A Novel Method for Optimizing Parameters influencing the Bearing Capacity of Geosynthetic Reinforced Sand Using RSM, ANN, and Multi-objective Genetic Algorithm

One of the most prominent techniques for optimization using a quadratic regression model is the Box–Behnken design [55]. Each variable can take three levels in this method, with two of those levels representing the same extreme two factorial points in the full factorial design, while the mean value of each factor represents its third level. The Box–Behnken technique provides efficient designs for problems with less than four parameters, and the numbers of consecutive runs are comparable.

The RSM is a practical tool to determine mathematical models linking input parameters and output responses [26]. The main idea of the RSM is to approximate the output response using an explicit function of the input variables. The expression of this approximation is given below: (1) g(x)=a0+∑i=1nai.xi+∑i=1naibj.xixj+∑i=1nbixi2 g\left( x \right) = {a_0} + \sum\limits_{i = 1}^n {{a_i}.{x_i} + \sum\limits_{i = 1}^n {{a_i}{b_j}.{x_i}{x_j} + \sum\limits_{i = 1}^n {{b_i}x_i^2} } } where x_i are the input variables; n the number of input parameters; and a_i, b_i, a_ib_j are coefficients to be determined; and g(x) represents the output response.

Also, b₀ is the intercept of the mathematical model and the coefficients b₁, b₂ …, b_k and b₁₂, b₁₃, b_k−1 are the linear and interaction terms, respectively. g(x) represents the output (bearing capacity).

To check the predicted model's capability and ensure that it provides the best approximation for the treated problem, the normal probability plot must be almost within a straight line. ANOVA is also an efficient tool to determine the factors that primarily influence the response. In addition, the evaluation of the coefficients of determination (R²) and the adjusted R² also indicates the robustness of the predictive model.

The technique of ANN is a computational tool inspired by the neuron performance of the human brain and is able to find a mathematical formulation connecting the inputs data and the output results of a defined problem [56].

This technique has been used to solve complex problems due to its exceptional capacity. It is recommended to use these sorts of tools to approximate the response functions in evaluating complicated processes. ANN can be used in place of polynomial regression models to achieve better accuracy and robustness in resolving nonlinear fitting models [57, 58]. To obtain significant results, a user must consider the principal factors: network type, network architecture, and network training parameters [56, 59]. The network was designed by incrementally increasing the number of hidden layers and nodes until a suitable architecture could be found. According to Meddour et al. [59] and Kalman and Kwasny [60], the tangent hyperbolic function is employed in the hidden layers of the neural network, in which the use of this function speeds up network training compared to the sigmoidal function [61]. Big data is reserved for network training during the calculation, and the rest is utilized for the validation process.

The back propagation algorithm is used to train the network based on the descending gradient rule. This algorithm minimizes the mean square error (MSE) by introducing the input-output patterns sequentially to update weights every time. The minimization is achieved by adjusting weights from the output to the input layer [59].

To evaluate the fitness and the precision of the ANN obtained model, Ramezani and Afsari [62], Rajendra et al. [63], and, Garcia-Gimeno et al. [64]suggested four indicators, namely coefficient of determination(R²), root mean square error (RMSE), mean absolute error (MAE) and model predictive error (MPE) and, which are defined as follows: (2) R2=∑i=1n(yi,p−yi,e)∑i=1n(yi,p−yaverage)2 {{\rm{R}}^2} = {{\sum\nolimits_{{\rm{i}} = 1}^{\rm{n}} {\left( {{{\rm{y}}_{{\rm{i,p}}}} - {{\rm{y}}_{{\rm{i,e}}}}} \right)} } \over {\sum\nolimits_{{\rm{i}} = 1}^{\rm{n}} {{{\left( {{{\rm{y}}_{{\rm{i,p}}}} - {{\rm{y}}_{{\rm{average}}}}} \right)}^2}} }} (3) RMSE=∑i=1n(yi,e−yi,p)2n {\rm{RMSE}} = {{\sqrt {\sum\nolimits_{{\rm{i}} = 1}^{\rm{n}} {\left( {{{\rm{y}}_{{\rm{i,e}}}} - {{\rm{y}}_{{\rm{i,p}}}}} \right)^2} } } \over {\rm{n}}} (4) MAE1n∑i=1n|(yi,e−yi,p)| {\rm{MAE}}{1 \over {\rm{n}}}\sum\nolimits_{{\rm{i}} = 1}^{\rm{n}} {\left| {\left( {{{\rm{y}}_{{\rm{i,e}}}} - {{\rm{y}}_{{\rm{i,p}}}}} \right)} \right|} (5) MPE(%)=100n∑i=1n|(yi,e−yi,p)yi,p| {\rm{MPE(\% ) = }}{{100} \over {\rm{n}}}\sum\nolimits_{{\rm{i}} = 1}^{\rm{n}} {\left| {{{\left( {{{\rm{y}}_{{\rm{i,e}}}} - {{\rm{y}}_{{\rm{i,p}}}}} \right)} \over {{{\rm{y}}_{{\rm{i,p}}}}}}} \right|} where: n is the number of experiments; y_i,e, y_i,p are the experimental and the predicted values of the ith experiment, respectively; and y_average is the average value of experimental results.

The first step to obtain an ANN model is the selection of an appropriate architecture network. The objective is to construct a predicted ANN model by minimizing the model size and errors used during the training and validation [65]. In this study and in all cases, the learning rate adopted is 0.01. Moreover, to select the optimum number of neurons in each hidden layer, proceed by the iterations number variation until a high value of R² with a lower value of RMSE parameter is obtained.

Experimental tests were conducted in a scale model built in the laboratory, constituting of a wooden box whose walls were 20 mm thick. The front wall was made of 12 mm-thick transparent glass (Figure 1). The internal dimensions of the box were 1300 mm × 600 mm in plan and had a depth of 650 mm (Figure 2). The box was supported by a metal table directly fixed in the lab soil by pins (Figure 1). The table was firmly set in a steel frame that supported the system loading via a horizontal steel standard beam. The latter consisted of a manual hydraulic jack with a capacity of 100 kN equipped with a force reading manometer.

Figure 1:

Figure 2:

The footing was modeled by square rigid steel with a width of 100 mm and a thickness of 15 mm. A thin layer of sand was glued with epoxy glue on the footing base to ensure a certain roughness during tests.

Natural siliceous sand was used in this research work, with a nominal maximum particle size of 4 mm; it was extracted from the Oum Ali deposit located in Tebessa province (north-eastern Algeria). The specific gravity and fineness modulus of this sand were found to be 2.56 and 2.36, respectively.

Figure 3 presents the grain-size curve of the used sand. According to the United Soil Classification System, this sand is classified as poorly graded sand (SP). The values of effective sizes (D₁₀), (D₃₀) and (D₆₀) are 0.19, 0.30, and 0.45, respectively, while the uniformity coefficient (C_u)=2.368 and the coefficient of curvature (C_c)=1.053. According to the American Society for Testing and Materials (ASTM) standard, the minimum and the maximum dry densities were found to be ρ_dmin=1.606 kN/m³ and ρ_dmax=1.867 kN/m³ respectively.

Figure 3:

Two types of reinforcement, manufactured by the Algerian company Afitex Algeria were used for this study, as shown in Figure 4. The first type of geogrid was Afitex RTE 35–35–40, which was made of high-density polyethylene with an aperture size of 40 × 40 mm. The final tensile strength was 35 kN/m in the two orthogonal directions. The second type of reinforcement was a nonwoven geotextile type AS30, with a mass of 300 g/m², made from polypropylene filaments, and its final tensile strength was about 25 kN/m. The properties of the two reinforcements are listed in Table 1.

Figure 4:

A relative density of 35% was adopted for all tests corresponding to loose sand, and a dry density of 1.67 kN/m³ was obtained on using Equation 6. (6) ρd=ρd max*ρd minρdmax−Dr(ρd max−ρd min) {\rho _{\rm{d}}} = {{{\rho _{\rm{d}\max} }*{\rho _{{\rm{d}}\min }}} \over {{\rho _{\rm{d}\max}} - {\rm Dr}\left( {{\rho _{\rm{d}\max}} - {\rho _{{\rm{d}}\,\min }}} \right)}} The model box was filled with test material in six layers, approximately 100 mm thick for each layer. Light compaction was applied to each layer using a tamper to obtain the appropriate relative density. The upper surface of the sand was leveled, and special attention was given to obtain the same dry density for all six layers, and therefore a uniform sand bed with a homogeneous unit weight.

The loading system used in this study is shown in Figure 1. It comprises a hydraulic jack with 100 kN capacity, and a loading gauge for reading the applied load, which is attached directly to the reaction frame provided via a horizontal steel beam. The footing settlement was measured using two dial gauges located symmetrically at the ends of the footing. Loading was applied incrementally, and the load increment was restarted after stabilizing the settlement. This loading process was repeated until a settlement of 35 mm of the plate footing was reached. After each test, the test box was imputed and refilled with sand material for the following test at suitable reinforcement configurations.

The test results. including the reinforcement length variation (L). are presented in Figures 5 and 6 for both types of reinforcement. It is worth noticing that the bearing capacity increases with increasing reinforcement length at an embedment reinforcement depth of 0.25B; also, the increase generated with geotextile reinforcement was more effective compared to geogrid (Figure 5). However. as shown in Figure 6, from the reinforcement length L = 7.0B. no significant increase in the bearing capacity of the footing was noticed. The lengths 7.0B and 9.0B have almost the same ultimate load in comparison to the size of 5.0B, and this occurs for both types of reinforcement. This finding is consistent with previous studies[66,67,68]. which concluded that the optimal layer length of reinforced sand subjected to axial loads varies between 6.0B and 8.0B. Also, Cicek et al. [69], El Sawwaf and Nazir [70], and Abu El-Soud and Belal [71] recommended using a reinforcement length equal to 7.0B for the case of a strip footing reinforced by geogrid to obtain the maximum ultimate bearing capacity.

Figure 5:

Figure 6:

Figures 7 and 8 illustrate the variations of the bearing capacity based on the number of reinforcement layers (N) for both types of reinforcement. In addition. the number of reinforcement varies from one to three layers. It is clear from the load-settlement curves that for the case of N=3, a significant increase in the bearing capacity was observed compared to those with one and two layers. This observation is justified for both cases of reinforcement. It is interesting to note that the influence of the number of layers depends on the depth of the first reinforcement and the vertical spacing between them. These conclusions were also reported by previous studies of reinforced sand supporting strip and square footings and subjected to axial loads [72,73,74].

Figure 7:

Figure 8:

The model test results for the influence of embedment of the first reinforcement layer are presented in Figure 9 for the case of one layer of reinforcement. Figure 10 for the case of two layers, and Figure 11 for the case of three layers of reinforcement for both geogrid and geotextile reinforcement. The depth U was taken with the values of 0.25B, 0.50B, and 0.75B, with B representing the footing width.

Figure 9:

Figure 10:

Figure 11:

From the obtained results of the different configurations, it is apparent that the height values of the bearing capacity of the footing were acquired at an embedment depth of 0.25B. The depth parameter is directly influenced by the number of reinforcement layers. From Figures 10 and 11, it is worth noticing that the depth of 0.75B loses its influence with the increase in the number of layers. These results are in good agreement with many previous studies dealing with the impact of this parameter for the case of a square and circular footing [75,76,77].

The effect of vertical reinforcement spacing (X) on the bearing capacity was investigated for given values 0.5B, 0.75B, and 1.0B. Figure 12 presents the obtained results based on the load-settlement curve for the two types of reinforcements. It was noticed that the spacing of 0.5B produced the most significant effect.

Figure 12:

In addition, the variation of the bearing capacity attributed to the spacing layers variation is less significant than previously studied parameters (U and N). It is worth noting that previous researchers [69, 70, 78] found that the optimum value of the vertical spacing between layers is close to 0.4B.

A comparison between the RSM and ANN methods was made to determine the accuracy of the predictive models. In this step, some comparisons were required to evaluate the difference between experimental results and predicted values obtained by RSM and ANN models. Terms of comparison for the two predicted models are the highest coefficient of determination R² and low values of RMSE and mean prediction error (MPE). The differences between experimental and predicted responses using RSM and ANN models for both geogrid and geotextile reinforcement are presented in Figure 20. It is evident from this figure that the experimental and ANN predicted values are very close to each other compared to RSM predicted values.

Figure 20:

Indeed, coefficients R² obtained by RSM models are 0.972 for geogrid reinforcement and 0.961for the second reinforcement type, and their corresponding values, obtained using ANN models are 0.9991 and 0.9998, respectively (see Table 7). In addition, the RMSE and MPE estimated values based on ANN fitted models are more accurate than those of RSM models. The RMSE and MPE calculated values are 0.057% and 0.0414%, respectively, for the ANN model and their values for the RSM model are 0.3595% and 0.7215%, respectively, for geogrid reinforcement. Similar observations and conclusions can be made for geotextile reinforcement (Table 7). As a result and based on the above statements, the ANN models will be adopted for the optimization process later.

To better estimate the cost of reinforcement, costs of material and installation are taken into consideration. The material cost noted, cost (M), is controlled by the number (N) and the length (L) of the layer reinforcement. In contrast. the installation cost noted cost (I) would depend on the depth of the first reinforcement layer (U) and the vertical spacing between layers (X). The total cost of the reinforcements is estimated as follows: (19) Total Cost=Cost (M)+Cost(I) {\rm{Total}}\,{\rm{Cost}} = {\rm{Cost}}\,\left( {\rm{M}} \right) + {\rm{Cost}}({\rm{I}}) (20) Cost (M)=LxN.and Cost (I)=U+X*(N−1) {\rm{Cost}}\,{\rm{(M)}} = {\rm{LxN.and}}\,{\rm{Cost}}\,\left( {\rm{I}} \right) = {\rm{U}} + {\rm{X}}*\left( {{\rm{N}} - 1} \right)

The total cost can be estimated using the equation (21) Total Cost=LxN+U+X*(N−1) {\rm{Total}}\,{\rm{Cost}} = {\rm{LxN}} + {\rm{U + X*}}\left( {{\rm{N}} - 1} \right)

The multi-objective optimization aims to find a compromise between several criteria and compute the values of the input parameters that can be brought to the optimal values of the response outputs [83]. taking into account some requirements simultaneously. Many optimization methods are available for solving both constrained and unconstrained problems. GAs are the most widely used optimization methods. Primarily, creating an arbitrarily initial population named chromosomes will be considered an initial solution, whose principal performance is evaluating the fitness function. Then. according to the obtained results, several pairs among these potential solutions are generated using evolutionary techniques such as selection, crossing, and mutation. The procedure of the resolution system may be repeated until the best solution is achieved [84] (Figure 21).

Figure 21:

The present work performs a combination of the maximization of the bearing capacity of a shallow footing (q) and, at the same time, minimization of the construction cost. It was performed with four input variables (L, N, U and X). The length (L) varied between 5.0B and 9.0B, the number of reinforcement layers between one and three, the depth of the first layer between 0.25B and 0.75B and the vertical spacing between 0.5B and 1.0B. The multiobjective GA tool is used based on the mathematical model formulated by the ANN method. The constraints used in the present optimization are displayed in Table 8.

Creating a program file containing fitness functions was performed using Matlab software. The characteristics of the multiobjective GA, such as population. crossover distribution index, mutation distribution index. crossover probability, and mutation probability, were set using the GA toolbox implemented in Matlab software as presented in Table 9.

Table 10 presents the obtained results according to GA optimization for the reinforcement parameters and the responses for the two types of reinforcement.