Soil spatial variability impact on the behavior of a reinforced earth wall

Abstract

This article presents the soil spatial variability effect on the performance of a reinforced earth wall. The serviceability limit state is considered in the analysis. Both cases of isotropic and anisotropic non-normal random fields are implemented for the soil properties. The Karhunen-Loève expansion method is used for the discretization of the random field. Numerical finite difference models are considered as deterministic models. The Monte Carlo simulation technique is used to obtain the deformation response variability of the reinforced soil retaining wall. The influences of the spatial variability response of the geotechnical system in terms of horizontal facing displacement is presented and discussed. The results obtained show that the spatial variability has an important influence on the facing horizontal displacement as well as on the failure probability.

Keywords

reinforced earth wall / geosynthetic / random field / spatial variability / Monte Carlo simulation

Introduction

In recent years, the probabilistic analysis of geotechnical structures has devoted a great deal of effort. Various uncertain parameters were modeled using random variables based on some simplified methods in which the soil is considered as a uniform material. However, in nature, as a result of deposition and post-deposition processes, soil parameters (elastic properties, shear strength parameters, etc.) vary spatially (horizontal and vertical direction). The spatial variability of the soil properties will then affect the behavior of geotechnical structures (bearing capacity, foundation settlement, slope stability, etc.). Therefore, the need to represent the soil parameters considering random fields is essential. Advanced probabilistic approaches were proposed in the literature. The deterministic models used to represent geotechnical structures are frequently based on the finite difference method (FDM) or the finite element method (FEM). In these approaches, it is necessary to convert the random field discretization into a finite number of random variables so that the probability of failure can be determined. Several methods to generate random fields are already available in literature like: Expansion Optimal Linear Estimation (EOLE), Stochastic Spectral (SS), Karhunen-Loève (K-L)), Random Finite Element Method (RFEM), stochastic response surface method (SRSM), polynomial chaos expansion (PCE). The use of these methods has allowed several authors to study the effect of soil spatial variability on the behavior of geotechnical structures [¹–¹⁸].

A mechanically stabilized earth (MSE) wall is a composite material formed by the combination of soil and metallic or synthetic strips able to sustain significant tensile loads. The reinforcing strips give to the soil mass anisotropic properties in the direction perpendicular to the reinforcement Schlosser and Elias [¹⁹]. To improve the optimization of the design method, it is essential to understand the behavior of such structures. Several studies, experimental, theoretical or numerical, were carried out with this objective. Using numerical studies, two and three-dimensional models based on finite elements or finite differences (Abdelouhab et al. [²⁰,²¹], Riccio et al. [²²]) allowed to analyze the deformation and the influence of soil parameters of reinforced soil walls.

The reliability analysis of reinforced retaining walls permits to consider the uncertainties due to the soil properties variability [²³–²⁸]. These authors have included in their studies the uncertainties related to the reinforcement properties. Chalermyanont and Benson [²⁹] evaluated the Hasofer-Lind reliability index through the application of Monte Carlo simulations (MCSs). Reliability diagrams were also developed to select the geosynthetic material which is able to meet the safety requirements [³⁰]. However, the representation of such a realistic structure for reliability analysis by analytical mechanisms is not satisfactory. So the use of the continuum mechanics (FEM or FDM) is essential. Thurner and Schweiger [³¹] found some difficulties to apply reliability calculation methods such as the first order reliability method (FORM) or the second order reliability method (SORM) when the performance is not explicitly available in an analytical form (such as the displacements of the reinforced retaining wall). Sayed et al. [²⁵] assessed the maximum failure probability of MSE using the RSM; this model is based on numerical simulations using FEM model. Lin et al. [³²] analyzed the probabilistic maximum deformations of three reinforced soil walls using a combination of a FDM and the RSM technique. This work was extended by Yu and Bathurst [³³] using different geosynthetic reinforcement configurations. The response surface methodology optimized by genetic algorithm was used by Hamrouni et al. [³⁴] to study the reliability of reinforced earth walls.

In the literature, very few studies have been done to investigate the effect of spatial variability on the response of a reinforced earth wall. Previous studies only considered random variables. However, Dodagoudar et al. [³⁵] attempted to consider the influence of soil spatial variability on reinforced earth walls. A linear elastic behavior was assumed for the soil and the soil Young modulus was considered as a 2D isotropic random field and supposed to follow a normal distribution. Ning and Bathurst [³⁶] investigated random fields of soil properties on reinforced soil slopes. The Random Finite Element Method (RFEM) developed by Griffiths and Fenton [³⁷] was used. Their work was only focused on the reinforced slopes stability at the ultimate limit state.

This article presents a probabilistic analysis at the serviceability limit state (SLS) of a reinforced earth wall. The spatial variability of the soil is modeled using non-Gaussian anisotropic random fields. Random fields are discretized into a finite number of random variables using the K-L expansion. The facing maximum horizontal displacement is used to represent the system response. The deterministic model considered is based on a finite difference numerical approach (FLAC2D [²⁰,²¹]). The MCS methodology is used to determine the probability density function (PDF) of the system response variability of the reinforced soil retaining wall. The aim of this study is to determine the effect of the different governing statistical characteristics (autocorrelation distances and coefficient of variation) on the facing maximum horizontal wall displacement.

Presentation of the numerical model

Abdelouhab et al. [²⁰,²¹] used the Lagrangian explicit finite-difference code FLAC2D to study the behavior of a mechanically reinforced earth wall reinforced by geosynthetic strips. The wall of reference is 6m high and is made of 4 superimposed panels (beam elements). This wall is reinforced by 8 levels of 4 m long reinforcement layers (strip elements) (Fig. 1(a)). The real cruciform geometry of the panels (Fig. 1(b)), leads to a complex geometry of the wall facing. A real panel have 4 points of connection for the strips. This three-dimensional geometry and the staggered layout are simplified into a two-dimensional model using some simplifications. The panels are modeled as rectangular plates of 1.5 m by 1.5 m. The simplification of the geometry permits to use a two-dimensional model with continuous reinforcements (plane strain configuration). The parameters of these reinforcements are calculated by homogenizing their geometrical characteristics (Fig. 1(c)). The reinforcement surface is the ratio between the individual reinforcement area multiplied by the number of reinforcement and the model width.

For the boundary conditions, the horizontal displacements are blocked at the vertical limits of the model. Vertical and horizontal displacements are blocked at the model bottom.

Concerning the interface friction at the soil/reinforcement, the apparent coefficient of friction is used in order to take into account the effect of the dilatancy and the reinforcements shape.

The model consists of two soils (Fig. 1(a)). The embankment soil is made of a uniform fine sand, known as Hostun RF sand [³⁸,³⁹]. The constitutive model used for this sand is linear elasticity with perfectly plasticity (shear failure criterion of Mohr-Coulomb type). This constitutive model is characterized by five parameters: 2 elastic parameters (E: Young’s modulus, u: Poisson’s ratio) and 3 plastic parameters (j: friction angle, C: cohesion, y: dilatancy angle). The parameters of this constitutive model were defined for the embankment soil by calibration on triaxial tests carried out under confinement of 30, 60, and 90 kPa [²⁰,²¹]. For the soil foundation, a linear elastic model is used. This model is characterized by two elastic parameters (E: Young’s modulus and u: Poisson’s ratio) (Table 1).

Beam elements are used to model the concrete panels. These structural elements are isotropic and linear elastic. They can resist to bending moments, compressive, and tensile efforts (Table 1).

The beams were represented by pins to reproduce the flexibility of the real wall facing. Wall panel spacers made of ribbed elastomeric pads are inserted to prevent the panels from having contact points and peeling of the concrete. In numerical modeling, the elastomeric pads are considered by reducing artificially the beam section keeping the real inertia moment.

Geosynthetic strip type of reinforcement containing high-tenacity polyester yarns protected by polyethylene sheath are studied and modeled by the use of structural elements of “Strip” type [⁴⁰]. Its properties are presented in Table 1. These elements are specially designed to simulate the behavior of reinforcing strips used in reinforced earth embankments. Strip elements consider tensile strength and compression but cannot withstand bending moments [⁴¹].

The soil and the beam elements are linked using interface elements in order to simulate the frictional interaction between the backfilling soil and the concrete facing panels (Table 2). The FLAC recommendations are used to calculate the normal and the shear stiffness. The value of the friction angle is estimated to be equal to 2/3 of the soil friction angle [⁴⁰].

A nonlinear shear failure envelope that varies as a function of the confining pressure defines the shear behavior of the strip/soil interface. The interface parameters are the shear stiffness k_b at the soil/strip interface and the apparent friction coefficient f*. The values of the shear stiffness (k_b) and parameter the friction coefficient (f*) used were defined by calibration on laboratory pullout tests (Abdelouhab et al. [⁴¹]). According to this calibration, the parameters used in the numerical modeling are presented in Table 2.

A non-uniform mesh composed of around 2500 zones is used (Fig. 1(d)). These zones increase gradually away from the ends of the reinforced area until they become equal to 0.80 m in areas far from the retaining wall but below the maximum limit (1 m) proposed by Der Kiureghian and Ke [⁴²]. In the vertical direction, the backfill is divided into 24 areas refined uniformly. This mesh have a constant vertical width equal to 0.25 m and does not exceed a maximum of 0.50 m vertically (recommendation proposed by Der Kiureghian and Ke [⁴²]). These values are used in accordance with the projection condition of a random field on the finite difference mesh.

The compaction of the soil layers is not considered in the reference study. A calculation considering compaction by an equivalent loading pressure is performed in the parametric study.

Calculations were made under drained conditions as there is no water presence.

To reproduce the real building steps, the setting up of the embankment is modeled by layers of 0.375 m thickness in 3 stages:

Stage 1. Set up of the first concrete panel of the first and of the second soil layer and installation of the first strip between the two layers of the reinforced backfill (equilibrium under self weight).

Stage 2. Placement of the third and the fourth layer and installation of the second strip between the two layers of the reinforced backfill (equilibrium under self-weight).

Stage 3. Set up of the second beam, of the fifth and sixth layer and installation of the third strip between the two layers of the reinforced backfill.

These phases are repeated up till 6 m high.

Results (displacements)

Using the reinforcement reference parameters, the maximum displacement calculated on the MSE wall is equal to 7.29 cm. This value is located just behind the concrete facing, at the level of the 3rd reinforcing bed (Fig. 2). The extensibility of the synthetic strips (low elastic modulus) leads to this high wall deformation.

Determination of the significant soil parameters on the wall behavior

To reduce the number of variables to be considered in the reliability approach, a parametric study was used to define the input parameters influence on the wall behavior. The output parameter considered in this study is the facing maximum horizontal displacement noted as U_hmax = 7.29 cm which is calculated with the reference values. A parametric study is performed here to check the sensitivity of the facing maximum horizontal displacement to the five parameters of the linear elastic perfectly plastic constitutive model (E, u, c, j, and y) and of the panel/soil interface friction angle (°). When studying the influence of one parameter, the other parameters are kept at their reference value. A total of 18 simulations corresponding to three different values of each parameter see Table 3 are studied, as shown in Fig. 3. The Young’s modulus, the Poisson’s ratio, the dilation angle and the friction angle at the panel/soil interface have a negligible influence on the U_hmax value (difference versus the reference case inferior to 2%). However, two parameters have an influence on the output parameters: the friction angle j and the soil unit weight g. The soil friction angle influence is higher than the one of the soil unit weight respectively of 123% and 20%. In the following study, only the friction angle j will be considered as a random variable.

The soil Young’s modulus has a negligible effect on the horizontal displacement U_hmax. This is due to the rigidity of the reinforcements which is stiffer than the soil one.

Random field method used for discretization

As already mentioned, random fields are defined by a distribution law and an autocorrelation function. Random fields generated in this study are two-dimensional and follow a lognormal distribution law. The mean and standard deviation of the field in (Z) normally distributed are named m_Z,s_Z, respectively, while the average (m_ln) and standard deviation (s_ln) of the lognormal Z field are given by

(1){{\displaystyle \sigma }}_{\mathrm{l}\mathrm{n}}=\sqrt{\ln (\mathrm{1}+{{\displaystyle COV}}_Z^{\mathrm{2}})},

(2){{\displaystyle \mu }}_{\mathrm{l}\mathrm{n}}=\ln ({{\displaystyle \mu }}_Z)-{{\displaystyle \mathrm{0.5}\sigma }}_{\mathrm{l}\mathrm{n}}.

Among the available methods used to generate correlated parameters randomly, the Cholesky decomposition [⁴³,⁴⁴] and K-L expansion [⁴⁵,⁴⁶] are often used. Since the considered problem deals with spatially correlated fields, the K-L decomposition is recommended and was employed in this study.

The random field defined by the method of expansion in series of K-L presented by Spanos and Ghanem [⁴⁵] is used to express it through a truncated series using M number of terms [⁴⁷,⁴⁸]. The field is then divided into a determined portion represented by the average, and a random element depending of f_i eigenvalues and of l_i functions of the corresponding covariance function:

(3)Z(X,\theta )\approx \exp ({{\displaystyle \mu }}_{\mathrm{l}\mathrm{n}}+{\mathrm{\Sigma }}_{i=\mathrm{1}}^M\sqrt{{{\displaystyle \lambda }}_i}{{\displaystyle \phi }}_i(X){{\displaystyle \xi }}_i(\theta )),

with X the vector of coordinates of the field points where the values of the Z property are to be generated; q is the variable of the random draw and x_i(q) is a normal random variable centered and reduced.

The main advantage of this method lies in the decoupling of spatial and stochastic variables (x and q, respectively). The eigenvalues and eigenfunctions (f_i and l_i, respectively) of the covariance function C (X₁, X₂) are solutions of the following equation:

(4){{\displaystyle \int }}_{D}C({{\displaystyle X}}_{\mathrm{1}},{{\displaystyle X}}_{\mathrm{2}}){{\displaystyle \phi }}_i({{\displaystyle X}}_{\mathrm{1}})\mathrm{d}{{\displaystyle X}}_{\mathrm{1}}={{\displaystyle \lambda }}_i{{\displaystyle \phi }}_i({{\displaystyle X}}_{\mathrm{2}})

This integral can be analytically solved for some types of autocorrelation function. The analytical solution for the exponential autocorrelation function here adopted is given by Ghanem and Spanos [⁴⁹]. The stochastic dependency appears only through the random variables x_i. The spatial correlation is defined by the eigen modes (f_i and l_i) of the covariance kernel (Eq. (4)).

The expression of the covariance function appears in Eq. (5b) and introduces both the variance {\sigma }^{\mathrm{2}} of the Gaussian random field and the correlation length L. Each term C₁₂ of the covariance matrix C is the value of the covariance function computed between the two nodes 1 and 2 of the spatial mesh, whose position is given by X₁ and X₂.

(5a){{\displaystyle C}}_{\mathrm{12}}=C({{\displaystyle X}}_{\mathrm{1}},{{\displaystyle X}}_{\mathrm{2}})=V\exp \left(-{\displaystyle \frac{\Vert {{\displaystyle X}}_{\mathrm{1}}-{{\displaystyle X}}_{\mathrm{2}}\Vert }{L}}\right).

In our problem, two spatial directions (horizontal and vertical) are distinguished. Therefore, two correlation lengths are also introduced. When a 2D field is considered, the covariance function is given as:

(5b){{\displaystyle C}}_{\mathrm{12}}={{\displaystyle \sigma }}_{\ln }^{\mathrm{2}}\exp \left(-{\displaystyle \frac{\left|{{\displaystyle x}}_{\mathrm{1}}-{{\displaystyle x}}_{\mathrm{2}}\right|}{{{\displaystyle L}}_{\ln x}}}-{\displaystyle \frac{\left|{{\displaystyle y}}_{\mathrm{1}}-{{\displaystyle y}}_{\mathrm{2}}\right|}{{{\displaystyle L}}_{\ln y}}}\right).

Equation (5b) is the covariance between nodes 1 and 2, for which (x₁, y₁) are the coordinates of node 1 (id. for node 2). L_lnx and L_lny are the autocorrelation distances in the horizontal direction and in the vertical one, respectively.

Note that the number of terms M is selected depending on the targeted accuracy. The estimated error e_rr(X) values generated by the K-L expansion series for a number of terms is given by M [⁵⁰].

(6){{\displaystyle \epsilon }}_{\mathrm{r}\mathrm{r}}(X)=\mathrm{1}-\left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \sigma }}_{\mathrm{l}\mathrm{n}}}}\right){\mathrm{\Sigma }}_{i=\mathrm{1}}^M{{\displaystyle \lambda }}_i{{\displaystyle \phi }}_i^{\mathrm{2}}(X).

Choice of the number of terms of the K-L series

When a random field is generated, the values of this field should reflect the initial theoretical statistical distribution and autocorrelation function. However, the accuracy of the random fields generated depends on the number of terms M of the K-L series. To select the number of terms M to be used in the random field discretization of the friction angle, the average distribution functions and autocorrelation are drawn for many realizations of this random field as a function of the number of terms M, and compared to the target functions.

A random field generation of the friction angle is proposed using a lognormal distribution with a mean value m_j = 36 ° and a covariance COV_j = 15%. An exponential autocorrelation function with autocorrelation distances equal to 1 m in both directions is chosen to illustrate the influence of the number of terms M. A large number of outputs (350) are generated for each variation of the number of terms M. For each case, the cumulative probability function and the autocorrelation function of the generated values are plotted. The average of these functions is then compared to the target theoretical functions.

Figure 4 shows a comparison between the average cumulative probability functions (CDF) using 350 random field realizations for different numbers of terms M and the target cumulative probability function. Figure 5 shows a comparison between the autocorrelation function of the target (Eq. (7)) and of the average autocorrelation functions generated in the horizontal direction after 350 random field realizations (Eq. (8), considering different numbers of terms M). The good concordance between these two curves has permitted to validate the generated stochastic field.

Figures 4 and 5 show that more the number of K-L terms increases, more the functions generated get closer to the target theoretical functions, especially for 200 and 250 terms M. A number of terms M equal to 250 is adopted in the following calculations.

The domain and the model mesh must be selected to meet certain conditions (fine mesh in areas near the facing). A transfer of the stochastic data to the mesh is necessary. For this purpose, two transfer methods exist:

1) The middle method used by Ref. [⁴²,⁵¹,⁵²]: in this method, the random field is represented by its value at the centroid of the finite element. One seeks the position of the centroid in the stochastic mesh and assigned to the finite element mesh the value of the field corresponding to the position of its centroid in the stochastic mesh.

2) The local averaging method (or spatial averaging) proposed by Vanmarcke [⁵³,⁵⁴]: it consists in allocating to each cell of the domain a value determined by averaging the stochastic mesh field values that fall within the finite element mesh.

Der Kiureghian and Ke [⁴²] noted that the method of the middle over-represents the variability of the field while the local averaging method tends to under-represent the true variability of the field. They also demonstrated that the spatial averaging method is more suitable for Gaussian fields. Popescu [⁵⁵] conducted a comparative study between these two methods and concluded that the middle method is best suited for non-Gaussian random fields since it keeps the probability distribution of the initial field. In this study, the middle method was used since the random field representative soil properties are considered as non-Gaussian fields.

Random field of soil properties

The numerical model proposed in this study focuses on the spatial variability of the soil friction angle. The corresponding field is expected to follow a lognormal distribution, the soil friction angle being a strictly positive quantity. The mean and standard deviation are denoted by m_ln and s_ln, respectively. The dependency of the friction angle values for two distinct points of the soil mass is represented by an exponential autocorrelation function.

To model a realization of the friction angle random field and pass it to the proposed numerical model, a 2D field representing the soil mass is discretized into N_x× N_y surface elements, where N_x and N_y are the number of elements in the x and y directions. With the implementation of a function using MATLAB, a friction angle value is given to each soil element. In fact, each element is assumed to be homogeneous and is affected by a single friction angle value at its center using the K-L method of expansion in series, developed by Ghanem and Spanos [⁴⁹]. In other words, for each realization of the random field, a given configuration of the soil is generated: a friction angle value is assigned to the center of each of its elements depending of its position in space, following the distribution law and the selected autocorrelation function. It should be noted that for each realization, the values of the random field (j) were determined at the centroid of each element of the deterministic mesh (FLAC2D) using the middle method [⁵⁵].

Finally, it should be mentioned that a link between FLAC2D and MATLAB was performed to automatically exchange the data in both directions and thus to decrease the computation time.

The spatial variability of the soil friction angle can be defined by a lognormal distribution with m_φ = 36 °, COVj = 15% and a second-order exponential function with an autocorrelation distance equal to 1.0 m in both directions. The embankment is represented by a random field (Fig. 6).

The field is generated in two dimensions using the following dimensions: 20 m in the longitudinal direction X and 6 m in the vertical direction Y.

This domain is discretized into elements of 0.25 m × 0.25 m.

Probabilistic analysis with Monte Carlo

Selection of the optimal number of MCS

The effect of the soil spatial variability on the wall facing horizontal displacement is done using the Monte Carlo (MC) method for several combinations of autocorrelation distances in the two directions. This probabilistic analysis requires, for each combination studied, the generation of a large number of samples, each represented by a random field realization. The maximum facing horizontal displacement (U_max) is then calculated for each of these achievements and the average (m_U) and standard deviation (s_U) of all values of the maximum facing horizontal displacement are then compared. The number N_s of simulations required is the one for which the values of the two statistical moments converge.

Table 4 shows the values of the first critical pressure statistical moments obtained for different simulation numbers N_s, while varying the number of terms of K-L, M. The corresponding curves are shown in Figs. 7 and 8.

The optimal number of MCSs is equal to 300. This number will be used for the following calculations.

Effect of the soil spatial variability on the facing horizontal displacement

To study the soil spatial variability effect on the facing horizontal displacement, several combinations of the autocorrelation distances (L) in the two directions were used. In fact, when a small distance autocorrelation is considered in a given direction, the soil friction angle values are highly correlated and quickly change from one point to another. However, when the distance is large, the values change slowly with the distance and can become homogeneous.

Figure 9 illustrates a realization of an isotropic random field of the soil friction angle according to a lognormal distribution law with an autocorrelation distance of 1 m. The mean and coefficient of variation of the soil internal friction angle j adopted are equal to 36 degrees and 15%, respectively. Dark regions correspond to small values of j and the lighter ones to larger values where the values vary between 28° and 44°. Figure 10 shows a realization of a random field of the anisotropic soil internal friction angle (i.e., L_x = 10 m; L_y = 1 m).

The considered combinations are as follows:

1) Case 1D: L_x≠ L_y

These cases represent stratified soils knowing for each layer its friction angle value. The layers are parallel to the plane formed by the directions for which a large autocorrelation distance is adopted.

2) Case 2D: L_x= L_y

These cases represent heterogeneous soils with friction angle varying from one point to another.

Several cases of autocorrelation distances (L) are considered using L = 1, 2, 5, and 10 m.

The two first statistical moments of the maximum facing displacement are calculated for 300 realizations of the random field for each of the 3 considered cases. The results are then compared and evaluated.

Effect of the soil spatial variability on the average maximum facing displacement

Figure 11 shows the variation of the average on the maximum facing displacement as a function of the distance autocorrelation for different random fields types (1D and 2D).

All curves have the same shape. With the increase of the autocorrelation distance, the average of the maximum displacement increases to reach a maximum at a facing height of L≈H / 2, and then decreases. This maximum indicates that the soil behind the facing is at least stable for this autocorrelation distance. Similar results are observed by Ref. [⁶] for shallow foundations. This can be explained as follows:

1) For small autocorrelation distances, the soil variability is very high and it results in a mix of small and large friction angle values. This gives a displacement close to the deterministic value.

2) The movement of the ground is greater in areas where there is a low value of friction angle and especially in the reinforced area. For intermediate autocorrelation distances (L≈ length of strips), agglomerates of weak and strong areas are created. Their distribution within the reinforced zone varies from one realization to another.

3) For larger autocorrelation distances: for the 1D case, the layers and soil columns form alternating small and large values of the friction angle. The soil zones with a high friction angle value will constitute a stabilizing barrier for the soil movements. It induces a small value of the maximum displacement mean of the facing and a more stable structure. As for the 2D case, the soil becomes more homogeneous especially for the case where L_x = L_y = 1000 m. In this case, the mean of the maximum facing displacement has a difference of 5.7% compared to the deterministic displacement.

4) The cases considering a spatial variability in a single direction (vertical one), lead to values very close to the deterministic displacement of the facing but lower than those obtained for the case 1D according to the horizontal direction. This is due to the fact that the variation of the friction angle along the vertical axis leads to “layered” vertical soil, perpendicular to the reinforcement plane of symmetry, which facilitates their collapse and makes this case as safe as the case according to the horizontal direction.

5) Finally, for the case 2D (L_x = L_y), the mean maximum displacement of the facing is greater than the one of the 1D case according to the horizontal and vertical direction. This is due to the introduction of the vertical variability which reduces the deformations of the facing, as explained above. In contrast, the 2D case reveals lower movements of the facing compared to the 1D case.

6) Observing the effect of the spatial variability direction on the mean maximum displacement of the facing reveals that taking into account the same variation in the angle of friction in the two directions leads to a less stable facing. However, considering spatial variability in the longitudinal and/or vertical directions improves the deformations of the facing and therefore a smaller value of the displacement is required to avoid the failure of the structure.

Effect of the soil spatial variability on the standard deviation of the maximum facing displacement

Figure 12 shows that the variation of the standard deviation of the facing maximum displacement depends of the distance autocorrelation. All curves have the same tendency regardless the considered variability. The standard deviation increases with the autocorrelation distance increase.

Indeed, for small autocorrelation distances, the generated friction angle values are very heterogeneous, i.e., a mixture of large and small values, which leads to an “averaging” of the backfill properties values. The resulting facing maximum displacements are more homogeneous. For larger autocorrelation distances, the friction angle values, although more homogeneous within the same mass of soil, vary considerably from one realization to other, leading to more scattered values of the maximum facing displacements.

Figure 12 shows that the greatest variation of the maximum facing displacement is obtained for the case 2D and is followed by the 1D cases according to the Y axis. The smaller variation is observed for the case 1D according to the X axis. This can be explained in the following way:

- For very large autocorrelation distances (L_x = L_y = 1000 m), the standard deviation of the maximum facing displacement tends to a maximum constant value which corresponds to the case of a random variable. In fact, the friction angle values for one realization case are highly or even perfectly correlated, thus generating a homogeneous ground mass and affected everywhere by the same friction angle value. However, the friction angle values depend on the theoretical statistical distribution adopted. The soil mass thus has a wide variation of the friction angle values which has a direct effect on the system response. (i.e., maximum facing displacement).

The cases considering a spatial variability in one direction (case 1D), exhibit a smaller variability of the maximum facing displacement than the case of “random variables” discussed above, i.e., L_x = L_y = 1000 m. In fact, by introducing a variation of the friction angle in a given direction, all the maximum facing displacement values will tend to increase, thus excluding the small displacement values. Consequently, the maximum facing displacement variability will decrease compared to the case of homogeneous soil mass.

Influence of the random field coefficient of variation

Figure 13 presents the probability density function (PDF) of the maximum facing displacement for three different configurations of the random field’s coefficients of variation. Table 5 presents the statistical moments of the maximum facing displacements. Figure 13 and Table 5 show that the variability of the maximum facing displacements increases with the random field coefficient of variation.

Conclusions

The influence of the soil spatial variability on the behavior of a reinforced earth wall at the SLS is considered in this work. Both cases of isotropic and anisotropic non-normal random fields are considered for the soil properties. The K-L method is used for the discretization of the random field. A deterministic model based on numerical simulations using the explicit code Lagrangian finite-difference code FLAC2D is used to calculate the maximum horizontal facing displacement. The MCS technique was used to determine the deformation response variability of the reinforced soil retaining wall. The main results of this study are the following ones:

1) The main input parameter which influence the movement of the wall is the internal friction angle for the studied case.

2) The numerical results have shown that the variability of the maximum facing displacement increases when the random fields coefficient of variation increase.

3) A maximum probabilistic mean is reached for an intermediate value of the autocorrelation distance (roughly equal to the reinforcement length), whatever the studied scenario. This maximum indicates that the reinforced area is the least stable for this autocorrelation distance.

4) Considering a friction angle spatial variability along the transverse direction, it induces a decrease of the horizontal facing displacement. As a result, it leads to higher displacement values than those obtained in cases of spatial variability in the longitudinal and / or 2D directions. The last two scenarios would lead to a minimum facing displacement.

5) The standard deviation of the maximum facing displacement increases with the autocorrelation distance increase for all the considered scenarios,

Finally, the obtained results show the importance of considering the spatial variability of soil properties due to the fact that some observed phenomena (such as the non-symmetric soil failure) cannot be seen when homogenous soils are considered.

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