{\gamma }_{\text{emb}}. Moreover, within the ranges of \mu -{\gamma }_{\text{emb}} and COV-{\gamma }_{\text{emb}}values considered in this study, \mu -{\gamma }_{\text{emb}} has an impact on the reliability level, while COV-{\gamma }_{\text{emb}} does not. This implies that the reliability-based settlement analysis of the SCM piled embankment is more sensitive to the uncertainty of \mu -{\gamma }_{\text{emb}}. Hence, appropriate values of \mu -{\gamma }_{\text{emb}} should be carefully chosen for reliability-based settlement analyses in the design of SCM piled embankments.

7 Discussion

The RVM is employed to simulate the uncertainty of material properties (soft soil, SCM pile, and embankment fill) for SCM piled embankments on soft soils. In the event that a direct MCS method is adopted to conduct the reliability-based settlement analysis for this problem (reference TDM pile case, P_{\text{f}}^{\text{diff}} = 0.420), at least 750 model calls are required for a target error, COV-P_{\text{f}}, of less than 5% ( COV-P_{\text{f}} = 4.29%), requiring approximately 9.6 h to complete. By applying the framework proposed in this study to the aforementioned TDM piled embankment problem, the AK-MCS method requires a much smaller number of calls of the deterministic model used to construct the meta-model. This is highly significant for reducing the computational time of the reliability analysis. The P_{\text{f}}^{\text{diff}} value obtained with the AK-MCS method is 0.416, corresponding to a COV-P_{\text{f}} = 0.37% (much smaller than that obtained with the direct MCS method); moreover, a calculation time of only 0.4 hours is required. This indicates that using the AK-MCS method can significantly reduce the computational time by at least 9 hours compared to the direct MCS method, while providing reliability results with a higher confidence level. A similar tendency is also observed for the reliability analyses of geotechnical structures (e.g., earth dam [41], strip footing [51], and tunnel face stability [55]) in which the AK-MCS can provide a very close value of P_{\text{f}} with the direct MCS but consume a reasonably small computational time. Therefore, the AK-MCS method used in this study for considering the uncertainty of SCM piled embankment systems could serve as a suitable alternative, as it is very efficient.

According to the results presented in Subsection 5.2, the failure probability related to the differential settlement ( P_{\text{f}}^{\text{diff}}) is mainly controlled by {\alpha }_{\text{s}}. The piles with large {\alpha }_{\text{s}} can provide very small values of P_{\text{f}}^{\text{diff}}, particularly when {\alpha }_{\text{s}} is greater than 3.0 (TDM piles with large but thin pile caps). This result is attributed to the larger pile head diameter, which can increase the load transfer to the piles through arching effects [5,18]. Hence, a large pile settlement and a small soil settlement are obtained, resulting in a decrease in the differential settlement [25,28]. Meanwhile, TDM piles with large caps can induce large failure probabilities for the maximum settlement ( P_{\text{f}}^{max}). Again, this result is mainly caused by the higher area improvement at the shallow depth of the TDM piled embankment case. The embankment loads can effectively be transferred to the piles [18,25]. Hence, the settlements at the embankment top in the case of TDM piled embankment are significantly larger than DCM piled embankment case [25,26]. However, the increase in P_{\text{f}}^{max} with {\alpha }_{\text{s}} is relatively small when compared to the decrease in P_{\text{f}}^{\text{diff}}. For an accurate design, serviceability requirements in terms of the differential and maximum settlement criteria, must be taken into account to prevent undesired settlement characteristics. Hence, the system failure probability ( P_{\text{f}}^{\text{sys}}) introduced in this study is a good indicator for capturing the failure modes in terms of both the differential and maximum settlements. The results presented in Fig.7 indicate that compared to DCM piles with the same volume, the use of TDM piles can effectively enhance the reliability level of SCM piled embankments when the uncertainty of materials is taken into account. This finding is very useful for geotechnical engineering to reduce undesired performance in terms of the settlement of the piled embankment system.

Based on the parametric studies presented in Section 6, it is concluded that the mean values of all of the random variables ( \mu -s_{\text{u, soil}}, \mu -q_{\text{u, pile}}, and \mu -{\gamma }_{\text{emb}}) have important effects on P_{\text{f}}^{\text{sys}} when {\alpha }_{\text{s}} is greater than 2.2 (TDM piles with large caps). Particularly, for {\alpha }_{\text{s}} of greater than 3, the TDM piles can drastically reduce the influence of the material variability in terms of the \mu values on P_{\text{f}}^{\text{sys}}. This is because the TDM piles with large caps can markedly decrease the differential settlement for any value of \mu. On the other hand, for the P_{\text{f}}^{\text{sys}} considered in this study, the impacts of the three coefficients of variation ( COV-s_{\text{u, soil}}, COV-q_{\text{u, pile}}, and COV-{\gamma }_{\text{emb}}) are insignificant. However, as an individual failure mode, P_{\text{f}}^{max} is still affected by the \text{COV} values of all materials. This result is in good agreement with the reliability results obtained by Wijerathna and Liyanapathirana [38,39], who reported that the variation in COV-q_{\text{u, pile}} significantly affected the individual failure probabilities of the maximum settlement and lateral displacement of DCM pile-supported embankments on soft soils.

8 Conclusions

The main conclusions of this study can be drawn as follows.

1) The AK-MCS method can considerably decrease the computational time required to conduct a reliability analysis with a high confidence level, although the failure probability result is lower. For the preliminary design stage, instead of a more general reliability method (i.e., MCS), the AK-MCS can be effectively adopted for reliability assessments that require a large number of iterations to computationally expensive mechanical models.

2) The use of TDM piles with large caps ( {\alpha }_{\text{s}} > 3.0) is recommended to ensure the effectiveness of using TDM piles for the enhancement of the reliability level or reduction of the system failure probability. These TDM piles can provide significantly lower estimations of the differential settlement failure probability than DCM piles. However, TDM piles with large caps should be used carefully owing to a slight increase in the maximum settlement failure probability.

3) A high reliability level in terms of the system failure probability can be found for the case of TDM piles with large heads but thinner caps. This promotes the use of TDM piles to reduce the undesired performance of settlement considering all of the material variabilities for piled embankment systems.

4) The mean values of the materials considered in this study ( \mu -s_{\text{u, soil}}, \mu -q_{\text{u, pile}}, and \mu -{\gamma }_{\text{emb}}) are found to have a significant effect on the system failure probability of SCM piled embankments on soft soils. Importantly, these parameters should be carefully selected for consideration in reliability analyses in further investigations.

The above conclusions are drawn based on a simplified numerical analysis. To confirm and enhance the findings of the current study, a broader set of numerical investigations should be performed in more complex cases (e.g., complex subsoil profiles, a large number of embankment heights, and several values of the surcharge loading). Furthermore, a more complex model (three-dimensional numerical simulation) would also be suggested for the design of piled embankments over soft soils. This model could be used to capture other interesting failure modes related to piled embankments, such as lateral movement and the pile bending moment at the embankment toe and could also be used to explore the slope stability of the embankment. A reliability analysis considering the uncertainties of more material properties and/or the soil spatial variability (RFM) should be developed in future studies in order to better describe the input uncertainties and obtain more precise P_{\text{f}} estimates.

References

[1] Huang J, Han J, Oztoprak S. Coupled mechanical and hydraulic modelling of geosynthetic-reinforced column-supported embankments. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 135( 8): 1011– 1021

[2] Huang J, Han J. 3D coupled mechanical and hydraulic modeling of a geosynthetic-reinforced deep mixed column-supported embankment. Geotextiles and Geomembranes, 2009, 27( 4): 272– 280

[3] Yapage N N S, Liyanapathirana D S, Kelly R B, Poulos H G, Leo C J. Numerical modeling of an embankment over soft ground improved with deep cement mixed columns: Case history. Journal of Geotechnical and Geoenvironmental Engineering, 2014, 140( 11): 04014062

[4] Yapage N N S, Liyanapathirana D S, Poulos H G, Kelly R B, Leo C J. Numerical modeling of geotextile-reinforced embankments over deep cement mixed columns incorporating strain-softening behavior of columns. International Journal of Geomechanics, 2015, 15( 2): 04014047

[5] Jamsawang P, Yoobanpot N, Thanasisathit N, Voottipruex P, Jongpradist P. Three-dimensional numerical analysis of a DCM column-supported highway embankment. Computers and Geotechnics, 2016, 72 : 42– 56

[6] Chai J C, Shrestha S, Hino T, Ding W Q, Kamo Y, Carter J. 2D and 3D analyses of an embankment on clay improved by soil–cement columns. Computers and Geotechnics, 2015, 68 : 28– 37

[7] Huang J, Han J. Two-dimensional parametric study of geosynthetic-reinforced column-supported embankments by coupled hydraulic and mechanical modeling. Computers and Geotechnics, 2010, 37( 5): 638– 648

[8] Jamsawang P, Phongphinittana E, Voottipruex P, Bergado D T, Jongpradist P. Comparative performances of two- and three-dimensional analyses of soil-cement mixing columns under an embankment load. Marine Georesources and Geotechnology, 2019, 37( 7): 852– 869

[9] Yu Y, Bathurst R J, Damians I P. Modified unit cell approach for modelling geosynthetic-reinforced column-supported embankments. Geotextiles and Geomembranes, 2016, 44( 3): 332– 343

[10] Lai Y P, Bergado D T, Lorenzo G A, Duangchan T. Full-scale reinforced embankment on deep jet mixing improved ground. Proceedings of the Institution of Civil Engineers––Ground Improvement, 2006, 10( 4): 153– 164

[11] Bergado D T, Jamsawang P, Tanchaisawat T, Lai Y P, Lorenzo G A. Performance of reinforced load transfer platforms for embankments supported by deep cement mixing piles. In: GeoCongress 2008: Geosustainability and Geohazard Mitigation. New Orleans: Geo-Institute of The American Society of Civil Engineers, 2008, 628– 637

[12] Han J, Oztoprak S, Parsons R L, Huang J. Numerical analysis of foundation columns to support widening of embankments. Computers and Geotechnics, 2007, 34( 6): 435– 448

[13] Borges J L, Marques D O. Geosynthetic-reinforced and jet grout column-supported embankments on soft soils: Numerical analysis and parametric study. Computers and Geotechnics, 2011, 38( 7): 883– 896

[14] Cheng Q, Wu J, Zhang D, Ma F. Field testing of geosynthetic-reinforced and column-supported earth platforms constructed on soft soil. Frontiers of Structural and Civil Engineering, 2014, 8( 2): 124– 139

[15] Liu Z, Zhang A, Xu J, Zhou C, Zhang L. Calculation model and bearing capacity optimization method for the soil settlement between piles in geosynthetic-reinforced pile-supported embankments based on the membrane effect. PLoS One, 2021, 16( 8): e0256190

[16] Okyay U S, Dias D. Use of lime and cement treated soils as pile supported load transfer platform. Engineering Geology, 2010, 114( 1−2): 34– 44

[17] Ishikura R, Yasufuku N, Brown M J. An estimation method for predicting final consolidation settlement of ground improved by floating soil cement columns. Soil and Foundation, 2016, 56( 2): 213– 227

[18] Liu S Y, Du Y J, Yi Y L, Puppala A J. Field investigations on performance of T-shaped deep mixed soil cement column-supported embankments over soft ground. Journal of Geotechnical and Geoenvironmental Engineering, 2012, 138( 6): 718– 727

[19] Yi Y L, Liu S Y, Puppala A J. Laboratory modelling of T-shaped soil–cement column for soft ground treatment under embankment. Geotechnique, 2016, 66( 1): 85– 89

[20] Yi Y L, Liu S Y, Puppala A J, Jing F. Variable-diameter deep mixing column for multi-layered soft ground improvement: laboratory modeling and field application. Soil and Foundation, 2019, 59( 3): 633– 643

[21] Yi Y L, Liu S Y, Puppala A J, Xi P S. Vertical bearing capacity behaviour of single T-shaped soil–cement column in soft ground: laboratory modelling, field test, and calculation. Acta Geotechnica, 2017, 12( 5): 1077– 1088

[22] Yi Y L, Liu S Y, Puppala A J. Bearing capacity of composite foundation consisting of T-shaped soil-cement column and soft clay. Transportation Geotechnics, 2018, 15 : 47– 56

[23] Yi Y L, Ni P, Liu S Y. Numerical investigation of T-shaped soil–cement column supported embankment over soft ground. In: Proceedings of China−Europe Conference on Geotechnical Engineering. Vienna: Springer, 2018, 1068– 1071

[24] Phutthananon C, Jongpradist P, Yensri P, Jamsawang P. Dependence of ultimate bearing capacity and failure behavior of T-shaped deep cement mixing piles on enlarged cap shape and pile strength. Computers and Geotechnics, 2018, 97 : 27– 41

[25] Phutthananon C, Jongpradist P, Jamsawang P. Influence of cap size and strength on settlements of TDM-piled embankments over soft ground. Marine Georesources and Geotechnology, 2020, 38( 6): 686– 705

[26] Phutthananon C, Jongpradist P, Jongpradist P, Dias D, Baroth J. Parametric analysis and optimization of T-shaped and conventional deep cement mixing column-supported embankments. Computers and Geotechnics, 2020, 122 : 103555

[27] Phutthananon C, Jongpradist P, Dias D, Jamsawang P. Numerical study of the deformation performance and failure mechanisms of TDM pile-supported embankments. Transportation Geotechnics, 2021, 30 : 100623

[28] Phutthananon C, Jongpradist P, Jongpradist P, Dias D, Jamsawang P, Bergado D T. Performance-based design optimization of embankments resting on soft soil improved with T-shaped and conventional DCM columns. Acta Geotechnica, 2021, 16( 10): 3301– 3326

[29] Jamsawang P, Voottipruex P, Jongpradist P, Bergado D T. Parameters affecting the lateral movements of compound deep cement mixing walls by numerical simulations and parametric analyses. Acta Geotechnica, 2015, 10( 6): 797– 812

[30] Omine K, Ochiai H, Yasufuku N. Evaluation of scale effect on strength of cement-treated soils based on a probabilistic failure model. Soil and Foundation, 2005, 45( 3): 125– 134

[31] Larsson S, Stille H, Olsson L. On horizontal variability in lime-cement columns in deep mixing. Geotechnique, 2005, 55( 1): 33– 44

[32] Namikawa T, Koseki J. Effects of spatial correlation on the compression behavior of a cement-treated column. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 139( 8): 1346– 1359

[33] Liu Y, Lee F H, Quek S T, Chen E J, Yi J T. Effect of spatial variation of strength and modulus on the lateral compression response of cement-admixed clay slab. Geotechnique, 2015, 65( 10): 851– 865

[34] Zhang R J, Hasan M S M S, Zheng J J, Cheng Y S. Effect of spatial variability of engineering properties on stability of a CSMC embankment. Marine Georesources and Geotechnology, 2018, 36( 1): 91– 99

[35] Al-Naqshabandy M S, Bergman N, Larsson S. Strength variability in lime-cement columns based on cone penetration test data. Proceedings of the Institution of Civil Engineers––Ground Improvement, 2012, 165( 1): 15– 30

[36] Al-Naqshabandy M S, Larsson S. Effect of uncertainties of improved soil shear strength on the reliability of embankments. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 139( 4): 619– 632

[37] Navin M P, Filz G M. Reliability of deep mixing method columns for embankment support. In: GeoCongress 2006: Geotechnical Engineering in the Information Technology Age. Atlanta: American Society of Civil Engineers, 2006, 1– 6

[38] Wijerathna M, Liyanapathirana D S. Reliability-based performance of embankments improved with deep mixing considering spatial variability of material properties. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems. Part A, Civil Engineering, 2018, 4( 4): 04018035

[39] Wijerathna M, Liyanapathirana D S. Significance of variability of deep cement mixed columns on the reliability of column supported embankments. International Journal of Geomechanics, 2019, 19( 8): 04019087

[40] Guo X, Pham T A, Dias D. Probabilistic analysis of geosynthetic-reinforced and pile-supported embankments. Computers and Geotechnics, 2022, 142 : 104595

[41] Guo X, Dias D. Kriging based reliability and sensitivity analysis—Application to the stability of an earth dam. Computers and Geotechnics, 2020, 120 : 103411

[42] Soubra A H, Al-Bittar T, Thajeel J, Ahmed A. Probabilistic analysis of strip footings resting on spatially varying soils using kriging metamodeling and importance sampling. Computers and Geotechnics, 2019, 114 : 103107

[43] Zhang T, Guo X, Dias D, Sun Z. Dynamic probabilistic analysis of non-homogeneous slopes based on a simplified deterministic model. Soil Dynamics and Earthquake Engineering, 2021, 142 : 106563

[44] Lü Q, Xiao Z P, Ji J, Zheng J. Reliability based design optimization for a rock tunnel support system with multiple failure modes using response surface method. Tunnelling and Underground Space Technology, 2017, 70 : 1– 10

[45] Pan Q, Dias D. Probabilistic evaluation of tunnel face stability in spatially random soils using sparse polynomial chaos expansion with global sensitivity analysis. Acta Geotechnica, 2017, 12( 6): 1415– 1429

[46] Engelund S, Rackwitz R. A benchmark study on importance sampling techniques in structural reliability. Structural Safety, 1993, 12( 4): 255– 276

[47] Au S K, Beck J L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 2001, 16( 4): 263– 277

[48] Guo X, Dias D, Carvajal C, Peyras L, Breul P. A comparative study of different reliability methods for high dimensional stochastic problems related to earth dam stability analyses. Engineering Structures, 2019, 188 : 591– 602

[49] Hurtado J E, Alvarez D A. Neural-network-based reliability analysis: A comparative study. Computer Methods in Applied Mechanics and Engineering, 2001, 191( 1−2): 113– 132

[50] Echard B, Gayton N, Lemaire M A K M C S. An active learning reliability method combining Kriging and Monte Carlo Simulation. Structural Safety, 2011, 33( 2): 145– 154

[51] Al-Bittar T, Soubra A H, Thajeel J. Kriging-based reliability analysis of strip footings resting on spatially varying soils. Journal of Geotechnical and Geoenvironmental Engineering, 2018, 144( 10): 04018071

[52] Li T Z, Yang X L. An efficient uniform design for Kriging-based response surface method and its application. Computers and Geotechnics, 2019, 109 : 12– 22

[53] El Haj A K, Soubra A H, Fajoui J. Probabilistic analysis of an offshore monopile foundation taking into account the soil spatial variability. Computers and Geotechnics, 2019, 106 : 205– 216

[54] Zhao H, Ru Z, Chang X, Yin S, Li S. Reliability analysis of tunnel using least square support vector machine. Tunnelling and Underground Space Technology, 2014, 41 : 14– 23

[55] Zhou S, Guo X, Zhang Q, Dias D, Pan Q. Influence of a weak layer on the tunnel face stability—Reliability and sensitivity analysis. Computers and Geotechnics, 2020, 122 : 103507

[56] Guo X, Dias D, Pan Q. Probabilistic stability analysis of an embankment dam considering soil spatial variability. Computers and Geotechnics, 2019, 113 : 103093

[57] Kroetz H M, Do N A, Dias D, Beck A T. Reliability of tunnel lining design using the Hyperstatic Reaction Method. Tunnelling and Underground Space Technology, 2018, 77 : 59– 67

[58] Jongpradist P, Jamsawang P, Kongkitkul W. Equivalent void ratio controlling the mechanical properties of cementitious material-clay mixtures with high water content. Marine Georesources and Geotechnology, 2019, 37( 10): 1151– 1162

[59] Jongpradist P, Homtragoon W, Sukkarak R, Kongkitkul W, Jamsawang P. Efficiency of rice husk ash as cementitious material in high-strength cement-admixed clay. Advances in Civil Engineering, 2018, 2018 : 8346319

[60] Huang J, Kelly R, Sloan S W. Stochastic assessment for the behaviour of systems of dry soil mix columns. Computers and Geotechnics, 2015, 66 : 75– 84

[61] Li D Q, Tang X S, Phoon K K. Risk and Reliability in Geotechnical Engineering. Boca Raton: CRC Press, 2014, 77– 128

[62] Schöbi R, Sudret B, Marelli S. Rare event estimation using Polynomial-Chaos Kriging. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems. Part A, Civil Engineering, 2017, 3( 2): D4016002

[63] Marelli S, Sudret B. UQLab: A framework for uncertainty quantification in Matlab. In: Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014). Liverpool: American Society of Civil Engineers, 2014, 2554– 2563

[64] Kasama K, Whittle A J, Zen K. Effect of spatial variability on the bearing capacity of cement-treated ground. Soil and Foundation, 2012, 52( 4): 600– 619

[65] Bhasi A, Rajagopal K. Geosynthetic-reinforced piled embankments: Comparison of numerical and analytical methods. International Journal of Geomechanics, 2015, 15( 5): 04014074

[66] Brinkgreve R B J Kumarswamy S Swolfs W M Zampich L Ragi Manoj N. PLAXIS 2D Material Model Manual 2019. Delft: Plaxis bv., 2019

[67] Schanz T, Vermeer A, Bonnier P. The hardening soil model: Formulation and verification. In: Proceedings of 1st International PLAXIS Symposium on Beyond 2000 in Computational Geotechnics. Amsterdam: CRC Press, 1999, 281– 296

[68] Jamsawang P, Voottipruex P, Tanseng P, Jongpradist P, Bergado D T. Effectiveness of deep cement mixing walls with top-down construction for deep excavations in soft clay: Case study and 3D simulation. Acta Geotechnica, 2019, 14( 1): 225– 246

[69] Waichita S, Jongpradist P, Jamsawang P. Characterization of deep cement mixing wall behavior using wall-to-excavation shape factor. Tunnelling and Underground Space Technology, 2019, 83 : 243– 253

[70] Wonglert A, Jongpradist P, Jamsawang P, Larsson S. Bearing capacity and failure behaviors of floating stiffened deep cement mixing columns under axial load. Soil and Foundation, 2018, 58( 2): 446– 461

[71] Surarak C, Likitlersuang S, Wanatowski D, Balasubramaniam A, Oh E, Guan H. Stiffness and strength parameters for hardening soil model of soft and stiff Bangkok clays. Soil and Foundation, 2012, 52( 4): 682– 697

[72] Jamsawang P, Voottipruex P, Boathong P, Mairaing W, Horpibulsuk S. Three-dimensional numerical investigation on lateral movement and factor of safety of slopes stabilized with deep cement mixing column rows. Engineering Geology, 2015, 188 : 159– 167

[73] Goh A T C, Zhang F, Zhang W, Zhang Y, Liu H. A simple estimation model for 3D braced excavation wall deflection. Computers and Geotechnics, 2017, 83 : 106– 113

[74] Hsiung B C B, Yang K H, Aila W, Ge L. Evaluation of the wall deflections of a deep excavation in Central Jakarta using three-dimensional modeling. Tunnelling and Underground Space Technology, 2018, 72 : 84– 96

[75] Waichita S, Jongpradist P, Schweiger H F. Numerical and experimental investigation of failure of a DCM-wall considering softening behaviour. Computers and Geotechnics, 2020, 119 : 103380

[76] Ma H, Luo Q, Wang T, Jiang H, Lu Q. Numerical stability analysis of piled embankments reinforced with ground beams. Transportation Geotechnics, 2021, 26 : 100427

[77] Zhuang Y, Wang K. Finite element analysis on the dynamic behavior of soil arching effect in piled embankment. Transportation Geotechnics, 2018, 14 : 8– 21

[78] Hamrouni A, Dias D, Sbartai B. Soil spatial variability impact on the behavior of a reinforced earth wall. Frontiers of Structural and Civil Engineering, 2020, 14( 2): 518– 531

[79] Das B M. Fundamentals of Geotechnical Engineering. Boston: Cengage Learning, 2007

[80] Kulhawy F H Mayne P W. Manual on Estimating Soil Properties for Foundation Design. Rep. No. EPRI-EL-6800. 1990

[81] Phoon K K, Kulhawy F H. Characterization of geotechnical variability. Canadian Geotechnical Journal, 1999, 36( 4): 612– 624

RIGHTS & PERMISSIONS

Higher Education Press 2022