Unconfined compressive strength prediction of soils stabilized using artificial neural networks and support vector machines

Abstract

This study aims to improve the unconfined compressive strength of soils using additives as well as by predicting the strength behavior of stabilized soils using two artificial-intelligence-based models. The soils used in this study are stabilized using various combinations of cement, lime, and rice husk ash. To predict the results of unconfined compressive strength tests conducted on soils, a comprehensive laboratory dataset comprising 137 soil specimens treated with different combinations of cement, lime, and rice husk ash is used. Two artificial-intelligence-based models including artificial neural networks and support vector machines are used comparatively to predict the strength characteristics of soils treated with cement, lime, and rice husk ash under different conditions. The suggested models predicted the unconfined compressive strength of soils accurately and can be introduced as reliable predictive models in geotechnical engineering. This study demonstrates the better performance of support vector machines in predicting the strength of the investigated soils compared with artificial neural networks. The type of kernel function used in support vector machine models contributed positively to the performance of the proposed models. Moreover, based on sensitivity analysis results, it is discovered that cement and lime contents impose more prominent effects on the unconfined compressive strength values of the investigated soils compared with the other parameters.

Graphical abstract

Keywords

unconfined compressive strength / artificial neural network / support vector machine / predictive models / regression

Introduction

The construction of soft soils requires soil improvement. One reliable method for ground modification is soil stabilization using cementitious materials such as cement and lime. In the presence of cement and lime, pozzolanic reactions can improve the mechanical properties of soils [1]. It is noteworthy that cement and lime are energy-intensive products that can increase the material cost for ground modification. Furthermore, the use of agricultural waste as a cementation material has been proposed to address environmental and economic concerns [2]. A vast proportion of agricultural lands constitutes paddy farms.

In terms of weight, 72% of the product is rice. Additionally, 20%–22% and 5%–8% of the cultivation are husk and bran, respectively. Therefore, a large volume of rice husks is available [3]. If rice husk is burned at an appropriate temperature, the achieved rice husk ash (RHA) will have a significant silica content. As a type of pozzolan, RHA can be used together with lime and cement mixtures to improve soil behavior. Pozzolanic stabilizers bind soil particles to each other, resulting in the reduced water absorption capacity of clay particles [4].

Generally, extensive and time-consuming experimental studies are required to evaluate the effect of different cementitious materials on soil behavior. Therefore, intelligence-based techniques such as artificial neural networks (ANNs) or support vector machines (SVMs) have been utilized to determine soil stiffness and strength [5–7]. Other methods, including multivariate adaptive regression spline models, have also been used to estimate maximum dry density and unconfined compressive strength (UCS) values [8]. Regarding stiffness, He and Li [9] used an ANN to demonstrate that the shear modulus can be estimated reasonably using shear strain, confining pressure, curing time, and fiber and lime contents. With regard to strength, Gunaydin et al. [10] reported that the ANN performed better in estimating the UCS of soils compared with traditional regression or statistical models. They discovered that the flexibility and adaptability of the ANN model in generalizing data is its major advantage over multivariable regression models in predicting UCS values. Furthermore, ANNs have been used to predict the UCS values of a mixture comprising cement, clay, and peat [11,12]; soil mixed with air-foam and fishing net waste [13], bottom ash, jute, and steel fibers; as well as geopolymer stabilized soils [14,15]; micro silica-lime stabilized sand [16]; and liquefiable soil [17]. In general, the multilayer ANN architecture has demonstrated better performance in predicting the UCS values of two types of fat clay and one type of lean clay stabilized with various proportions of soil/water/cement in several curing times compared with the radial basis function or genetic algorithms. Furthermore, it has been demonstrated that strength predicted by models using the ANN is more representative than the stiffness predicted by such models [12]. As mentioned earlier, the SVM, which is dependent on a predetermined kernel function [18], has been used successfully to predict the UCS values of soils [19,20], jet grouting columns and volcanic rocks [21,22], slope reliability [23], bearing capacity of piles [24], soil permeability [25], and liquefaction [26–28]. A more detailed explanation will be provided in the following sections. It is noteworthy that these methods have not been used comparatively for treated soils. This study was conducted to predict the UCS values of silty sand and high plasticity silt treated with 5%–15% cement, lime, and rice husk ash (CLR) mixtures. To determine the UCS improvement rate, the curing time was varied from 7 to 60 d. Moreover, a set of treated samples without RHA was tested to identify its effect on soil strength. Two types of ANN architectures and an SVM method using two kernel functions were adopted to predict and compare the UCS values of the treated samples.

Materials and methods

Materials

The database used in this investigation was obtained from previous experimental studies [4], where two types of locally available soils were used. One type of soil was collected from Matang Kerat Telunjuk, Bandar Baharu in Kedah, and the other was collected from Nibong Tebal, USM Engineering campus, both in Malaysia. The gradation curves of the investigated soils are plotted in Fig. 1. The geotechnical properties of the soil samples determined based on the ASTM standard tests are summarized in Table 1.

The cement and lime used in this study were ordinary Portland cement and hydrated high-calcium lime (Ca(OH)₂), respectively. Moreover, the ash used in this study was procured by burning rice husks in a gas furnace. RHA demonstrates more prominent pozzolanic properties if incinerated under controlled conditions. The RHA used in this study contained as much as 90% silica, indicating the appropriate pozzolanic properties of this type of additive. In this study, the effects of various proportions of CLR on the strength of two types of soils under different conditions, including the dry unit weights and cutting times, were investigated.

To evaluate the mechanical behavior of CLR-treated soil (Table 2), UCS tests were conducted according to ASTM D2166.

Predictive model development

Database

To determine the strength characteristics of CLR-treated soil specimens under different conditions, soft computing techniques such as ANNs and SVMs were adopted. The data used to estimate the UCS values of CLR-treated soils were obtained by conducting UCS tests on 137 specimens (Appendix A).

ANNs

To mathematically simulate the brain learning method using mesh interconnection among cells, ANNs were used. ANNs are mathematical tools that attempt to imitate the neural network of the human brain as well as the nervous system. These tools can define the interactive effects of different variables in a complicated process. Their design is based on the neural structure of the brain, in which living neurons and dendrites are represented by silicon and wires. The brain learns from experience [29] and stores information in a systematic pattern. The process of storing information as patterns and using them in solving problems are imitated in the field of computational mechanics. In a neural network, a neuron can be considered as the fundamental processing element. A biological neuron receives inputs from other sources. Generally, after combining these inputs, a nonlinear operation is conducted on the result, and the final output is provided. Similar to biological neurons of the human brain, ANNs comprise a large number of nodes. Neurons are connected by links and interact with each other. The nodes representing neurons receive data as input and perform simple operations on them. Subsequently, they pass the outcome to the other nodes. The output determined at each node is known as its activation or node value. Feedforward and feedback are two types of ANN topologies. The efficiency of an ANN in predicting new events based on previous history is affected by their architecture, learning algorithm, and robustness. Furthermore, ANNs have been used for function approximation, regression analysis, classification, data processing, robotics, and control [29]. In typical deep learning applications, neural networks are developed based on a number of inputs and outputs to minimize the difference between predicted and real values. This process, which is typically performed manually, requires a large number of datasets and is often susceptible to different types of errors [30]. Hence, a deep neural network was introduced for the bending analysis of Kirchoff plates exhibiting various shapes and subjected to different loads and boundary conditions. This technique is regarded as “mesh free” and can be employed to estimate continuous functions. Therefore, it is considered an appropriate method to analyze thin-plate bending problems [31].

In this study, ANN models based on feedforward multilayer perceptron neural networks [12] were adopted. Among the abovementioned experimental data, 60% was randomly employed for training, 20% was randomly employed for validation, and the remainder was used to test the proposed models. To perform the training, validating, and testing process, a computer program was developed in MATLAB. Generally, two methods can be used to avoid overfitting a model. This problem can be avoided by training the model using more data or by altering the complexity of the network. In this study, a sufficient input sample was provided to evade the overfitting problem. Subsequently, the appropriate network structure was selected to avoid pruning. As a typical approach in artificial intelligence, the “early stopping” method is used to address the overfitting problem. In this regard, the number of epochs increases when the network error for the test data (validation set) decreases. By increasing the number of epochs beyond a certain level, the neural network error will no longer decrease, demonstrating the overfitting problem. In this study, this problem was avoided by investigating different numbers of epochs and their corresponding neural network errors. Moreover, it has been established that the underfitting problem will not occur owing to the number of used data and the proposed architecture of the neural network.

In these proposed models, soil type, curing time, dry unit weight (γ_d), cement, RHA, and lime contents were considered as input parameters, whereas the UCS value was used as the output. To achieve the best predictive model, an appropriate ANN architecture must be selected. Two ANN models, ANN1 and ANN2, were selected to predict the UCS values of CLR-treated soil specimens. The architectures of ANN1 and ANN2 are illustrated in Figs. 2 and 3, respectively. As shown, the ANN1 model was constructed using seven neurons in a single hidden layer, whereas the ANN2 model comprised two hidden layers with four neurons in each layer. Table 3 shows the characteristics of the ANN models used in this investigation. In these figures, W and b represent the weight and bias, respectively.

The performance of the developed ANN models was evaluated based on the average of absolute percentage error (AAPE) and correlation coefficient (R). These indices are provided in Eqs. (1) and (2), respectively.

(1)AAPE=\frac{{\displaystyle \sum _{i=1}^n\frac{\left|{(x_{\mathrm{t}})}_i-{(x_{\mathrm{p}})}_i\right|}{{(x_t)}_i}}}{n}\times 100,

(2)R=\frac{{\displaystyle \sum _{i=1}^n\left[{\left(x_{\mathrm{t}}\right)}_i-{\left({\overline{x}}_{\mathrm{t}}\right)}_i\right]}\left[{\left(x_{\mathrm{p}}\right)}_i-{\left({\overline{x}}_{\mathrm{p}}\right)}_i\right]}{\sqrt{{\displaystyle \sum _{i=1}^n{\left[{\left(x_{\mathrm{t}}\right)}_i-{\left({\overline{x}}_{\mathrm{t}}\right)}_i\right]}^2}{\left[{\left(x_{\mathrm{p}}\right)}_i-{\left({\overline{x}}_{\mathrm{p}}\right)}_i\right]}^2}},

where x_{\mathrm{t}}and x_{\mathrm{p}}are the target and ANN predicted results, respectively; {\overline{x}}_{\mathrm{t}} and {\overline{x}}_{\mathrm{p}} are the means of the target and predicted values, respectively; n is the number of data points.

To evaluate the generalizing capacity of the ANN, information criteria (such as the Akaike information criterion (AIC) and Bayesian information criterion) are used to determine the “insample” fit (i.e., fit to the training data) and penalize the complexity of the models. Nonetheless, the best criterion to determine the most appropriate ANN model is yet to be elucidated. According to the literature, the weighted information criterion is used more frequently than the root mean square error. Furthermore, the AIC is the most popular method used in linear and nonlinear models. In the proposed ANN models, the AIC was introduced as an evaluation index. The popular form of the AIC is as follows:

(3)AIC=\log \left(\frac{{\displaystyle \sum _{i=1}^T{\left[{\left(x_{\mathrm{t}}\right)}_i-{\left(x_{\mathrm{p}}\right)}_i\right]}^2}}{T}\right)+\frac{2m}{T},

where T and m indicate the amount of data and the weight, respectively [32].

Support vector machine

SVM analysis, as a new type of learning algorithm, was first introduced by Vapnik et al. [33] in 1992 and then continuously used by other researchers. The SVM is a widely used machine learning tool for classification and regression analyses. It operates based on the structural risk minimization (SRM) principle. According to this principle, the complexity of the model is balanced with its success in fitting the training data [18,33]. In this method, the SRM principle is used to minimize errors, while empirical risk minimization is used in other methods such as AANs. The main objective of SRM is to minimize the empirical risk and maximize the generalizing ability of the model simultaneously [34]. The SVM can be easily analyzed mathematically because it corresponds to a linear method in a highly nonlinear input space. An SVM is a special class of algorithms. The use of kernel functions, absence of local minima, sparseness of the solution, and capacity control achieved from acting on the margin or on the number of support vectors are among its major specifications. The capacity of the system is dominated by parameters that are independent of the featured-space dimensions [35]. During classification, the data are segmented into two sets, i.e., training and testing sets. The training set comprises one target and several attributes. The SVM is primarily aimed at providing a model using the training data. Subsequently, it predicts the target value of the test data based on their attributes [36]. The support vector regression (SVR) technique is typically used in linear regression problems. Nonetheless, this method can be used in nonlinear regression problems if the training patterns are first mapped into a higher-dimensional feature space where linear regression is applicable.

SVR and the SVM use similar principles for classification. However, Vapnik et al. [33] introduced a new type of loss function. Using this function, known as the ε-insensitive loss function, the concept of margin is incorporated into SVM regression techniques. According to the basic assumption of the ε-insensitive loss function, if the error is within the range of ε, the model will not contain any prediction error. In other words, the error is acceptable if it is within the range of ε. The width of the ε-insensitive zone is determined by the parameter ε. This parameter is used to fit the training data, and its value is specified based on the number of support vectors used to establish the regression function [34]. To achieve larger values of ε, fewer support vectors are selected. In other words, larger values of ε will result in more flat approximations.

The main objective is to obtain a function f(x) with a deviation of ε from the actual target, i.e., y, using the training data. This function must be as flat as possible. A loss function with an ε- insensitive zone is expressed as shown in Eq. (4).

(4)L_{ϵ}\left(y\right)=|y-f(x){|}_{ϵ}=\{\begin{array}{ll}\mathrm{0},\hfill & |y-f(x\left)\right|\le ϵ,\hfill \\ |y-f(x\left)\right|-ϵ,\hfill & \mathrm{othewise},\hfill \end{array}

where L_{\epsilon }(y) is the loss function.

The function must have the largest ε from the target achieved for all training data. Moreover, it must be as flat as possible. Therefore, the function

(5)\phi (w,{\xi }^{*},\xi )=\left({\displaystyle \frac{{\Vert w\Vert }^2}{\mathrm{2}}}\right)+C{\displaystyle \sum _{i=\mathrm{1}}^n({\xi }_i+{\xi }_i^{*})},

is minimized, subject to

(6)\{\begin{array}{l}{y}_i-[(w{x}_i)+b]\le \epsilon +{\xi }_i^{*},\hfill \\ [(w{x}_i)+b]-y_i\le \epsilon +{\xi }_i^{*},\hspace{1em}i=\mathrm{1},\mathrm{2},\mathrm{3},\dots \dots \dots ,n\hfill \\ {\xi }_i,{\xi }_i^{*}\ge \mathrm{0},\hfill \end{array}\mathrm{.}

In this function, {\Vert w\Vert }^2 is the Euclidian norm of the weight vector, and the constant C\succ0 is the penalty parameter. In addition, \xi and {\xi }^{*} are slack variables. In the regularized form of the risk function (Eq. (5)), the first term {\displaystyle \frac{{\Vert w\Vert }^2}{\mathrm{2}}}is the structure risk and is used to control the smoothness or complexity of the function. The second term C{\displaystyle \sum }_{i=\mathrm{1}}^n({\xi }_i+{\xi }_i^{*}) indicates the empirical risk. Therefore, both terms must be minimized [34,37].

The constant ε used in the loss function and penalty parameter “C” is determined by the user. The parameter “C” controls the tradeoff between the complexity of the model and a specific tolerance that is acceptable and can be considered as a regulatory factor [37]. The use of kernel functions causes SVR to be regarded as a nonparametric method. Meanwhile, computational problems arising from the high dimensionality of the feature space can be avoided using such functions [35]. The kernel function is regarded as the dot product of the training patterns, and it is less expensive and much faster for solving computational problems. Hence, SVR can be performed in a higher-dimensional feature space without overcomputing the feature vectors in that particular space. Furthermore, using SVR in convex optimization problems results in a unique and global solution. It is noteworthy that a kernel function is required for the implicit mapping of training patterns into a feature space in nonlinear regression problems. Thus far, different types of kernel functions, including linear, polynomial, radial basis, and sigmoid ones, have been presented [37]. However, polynomial and radial basis functions demonstrated higher efficiency in geotechnical engineering [38,39].

In this investigation, SVM models using polynomial functions (SVM-poly) and Gaussian radial basis functions (SVM-RBF) were implemented in MATLAB. These two functions are expressed as shown in Eqs. (7) and (8), respectively.

(7)k\langle x_k\cdot x\rangle ={\left(\langle x_k\cdot x\rangle +1\right)}^d,

(8)k\langle x_k\cdot x\rangle =\exp \left({\displaystyle \frac{-{\Vert x_k-x\Vert }^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}}\right)\mathrm{.}

In these equations, x is the input vector; d and \sigma indicate the degree of polynomial and width of RBF functions, respectively, where both are defined by the user.

Details regarding the SVM technique are available in [33,35]. To evaluate the efficacy of the model, two statistical parameters, including the R, AAPE, and AIC were used.

In this study, to achieve the optimum SVM models, the parameter C was determined using the training and test (validation) datasets. The value of \epsilon is effective for the number of support vectors used to construct the regression function. In other words, less support vectors are selected to achieve larger values of \epsilon. Moreover, the larger values of \epsilon will result in more flat approximations. Therefore, both C and \epsilon will be effective for the complexity of the model (but in different respects). Different combinations of kernel parameters, including d, \sigma, penalty parameter C, and loss function factor \epsilon were adopted on a trial-and-error basis. Subsequently, the performance of the SVM models was determined for each combination of the abovementioned parameters. The best performance of the investigated models using different kernel functions corresponded to the optimal values of the kernel and loss function parameters, which are summarized in Table 4.

Results and discussion

The performances of the ANN1 and ANN2 models were compared, the results of which are shown in Table 5. In the ANN1 model, the values of AAPE for the training, validating, and testing datasets were determined to be 3.979, 11.598, and 12.349, respectively, whereas the corresponding values obtained by the ANN2 model were 4.394, 8.247, and 10.325, respectively. In the ANN1 model, the values of R corresponding to the training, validating, and testing datasets were 0.9998, 0.9976, and 0.9957, respectively. Meanwhile, in the ANN2 model, the values of R corresponding to the training, validating, and testing datasets were calculated to be 0.9997, 0.9952, and 0. 9979, respectively.

The AIC values for the testing datasets were determined to be 4.289 and 3.979 for the ANN1 and ANN2 models, respectively. According to the results, the values of AIC are typically lower than those of AAPE as an evaluation index in soft computing techniques. These results indicate that the ANN1 and ANN2 models can predict the UCS values both similarly and reasonably.

Figures 4 and 5 show scatter plots corresponding to the ANN1 and ANN2 models for the training and testing datasets, respectively. As shown, the points were distributed well around the line of equality, indicating that the UCS values estimated by the ANN1 and ANN2 models did not differ significantly from the experimental results.

The statistical performance of SVM models using different kernel functions was evaluated based on the values of R and AAPE (Table 6). The results indicate that the SVM- RBF model performed slightly better than the SVM-poly model. However, the difference between the R values of the training and testing datasets was not significant, demonstrating the high generalizing capacity of the developed models. Moreover, the AAPE value of the SVM-RBF, calculated for the testing datasets, was approximately 13% lower than that of the SVM-poly. Therefore, it can be concluded that the correlation coefficient and prediction error of the SVM model depended on the type of kernel function used in the model.

Figures 6 and 7 show the UCS values estimated by the SVM model using two different kernel functions as well as laboratory results used as training and testing datasets. As shown, both functions exhibited a reasonable level of precision in predicting the UCS values. However, the values predicted by the SVM-RBF model were much more consistent compared with those obtained by the SVM-poly model.

Using a large number of support vectors might signify the overfitting (low level of generalizing capacity) of the testing data [18]. In this study, the numbers of support vectors were determined to be 58 and 67 for the radial basis and polynomial kernel functions, respectively. This indicates the relative superiority of radial basis functions. Furthermore, it shows a tradeoff between the complexity of the model and its generalizing capacity. Figure 8 shows the scatter plots corresponding to the testing datasets used by the AANs and SVM. As shown, the developed models demonstrated reasonable accuracy in generalizing the testing datasets. Similar to models already developed, the use of these models is often limited and case dependent. The AAPE values of the testing datasets used by the SVM-RBF, SVM-poly, ANN2, and ANN1 models were determined to be 4.606, 5.286, 10.325, and 12.349, respectively. By comparing the values of R and AAPE obtained by the ANN and SVM models, it can be concluded that the SVM models performed better than the ANN models.

In this investigation, a whisker plot box was plotted to evaluate the error distribution of the predicted values by the ANN and SVM models corresponding to the testing datasets. Box plots, as exploratory data analysis tools, are generally used to provide the statistical summaries of the underlying prediction error distribution [40]. They are used to display the overall pattern of response in a group and can be considered as a suitable method to display characteristics such as the response range. Furthermore, the upper and lower boundaries of the whisker box represent the 0.25 and 0.75 absolute percentage errors, respectively. Moreover, the whiskers are lines extending from each end of the box to 1.5 times the interquartile range and also extending from the top and bottom of each box. In addition, the error distribution is determined by the length of the interquartile range, and the error median is displayed by a thick solid line in the box [18]. The cross symbols shown in the plot represent the values beyond the ends of the whiskers, i.e., the outlier data.

As shown in Fig. 9, the smallest length of the interquartile range was related to the ANN model, suggesting less variability in the error distribution compared with other predictive models. Furthermore, it was discovered that the SVM-RBF exhibited more variabilities in its error distribution, while exhibiting the lowest error median. The variations in the interquartile range among the ANN1, ANN2, and SVM-poly models were similar. In the SVM-poly model, the error median was located almost at the middle of the interquartile range, and no outlier data were observed. It is noteworthy that the multilayer perceptron neural network contains more outlier data than the SVM. Because the error median is less than the mean obtained by the ANN1, ANN2, and SVM-RBF models, the error distribution magnitude was reduced. In addition, the error mean obtained by SVM-poly was approximately equal to the error median, demonstrating that the error distribution was symmetric and the distribution normal. In all models, the two ends of whiskers (minimum and maximum) were unequal to the median. In general, it can be concluded that the SVM-poly performed better, with a symmetric error distribution, smaller error, and lower error variability compared with the other models. The minor difference observed in the performance of the kernel functions might be attributed to their flexibility in mapping data as well as the formation of optimal regression functions that depended on the distribution of the data in the featured space [18].

Regression model and sensitivity analysis

Multiple regression analysis is extensively used to develop models for geotechnical characteristics, such as the strength parameters of soils. In this study, the efficiencies of two artificial intelligence–based models, namely, the ANN and SVM, were compared with that of regression analysis. Therefore, a linear regression model was assumed to predict the UCS values using 137 experimental datasets (Eq. (9)).

(9)UCS=k_{\mathrm{0}}+k_{\mathrm{1}}D+k_{\mathrm{2}}T+k_{\mathrm{3}}C+k_{\mathrm{4}}L+k_{\mathrm{5}}R\mathrm{.}

In the equation above, the dry unit weight (kN/m³), curing time (d), as well as cement (%), lime (%), and rice husk ash (%) contents are independent variables denoted by D, T, C, L, and R, respectively. In addition, k₀ to k₅ are regression coefficients. Based on observations, the regression models used to estimate the UCS values (kPa) of the investigated soils were obtained using Eqs. (10)–(11).

(10)\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{M}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l}:\hspace{1em}UCS=-\mathrm{4601.940}+\mathrm{312.884}D+\mathrm{2.337}T-\mathrm{19.667}\mathrm{e}\mathrm{16}C+\mathrm{9.833}\mathrm{e}\mathrm{16}L+\mathrm{20.944}R,

(11)\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{M}\mathrm{H}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l}:\hspace{1em}UCS=-\mathrm{856.391}+\mathrm{70.594}D+\mathrm{2.299}T-\mathrm{1.618}\mathrm{e}\mathrm{16}C+\mathrm{0.809}\mathrm{e}\mathrm{16}L-\mathrm{3.74}R\mathrm{.}

Based on the results, the R values obtained by the regression models were 0.9080 and 0.8933 for SM and MH soils, respectively, which were lower than the values determined using the ANN and SVM. Therefore, the two artificial intelligence–based models are superior to regression analysis in predicting UCS values. Figures 10 and 11 show the relationship between the experimental data and the corresponding UCS values estimated using the regression model for the soils investigated.

Sensitivity analysis can be considered as a valuable method for quantitatively determining the effect of any change in input parameters on the output data. Following the determination of regression models for predicting the UCS values of soils treated with CLR, a sensitivity analysis was performed to estimate the effect of input variables on the estimated UCS value as the target parameter.

To determine the effective degree of the investigated parameters corresponding to SM and MH soil, Eqs. 12 and 13 were used. Using these equations, the sensitivity of the input variables was evaluated.

(12)N_i=f_{xi}^{\max }-f_{xi}^{\min },

(13)S_i={\displaystyle \frac{N_i}{{\displaystyle \sum _{i=\mathrm{1}}^n{N}_i}}}\times \mathrm{100},

where f_{xi}^{\max } and f_{xi}^{\min } are the highest and lowest estimated values corresponding to the ith input domain, respectively, if all other input parameters are assumed to be their mean values. Moreover, n and S_i are the number of input parameters and the effective degree of the ith parameter, respectively [41].

Table 7 shows the results of the sensitivity analysis for the suggested regression models. As shown, among the different parameters analyzed in this investigation, the cement and lime contents exerted the highest effect on the UCS. These findings agree well with those of previous investigations [4]. Although the RHA affected the UCS less compared with the cement and lime contents, it is still considered as an important parameter. This significance can be attributed to its considerable effect on problematic soils as well as its environmental benefits.

Conclusions

The aim of this study was to evaluate the strength behavior of SM and MH soils treated with CLR by conducting UCS tests. Hence, different soil samples were prepared with various CLR contents. The UCS test results indicated that adding RHA to both types of soil (SM and MH) resulted in a considerable increase in the strength of the investigated soils. However, the rate of strength increase of the CLR-treated soil specimens was higher than that of the CL-treated specimens. Using RHA as a waste material will reduce the consumption of cement and lime, thereby demonstrating its environmental and economic advantages. In addition, the feasibility of using ANN and SVM methods to estimate the UCS values of CLR-treated soils was investigated in this study. In the ANN models, different numbers of hidden layers and architectures were used. Furthermore, two types of kernel functions, i.e., polynomial and Gaussian, were used in the SVM models. In these models, soil type, dry unit weight, cement, lime, RHA content, and curing time were considered as input variables.

A database containing the results of UCS tests conducted on CLR-treated soil specimens was used to develop the models. Subsequently, the values of R and AAPE corresponding to the training and testing datasets were compared. By evaluating two popular shallow machine learning architectures, it was discovered that the SVM models performed better compared with the ANN models. Moreover, the type of kernel function used was effective in improving the accuracy of the developed models. By comparing the values of R and AAPE obtained by these models, it can be concluded that the SVM-RBF exhibited higher accuracy in predicting the UCS values. Additionally, these findings indicated that the two artificial-intelligence-based models (ANN and SVM) possessed higher capability compared with multiple regression analysis in predicting the UCS values. Moreover, based on the sensitivity analysis results obtained by statistical software, it was discovered that the cement and lime contents exerted a more prominent effect on the UCS values of the investigated soils compared with other parameters.

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