FEM and ANN analyses of short and tall geosynthetic reinforced soil walls: Evaluation of various effective aspects and ASSHTO acceptable design methods

Abstract

In addition to confined investigations on tall geosynthetic reinforced soil (GRS) walls, a remarkable database of such walls must be analyzed to diminish engineers’ concerns regarding the American Association of State Highway and Transportation Officials (AASHTO) Simplified or Simplified Stiffness Method in projects. There are also uncertainties regarding reinforcement load distributions of GRS walls at the connections. Hence, the current study has implemented a combination of finite element method (FEM) and artificial neural network (ANN) to distinguish the performance of short and tall GRS walls and assess the AASHTO design methods based on 88 FEM and 10000 ANN models. There were conspicuous differences between the effectiveness of stiffness (63%), vertical spacing (22%), and length of reinforcements (14%) in the behavior of short and tall walls, along with predictions of geogrid load distributions. These differences illustrated that using the Simplified Method may exert profound repercussions because it does not consider wall height. Furthermore, the Simplified Stiffness Method (which incorporates wall height) predicted the reinforcement load distributions at backfill and connections well. Moreover, a Multilayer Perceptron (MLP) algorithm with a low average overall relative error (up to 2.8%) was developed to propose upper and lower limits of reinforcement load distributions, either at backfill or connections, based on 990000 ANN predictions.

Graphical abstract

Keywords

geosynthetics / short and tall GRS walls / artificial intelligence / MLP / AASHTO Simplified and Simplified Stiffness Methods

1 Introduction

The use of reinforced soil (GRS) walls instead of conventional retaining ones (i.e., rigid gravity walls described in American Association of State Highway and Transportation Officials (AASHTO) [1]) has increased worldwide (over 200000 cases until 2018 [2]). This is due to the lower construction costs and time required, more deformability and comfortable installation, and the possibility of construction on weak soils [3–66]. A summary of several numerical studies on the different aspects of the performance of GRS walls through the past two decades is presented in Tab.1. It illustrates the necessity of more detailed studies on the effectiveness of various reinforcement parameters and evaluation of current acceptable design methods of such walls. Tab.2 describes some significant studies focused on the reinforcement parameters (which ensure the internal and external stability and construction costs of GRS walls), including length, vertical spacing, layout, stiffness, ultimate strength, corrosion, creep and pull-out resistance.

The possibility of constructing tall walls is one of the most significant reasons for using reinforced soil systems instead of weighted concrete walls [3]. Although the performance of these walls should be examined separately from regular height walls, research in this field is relatively limited. Tab.3 summarizes research about the behavior of tall walls in the literature.

The use of tall GRS walls is expanding swiftly in mining areas, airport runways, highway structures, and in space-limiting circumstances (Sankey and Soliman [6]). However, there is uncertainty about the effectiveness of the wall height and reinforcement parameters around the performance of such walls. It is a critical issue that the wall height should be considered in the Simplified Method, which is still used in practice and is an acceptable design method in AASHTO [1] for designing GRS walls. Therefore, the effectiveness of the wall height parameter should be analyzed on the contrast in performance between short and tall GRS walls and their improvement methods. More investigations with various scenarios of GRS wall conditions should be conducted on the performance evaluation of the stiffness-based methods. It is particularly important to expand the simulations and assessments of tall GRS wall cases to diminish engineers’ concerns about implementing such new allowable design methods (i.e., Stiffness-based Methods) in projects.

Furthermore, there are uncertainties regarding reinforcement load distributions of GRS walls at the facing connections for both short and tall walls in the literature, but most previous investigations only concentrated on the load distributions at backfill. In addition, investigations on GRS walls in previous research were confined to limited cases, even in the cases of numerical modeling. In this regard, artificial intelligence can achieve breakthroughs for expanding investigated GRS models. Therefore, this paper focuses on illustrating the potential discrepancies between the performance of short and tall GRS walls and considering the effectiveness of reinforcement parameters based on finite element method (FEM) simulations through PLAXIS 2D [7]. The effectiveness of the backfill soil types and varying L/H (normalized geogrid length with wall height), S_v (vertical spacing of reinforcements), and J (geogrid stiffness) on the performance (wall deformations and overturning moments and reinforcement loads) of 46 short (8 m) and 42 tall (20 m) GRS walls were evaluated and compared with each other.

In addition, an extensive database of 88 FEM and 10000 artificial neural network (ANN) GRS wall models was developed to distinguish the performance of short and tall GRS walls and assess the allowable design methods of GRS walls described by AASHTO [1], including the Simplified and Simplified Stiffness Methods. The distributions of the normalized maximum reinforcement tensile loads at backfill and even normalized connection forces were compared with the predictions of these design methods. Next, 552 FEM results (based on 88 simulations of different scenarios of 8 and 20 m walls) were used to train and test an Multilayer Perceptron (MLP) algorithm (as an ANN method) with low AORE values (2.5% and 2.8% for training and testing data sets, respectively). Over 10000 scenarios of GRS walls with various H, L/H, S_v, and J values were generated to establish 990000 ANN simulations of reinforcement load distributions at backfill and connections. Therefore, the current study has minimized the uncertainties and limitations of considering a wide range of input parameters of GRS walls by applying artificial intelligence to a geotechnical problem. Finally, the upper and lower limits of reinforcement load distributions, either at backfill or connections, are proposed based on 552 FEM and 990000 ANN predictions to fill the gap in the literature.

2 Numerical model verification

Wall 1 of physical model tests at the Royal Military College of Canada (RMC) was considered for numerical modeling verification [8]. The FEM-based program PLAXIS 2D V20 [7] was used for the numerical modeling of this wall, and Fig.1 presents the configuration and support conditions of the corresponding model. The vertical direction of the first facing block was restrained and a fixed-end anchor with high stiffness was used to control the horizontal movement, the same as the physical test program. Furthermore, staged construction was considered in numerical modeling and a uniform vertical stress of 8 kPa (suggested by Hatami and Bathurst [9]) was applied to compact each soil layer and removed before modeling a new layer. Moreover, the linear elastic–plastic model was employed for reinforcement modeling, as proposed by Askari et al. [10]. Tab.4 shows the tensile stiffness and yield strength of reinforcements of RMC Wall 1 (derived from Askari et al. [10]).

Backfill soil was modeled using a hardening soil (HS) constitutive model with small-strain stiffness (HS-small) proposed by Benz [11], which is an enhanced version of the HS model [12] and has been successfully used in geotechnical applications (e.g., Brinkgreve et al. [13], Obrzud [14], Ezzeldin and EI Naggar [15]). The HS-small model employs a hyperbolic function for the stress–strain relationship due to primary loading as follows [7]:

(1)

(2)q_{\mathrm{f}}=(c\mathrm{c}\mathrm{o}\mathrm{t}\phi -{\sigma }_3^{\prime })\frac{2\mathrm{s}\mathrm{i}\mathrm{n}\phi }{1-\mathrm{s}\mathrm{i}\mathrm{n}\phi },

where ε₁ is the axial strain, q is the deviatoric stress, q_a is the asymptotic value of the shear strength, q_f is the ultimate deviatoric stress, E_i is the initial stiffness, {\sigma }_3^{\prime } is the minor principal effective stress, c is the cohesion, and φ is the internal friction angle. All stiffnesses in this model are stress-dependent. For example, the stress dependency of the initial shear modulus (G₀) is described in Eq. (3) [7]:

(3)G_0=G_0^{\mathrm{r}\mathrm{e}\mathrm{f}}{\left(\frac{c\mathrm{c}\mathrm{o}\mathrm{s}\phi -{\sigma }_3^{\prime }\mathrm{s}\mathrm{i}\mathrm{n}\phi }{c\mathrm{c}\mathrm{o}\mathrm{s}\phi +p^{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{s}\mathrm{i}\mathrm{n}\phi }\right)}^m,

where p^{\mathrm{r}\mathrm{e}\mathrm{f}} is the reference stress for stiffness, G_0^{\mathrm{r}\mathrm{e}\mathrm{f}} is the initial shear modulus related to reference confining pressure at small strains (, and m is the constant representing the power of stress-level dependency.

Tab.4 presents the detailed parameters of the RMC wall numerical model for all materials (plane strain parameters were used in the numerical modeling). Numerical modeling of GRS walls in PLAXIS 2D software consisted of considering the plane strain condition (two-dimensional analysis of problems) and using 15-noded elements. The strength and stiffness properties of soils are higher in the plane strain-based analyses than when considering the triaxial conditions due to the additional confinement of strains in one direction. Previous studies have proposed various correlations between soil parameters for the triaxial and plane strain conditions in the absence of test data (e.g., Hatami and Bathurst [9], Lade and Lee [16], Marachi et al. [17], Mohamed et al. [18]). Regarding soil modulus, Hatami and Bathurst [9] and Marachi et al. [17] indicated that the soil plane strain modulus is stiffer than the soil triaxial ones for the same material under the same minor principal stress conditions. To consider plane strain conditions and obtain corresponding parameters in the numerical modeling of RMC sand in the current study, the procedure and recommendations proposed by Mohamed et al. [18] and Hatami and Bathurst [9] were used. The strength parameters (c and φ) were derived from Hatami and Bathurst [9] calibrated with plane strain tests. The plane strain secant modulus was also deduced from the triaxial modulus multiplied by ({\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }_{\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}}{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}}}) to consider the same failure strains in triaxial and plane strain conditions [18].

Adjacent soil material properties were considered for interfaces between reinforcements and other materials, which mobilized the maximum friction angle and prevented geogrids from slipping through. Previous studies indicated that a perfect adherence between reinforcements and adjacent materials is an acceptable assumption under working stress conditions [19,20]. In addition, interactions between block-block and soil-block were modeled, using interface layers with normal and shear stiffness values recommended in previous studies [9,10,21,22] (Tab.4).

The accuracy of various aspects of numerical modeling of the RMC Wall 1 is presented in Fig.2 and Fig.3. The regenerated soil stress–strain behavior of RMC sand in triaxial tests in Fig.2(a) and 2(b) illustrates a good prediction of experimental results using the HS-small model for backfill soil, especially for confining pressures over 50 kPa and axial strains lower than 5%–6%. Fig.2(c) represents measured and predicted values of stress–strain behavior of RMC sand under the plane strain condition, which indicates good predictions of the numerical model. Predictions of vertical pressures on the foundation, normalized to the weight of soil layers at the end of construction, were in fair agreement with the measured data and the trend of foundation pressures (Fig.2(d)). Comparison between the measured and calculated values of facing displacements at the end of construction and under various surcharges in Fig.2(e) and Fig.2(f) demonstrates acceptable predictions of deformations by using the HS-small model, especially with increasing surcharge loads.

It should be noted that values reported in Fig.2(e) are based on data recorded after the reinforcement placement at each layer, and facing displacements shown in Fig.2(f) are after the end of construction. Additionally, in the case of reinforcement strain distributions at the end of construction (Fig.3), the measured and numerical results were in fair agreement in most layers. They indicated that reinforcement strains were less than 1% in all layers.

3 Finite element method simulations

A series of FEM simulations was performed to investigate the performance of GRS walls under various conditions of reinforcements (L/H, S_v, and J). Fig.4 and Tab.5 describe the descriptions of 46 short and 42 tall GRS walls modeled in the PLAXIS 2D program [7]. The detailed properties of all materials used in these models are also presented in Tab.4.

More than 71% of the 320 cases of MSE wall failures reported by Koerner and Koerner [2] were between 4 and 12 m high, and 17% and 6% were taller than 12 m and 20 m, respectively. Thus, 8 m and 20 m high GRS walls were selected to represent short and tall walls in FEM simulations. The GRS wall design guidelines proposed various allowable values for minimum L/H, including 0.7 [1,23] and 0.6 [24]. Liu and Evett [25] specified 0.8 for the minimum L/H value, but some studies showed that values less than 0.7, as low as 0.5, were possible in some cases [26–28]. Consequently, L/H values were considered from 0.3 to 1.2 for short walls and from 0.5 to 1.2 for tall ones (with 0.1 increments per step).

Additionally, the S_v was varied from 20 cm (one block height) to 120 cm for short walls and from 50 cm (one-third of the CP height) to 250 cm for tall ones. All models considered stage construction and interface properties the same as the validated model (reported in Tab.4). To study the effect of retained soil type, two scenarios were considered for the backfill and retained soils. The first scenario assigned two different soils to the reinforced and retained zones. Moreover, the same soil (Hostun sand) was considered for both reinforced and retained zones in the second scenario. The Hostun sand with a coefficient of uniformity (C_u) of 1.7 and medium grain size (D₅₀) of 0.35 mm was modeled as reinforced soil using the HS-small model. In addition, properties of retained and foundation soil constitutive models (i.e., Mohr–Coulomb model) were derived from a constructed 6 m high GRS wall in Kansas, USA [29,30] to improve the simulation of actual GRS wall conditions in numerical models. CBs are widely used as facing in short walls due to their lower costs, easy application, and availability [3]. In the case of tall walls, the facing type is limited to concrete precast panels or full rigid facing in practice to restrain deformations. Therefore, CBs with dimensions of 20 cm (height) \times 40 cm (width) in short wall models and 1.5 m high CPs with 0.5 m wide in tall ones were considered facing types.

4 Artificial neural network simulations

Previous research has proposed various applications and detailed descriptions of analyzing methods of problems based on artificial intelligence and machine learning (e.g., Anitescu et al. [31], Samaniego et al. [32]). Additionally, several machine learning methods have been implemented into geotechnical and civil engineering applications as a solution in recent years (e.g., Wang et al. [33], Dang et al. [34], Nghia−Nguyen et al. [35], Nguyen and Abdel Wahab [36], Tran et al. [37]). ANN is a machine learning mechanism and a subcategory of artificial intelligence that establishes a nonlinear mapping between inputs and outputs (e.g., Gordan et al. [38], Pramanik and Babu [39]). The ANN-based methods, such as MLP, can develop quick and practical estimation models if adequate databases are available [40].

The Simplified Method described by Allen et al. [41] for predicting maximum reinforcement loads in GRS walls has been developed based on the Rankine failure wedge, limit equilibrium analyses, and considering a triangular distribution for the maximum axial loads in reinforcement layers at backfill (T_max) against wall height. However, previous studies indicated that reinforcement stiffness had a major role in controlling reinforcement loads, especially under operating conditions (Bathurst and Naftchali [42]). Therefore, stiffness-based methods have been developed and revised over time to enhance predictions of the distribution of maximum reinforcement loads over wall height, including the K-stiffness Method (Allen et al. [43]), the Modified K-stiffness Method (Bathurst et al. [44]), and the Simplified Stiffness Method (Allen and Bathurst [45]). These methods consider nearly bi-linear (Simplified Stiffness Method, the upper portion of the graph is curved) or trapezoidal distributions (K-stiffness Method and Modified K-stiffness Method) for maximum reinforcement loads using the D_tmax factor. The D_tmax is the ratio of T_max to the maximum value of T_max in all reinforcement layers of the wall (T_max). The same procedure can be used to evaluate the distribution of connection loads (T_c) in reinforcements over wall height by considering the T_c/T_cmax factor, in which, T_cmax represents the maximum connection load in all reinforcement layers. It should be noted that the effect of the wall height parameter is only considered in the Simplified Stiffness Method.

In the current study, the MLP procedure was considered to predict the D_tmax and T_c/T_cmax values (dependent variables) based on z/H (z presents reinforcement elevation from wall base), L/H, S_v, J, and H values (independent variables) in GRS walls using SPSS software [46]. The results of FEM simulations were used as training and testing data for the MLP predictive model. Next, the validated MLP algorithm was considered to predict the D_tmax and T_c/T_cmax values for 990000 selected scenarios of reinforcement layers of GRS walls with various input parameters. Tab.6 shows the properties of these selected scenarios used for predictions of D_tmax and T_c/T_cmax values by the MLP procedure.

5 Results and discussion

5.1 Effectiveness of L/H, S_v, and J

The effectiveness of L/H, S_v, and J variations on the normalized facing horizontal displacements (FHD) over wall height is illustrated in Fig.5(a)–5(d), Fig.5(e)–5(h), and Fig.5(i)–5(l), respectively. Models 1 (short wall, various backfill-retained soils, L/H = 0.3), 73 (tall wall, various backfill-retained soils, J = 300 kN/m), and 81 (tall wall, same backfill-retained soils, J = 300 kN/m) were statically unstable and eliminated from the results. However, the deformation mode of an MSE wall is a combination of base sliding, bulging, and overturning in real field projects; the failure mechanism and predominant deformation mode of such walls depend on their relative flexibility and rigidity [4]. Reinforcing soil walls with extensible reinforcements (e.g., geogrids) instead of inextensible ones (e.g., steel strips) allows more wall deformations. In the case of flexible GRS walls, more deformations would occur at the middle parts of the wall, and an internal failure surface within the reinforced zone is usually observed. The relatively low reinforcement stiffness in the investigated models of this study led to the development of a general bulging deformation mode of facing (compared to overturning mode) in all walls in Fig.5. The intensity of this observed bulging mode was also higher in the tall GRS wall models. However, increasing the model stiffness, mainly due to increasing J values, reduced the severity of this bulging (Fig.5(i)–5(l)). Variation of J values had the most influence on normalized FHD values, especially in tall models (up to about 80% reductions), and changing S_v and L/H stood in the following ranks. The elevation of the maximum facing deformation in the tall walls was about 0.1H higher than in the short ones. Increasing J and decreasing S_v values reduced the elevation of maximum facing deformations for short models (0.5H to 0.3H) and tall ones (0.55H to 0.35 H). Considering models with the same parameters (L/H, S_v, or J) in Fig.5 demonstrated that the normalized FHD values in tall models were 2.3–3.5 times those in short ones, which indicates a notable difference between the performance of short and tall GRS walls. Hence, Fig.5 illustrates that the wall height parameter affects the wall deformations (i.e., FHD values) and controls the efficiency of the treatment methods for reducing these displacements (i.e., L/H, S_v, or J). Moreover, using the same or various backfill-retained soils had a minor effect on the facing deformations of short and tall models, as shown in Fig.5. Therefore, the effects of varying L/H, S_v, or J values dominated the effects of the parameters of retained soil in the current study.

Three parameters were defined and investigated in Fig.6–Fig.8 to evaluate different aspects of the performance of short and tall GRS walls, including D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R. The D_Max-Facing/H factor represents the maximum facing deformation normalized with wall height. The equivalent force of horizontal soil pressures on the facing and the corresponding effect point (concerning the wall base) have a significant role in the overturning stability of walls. Therefore, the M_Overturning/M_R parameter practically assesses the overturning moment around the bottom of the wall (M_Overturning), normalized with the Rankine overturning moment (M_R). Equations (4) and 5 show the overturning moment due to the horizontal soil pressures based on the Rankine theory, as follows [3]:

(4)M_{\mathrm{R}}=F_{\mathrm{R}}\times \frac{H}{3},

(5)F_{\mathrm{R}}=0.5K_{\mathrm{a}}\times \gamma \times H^2,

where F_R is the soil equivalent horizontal force toward facing based on the Rankine theory, and K_a is the active earth pressure coefficient. Furthermore, the F_Max/F_R factor presents the maximum axial force of reinforcements normalized with the Rankine soil equivalent force. It should be noted that the application of the Rankine theory parameters (i.e., M_R and F_R) is limited to the assumption of a rigid block wall, which is not the case in the too-flexible walls (e.g., GRS walls). Therefore, the only reason for using M_R and F_R here is to normalize moments and forces derived from numerical models with well-known and practical parameters.

Fig.6-Fig.8 illustrate the effectiveness of reinforcement conditions on various aspects of short and tall models. In addition, all graphs in these figures are plotted with the same scales in these figures to highlight the differences visually. Fig.6 shows the effect of L/H variations on the D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R values. Increasing L/H up to 1.2 decreased the D_Max-Facing/H values by 13% in tall models (Fig.6(d)), which was less than half of the reductions observed in short ones (27%–32% in Fig.6(a)). Furthermore, the majority of these reductions occurred due to increasing L/H up to 1 and further increases in L/H values had a negligible effect. As a result, the length of reinforcements should at least be limited to an optimum value. The trends of M_Overturning/M_R and F_Max/F_R graphs were slightly descending (Fig.6(b) and 6(e)) and nearly constant (Fig.6(c) and 6(f)) with variations in L/H values, respectively. Hence, increasing reinforcement length, especially in models with L/H values of more than 0.5, cannot be a practical method to control either M_Overturning/M_R or F_Max/F_R values in short and tall walls. Using the same or various backfill-retained soils in all models had a negligible effect on the M_Overturning/M_R and F_Max/F_R graphs (less than a 5% discrepancy).

The effect of S_v variations on the D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R parameters is presented in Fig.7. Decreasing S_v reduced D_Max-Facing/H values in both short (45%–52% in Fig.7(a)) and tall models (67% in Fig.7(d)) due to increasing the global stiffness of models (which is the summation of all reinforcement stiffness normalized with wall height: S_{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}={\displaystyle \frac{{\sum }_1^n{J}_i}{H}} , where J_i is the reinforcement stiffness of layer i). Furthermore, a decrease in facing deformation might lead to more soil horizontal pressure toward the wall due to getting away from the active earth pressure mode and consequently increasing the soil horizontal pressure coefficient. Therefore, reductions in facing deformations and stiffer soil mass (increasing S_global) led to increases in M_Overturning/M_R values in both short (14% in Fig.7(b)) and tall walls (22%–26% in Fig.7(e)). The number of geogrid layers increased during decreasing S_v values. Thus, soil pressure was tolerated by more reinforcements, and as a result, the F_Max/F_R values were reduced by about 70% in all models (Fig.7(c) and 7(f)). Moreover, the retained soil type had no significant impact on the graphs in Fig.7. Generally, in short GRS walls, the effectiveness of decreasing S_v on D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R values was more intense than the effectiveness of increasing L/H values (about 18%–20%, 14%, and 50%–70%, respectively). In the case of tall models, this discrepancy in effectiveness continued for D_Max-Facing/H (54%), M_Overturning/M_R (22%–26%), and F_Max/F_R (71%) values. Hence, considering two approaches to enhancing the performance of short and tall GRS walls, including increasing L/H and decreasing S_v, demonstrated that reducing S_v is much more effective regarding either wall deformations or maximum reinforcement loads.

Fig.8 demonstrates the effect of J variations on the D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R factors. The D_Max-Facing/H values severely decreased (about 74%) with increasing J up to 4000 kN/m in both short and tall models (Fig.8(a) and 8(d)), while further increase in J values had minor effects (about 6% reductions). As expected, the reductions in D_Max-Facing/H resulted in the ascending trends of M_Overturning/M_R values (about 26%) in both graphs for short and tall models (Fig.8(b) and 8(e)). Consequently, increasing overturning moments toward facing and S_global resulted in 103% and 40% increases in F_Max/F_R values of short (Fig.8(c)) and tall models (Fig.8(f)), respectively. Note that the elevation of F_Max decreased from about 0.45H to 0.25H due to increasing J up to 2000 kN/m in all models and remained constant with further increases in J values. Moreover, using the same or various backfill-retained soils had a minor influence (less than 5%) on the investigated parameters in Fig.8.

Tab.7 presents a summary of the change percentage of D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R values (relative to the results of the first model) due to changing L/H, S_v, and J to illustrate the effectiveness of the investigated parameters. To compare all results properly, unstable models were eliminated, and the same range was considered for variations in L/H, S_v, and J values in Tab.7. Increasing J was the most effective method to reduce D_Max-Facing/H in short and tall walls, considering increasing L/H or decreasing S_v. Considering models with similar L/H, S_v, and J in Fig.6–Fig.8 illustrated that D_Max-Facing/H, M_Overturning/M_R, and F_Max/F_R values in tall GRS walls were about 2.3–4.25, 0.9–0.95, and 0.5–0.85 times those values in short ones, respectively. Therefore, FEM simulations illustrate significant differences between the behavior of short and tall GRS walls due to variations of L/H, S_v, and J values. The effectiveness of L/H, S_v, and J variations differed in short and tall-investigated models. Therefore, comparing the normalized results of this section sheds light on the necessity of implementing the wall height parameter in all aspects of the design processes of GRS walls in worldwide guidelines.

5.2 Distribution of maximum reinforcement loads

Fig.9 presents variations of the D_tmax (i.e., T_max/T_mxmx) and T_c/T_cmax factors against z/H in short and tall GRS wall models using the results of FEM simulations (552 data). To consider closer results to the actual field data, only models with various backfill-retained soils have been considered in Fig.9. Comparing normalized results of short and tall models in Fig.9 indicated that D_tmax had higher values of up to 20% in the upper parts (z/H > 0.6) of tall walls. In addition, tall models had more T_c/T_cmax values in the z/H > 0.4 than short ones. Therefore, reinforcements of upper layers are more involved in tolerating loads by increasing the wall height. Predictions of various design methods (Simplified Method, K-stiffness Method, Modified K-stiffness Method, and Simplified Stiffness Method) and corresponding parameters are also illustrated in the graphs. The Simplified Method and Simplified Stiffness Method are currently acceptable for GRS wall design purposes in AASHTO [1]. Although the Simplified Stiffness Method is a revised version of previous stiffness-based methods, this paper evaluates the accuracy of all mentioned methods to compare their performance.

In Fig.9(a) and Fig.9(b), D_tmax decreased at the bottom of all short and tall models (z/H ≤ 0.2), and maximum values occurred at nearly 0.15 ≤ z/H ≤ 0.3. Therefore, the D_tmax distribution is not triangular, as is considered in the AASHTO Simplified Method. This issue could be critical in elevations in which predictions of the Simplified Method for D_tmax and T_c/T_cmax values underestimate the measured values and result in unconservatism, including middle (0.2 < z/H < 0.6) and nearly whole parts (z/H > 0.2) of the wall for D_tmax and T_c/T_cmax, respectively. The prediction of the Simplified Stiffness Method for D_tmax and z/H > 0.2 was better in tall models (Fig.9), for the Z_b/H parameter (Z_b is the depth from the top wall where D_tmax first become 1 (Allen and Bathurst [45])). However, due to less complexity and more conservatism in the elevations close to the wall toe, this method considers a constant linear distribution for geogrid loads of bottom layers (with z/H < 1 − Z_b/H ), and thus maximum D_tmax values (equals to 1) continue up to the bottom of the walls. Moreover, the predictions of the K-stiffness Method and Modified K-stiffness Method were so over-conservative for z/H > 0.35 in all models (Fig.9(a) and Fig.9(b)) and the Simplified Stiffness method reduced this conservatism, especially in tall models. In the case of T_c/T_cmax, the Simplified Stiffness Method had a good prediction for z/H > 0.2 in short walls (Fig.9(c)). Furthermore, the K-stiffness Method also had acceptable predictions in both short and tall walls (Fig.9(c) and 9(d)). Therefore, although the predictions of stiffness-based methods are restricted to predict the T_max at locations away from the connections (Allen and Bathurst [45]), these methods could also capture the general trend of the connection load distribution (i.e., T_c/T_cmax).

In general, the AASHTO Simplified Method could not predict maximum reinforcement load distributions in both short and tall GRS walls, either at backfill or connections, and was even non-conservative in some parts. The K-stiffness and Modified K-stiffness Methods were over-conservative for predicting D_tmax. The Simplified Stiffness Method had the closest predictions of D_tmax for z/H > 0.2 with some conservatism. In addition, this method considered the effect of the wall height parameter on the D_tmax graphs. Moreover, the stiffness methods could also be used to predict T_c/T_cmax values with acceptable errors.

To consider many more GRS wall cases and expand the results of reinforcement load distributions for walls with various characteristics, an artificial intelligence-based method has been used in the current study. Hence, all predictions of D_tmax and T_c/T_cmax through FEM simulations (552 data) were used to set up an MLP procedure. About 70% and 30% of this FEM-based data set were dedicated to the training and testing process of the MLP algorithm, respectively. A feedforward architecture (connections in the network flow forward from the input layer to the output layer) was considered for the neural network structure. Moreover, the backpropagation algorithm was used for training the network (i.e., the weights of the neurons in the network are adjusted to minimize the difference between the predicted and actual output values). The automatic option in SPSS was enabled to optimize the number of neurons in the hidden layer. This option uses a cascade-correlation algorithm that iteratively adds hidden neurons one at a time until it achieves a desired level of accuracy on the testing data set. The automatic process determined proper activation functions based on the nature of the dependent variables (e.g., continuous numerical values), the number of neurons in each layer, and the type of problem (e.g., nonlinear regression). Therefore, the hyperbolic tangent function and the identity function were considered activation functions in hidden and output layers, respectively, as follows:

(6)\mathrm{H}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}},

(7)\text{Identity function:}f(x)=x,

where x is the weighted sum of the input values and the bias term for a given neuron. The structure of the considered MLP network with one hidden layer is shown in Fig.10.

In the validation process of an MLP procedure, the AORE is a normalized measure of the error, expressed as the average of the relative error values for all the observations in the data set. In addition, the relative error is calculated by dividing the absolute difference between the actual and predicted values by the actual value. The AORE of the considered MLP procedure is 2.5% for training and 2.8% for testing data sets. Thus, the MLP method well predicted the results of FEM simulations for D_tmax and T_c/T_cmax values, as can be seen in Fig.11.

The generated MLP algorithm was used to predict the D_tmax and T_c/T_cmax values in 990000 cases of GRS walls with various reinforcement conditions described in Tab.6. Fig.12 presents all predictions of D_tmax and T_c/T_cmax using ANN (990000 data) and FEM (552 data) simulations. The maximum D_tmax values occurred in the range of 0.15–0.3 of z/H (Fig.12(a)), while the T_c/T_cmax had maximum values in 0.15–0.55 of z/H (Fig.12(b)). Furthermore, T_c/T_cmax had greater values than D_tmax in the upper half of walls (z/H > 0.5) and fewer values in the lower elevations (z/H < 0.1). Finally, the recommended lower and upper limits of D_tmax and T_c/T_cmax values are illustrated in Fig.12. The results of ANN simulations indicate that increasing D_tmax and T_c/T_cmax values to 1 is unnecessary for z/H ≤ 0.15. Note that the proposed lower and upper limits are only recommended for the specific types of GRS walls and reinforcement properties investigated in the current study (Tab.6).

6 Conclusions

To date only a limited number of studies have investigated tall GRS walls, a conspicuous number of comprehensive FEM and ANN analyses of short and tall GRS wall models were conducted in the current study. Furthermore, the contrasts of short and tall GRS wall performances and the efficiency and priority of improvement methods for controlling such wall deformations and reinforcement loads were evaluated and reported. The accuracy and deficiencies of predictions of the AASHTO Simplified Method and stiffness-based methods for reinforcement load distributions at backfill and connections were also assessed and compared. In addition, an MLP algorithm was well-trained, using FEM results to predict these distributions for various GRS wall cases based on over 990500 simulated data. The significant outcomes of this research study can be listed as the following.

1) Facing deformation mode was general bulging of the facing (neither localized bulging between layers nor overall overturning mode) in all short and tall models due to the relatively low stiffness of geogrids and low S_{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}} values. This was more severe in tall walls. However, increasing model stiffness led to reductions in the intensity of bulging. Formation of the internal failure surface and corresponding bulging mode resulted in the occurrence of maximum values of D_tmax and T_c/T_cmax in the middle parts of the walls (0.15–0.3 of z/H and 0.15–0.55 of z/H, respectively).

2) Increasing J was the unequivocally most effective method (80% reduction) for reducing D_Max-Facing/H in all groups of models (short, tall, same, and various backfill-retained soils), making it superior to the alternatives of decreasing S_v (45%–67% reduction) or increasing L/H (13%–27% reduction). Hence, adopting J variations as the preferred approach for enhancing GRS wall performances could bring an avalanche of benefits regarding the control of wall deformations. Note that increasing L/H was more effective for short walls, while decreasing S_v affected tall walls more.

3) A decrease in facing deformations due to variations in S_v and J resulted in increasing M_Overturning/M_R values (14%–28%). Hence, increasing J increased F_Max/F_R values in short (about 100%) and tall (40%) models. However, decreasing S_v and using more geogrid layers in models decreased F_Max/F_R values in all short and tall walls (70%). Additionally, the variations of F_Max/F_R due to increasing L/H values were negligible (less than 5%).

4) Conspicuous discrepancies between the deformations, overturning moments, and reinforcement loads of short and tall GRS walls due to variations of L/H, S_v, and J values demonstrated remarkable differences between the performance of such walls. Therefore, it is recommended that the application of design methods that do not include the wall height parameter in the design procedures of the GRS wall be reconsidered. In particular, using the AASHTO Simplified Method might exert profound repercussions.

5) D_tmax and T_c/T_cmax distributions were nearly trapezoidal and different for short and tall walls. Worryingly, the AASHTO Simplified Method could not predict D_tmax and T_c/T_cmax values in both short and tall GRS walls and was even non-conservative in some parts. However, the Simplified Stiffness Method outperformed other methods in predicting D_tmax when z/H > 0.2, with the lowest errors and it properly considered the wall height parameter. Additionally, this method could be used for anticipating the T_c/T_cmax distribution, especially in the context of short walls.

6) Artificial intelligence was used to overcome the difficulties in numerical modeling of GRS walls with a wide range of input parameters. Hence, over 10000 various scenarios of short and tall GRS walls with different H, L/H, S_v, and J values were generated and implemented into an MLP algorithm (which was trained with 552 FEM results). The AORE of 2.5% and 2.8% were achieved for training and testing data sets, respectively. The resulting model comprehensively understood GRS wall behavior under various conditions. Based on the results of ANN simulations, it was evident that there is no need to increase D_tmax and T_c/T_cmax to the maximum values for the bottom of the walls (z/H ≤ 0.15). Finally, to improve the predictions of reinforcement load distributions in the literature, practical upper and lower limits are proposed for the D_tmax and T_c/T_cmax values in short to tall GRS walls (4 to 31 m) with various characteristics mentioned in the paper, based on 990000 ANN and 552 FEM predictions.

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