1 Introduction

Climate change is one of the most serious environmental problems that has been affecting the world in the last decades. In order to find a global solution to this issue, sustainable development is a must and includes waste management. Waste management not only consists of the collection, transportation, and recovery of natural resources, but it also covers the disposal of waste through recycling or reuse of the materials [1]. One of the solutions is the use of waste materials for soil improvement, which leads to the reduction of the settlement of structures, increase in the bearing capacity of the soil, reducing the voids in the soil, increasing the safety factor against slope failure, and controlling shrinkage and swelling.

There is a large amount of expanded polystyrene (EPS) waste since it cannot be degraded in nature. Globally, 350 million tons of plastic waste are produced every year, and EPS accounts for 6.26% of this plastic waste [2]. The use of expanded polystyrene beads (EPSb) for soil improvement purposes may directly reduce the amount of EPS waste in nature since soil improvement requires large amounts of material. Several researchers, such as Shirazi et al. [3], Illuri [4], Rocco [5], and Soundara and Selvakumar [6] investigate the effects of EPSb on the engineering properties of different soil types. These studies show that the addition of EPSb improves the swelling potential of the treated soils by reducing the free swell (FS) values.

Another material that is commonly discarded, glass has the lowest recycling percentage (about 4.43%) [7]. The main benefits of recycling glass are a 25% reduction in energy consumption, a 20% reduction in air pollution, an 80% reduction in mine waste, and a 50% reduction in water consumption. These values can be doubled by expanding the use of waste glass [8]. There are several studies in the literature that investigate the use of waste glass as an additive for soil improvement through laboratory tests [9,10,11,12,13,14]. These studies show that the addition of glass powder leads to an increase in the unconfined compressive strength (UCS) of the treated specimens and a decrease in the free swell (FS) values. Therefore, glass powder can be considered as a strength-swell modifier, while EPSb is a swell modifier only.

Although EPSb has the potential to improve the FS behavior of soils, there has been few research investigating the effect of both EPSb and WGP on the FS and UCS values. In the first part of this study, waste glass powder (WGP) and/or expanded polystyrene beads (EPSb) are used as additives, and their effects on the engineering properties of high plasticity clay (CH) are investigated both individually and together by performing a series of experiments. In these experiments, the liquid limit, plastic limit, plasticity index, optimum moisture content, maximum dry density, free swell, and unconfined compressive strength of the treated soil samples are obtained.

It is known that the ANN method can be widely and successfully used to estimate the mechanical properties of engineering materials [15,16,17,18,19,20,21,22,23,24,25]. Therefore, in the second part of the study, the FS and UCS of the treated soil samples are predicted by artificial neural network (ANN) methods. In addition, multiple linear regression analyses are performed to provide empirical equations for the prediction of FS and UCS. The training and testing datasets based on the test results and those obtained by relevant studies in the literature are gathered. These datasets consist of the Atterberg limits, additive amounts, and test results of untreated soils as the independent variables, as well as the FS or UCS of treated samples as dependent variables. Firstly, multiple linear regression (MLR) analyses are carried out and empirical correlations are proposed. Then, a previously developed ANN algorithm [15] is used to predict the FS and UCS of the treated samples. Finally, since the ANN models may heavily depend on the selection of datasets, a k-fold cross-validation analysis is carried out.

2 Materials

2.1 Bentonite

Bentonites, often referred to as clay rocks, are primarily composed of clay minerals. Renowned for their remarkable physical and chemical properties, they find extensive use across diverse industries. These properties encompass their crystal structure, chemical composition, high ion exchange capacity, hydration, and swelling behavior. Due to these attributes, bentonites are applicable in a broad spectrum of uses, ranging from clarification processes to applications in civil engineering [26]. The concentration of ions within these clay rocks determines the type of application that is most suitable for any given bentonite. Bentonites, in general, are divided into three main groups: sodium bentonite, calcium bentonite and sodium-calcium bentonite. Sodium bentonite exhibits the highest swelling capacity among these three types, expanding up to 10–15 times its volume. It finds primary use in the drilling industry. Sodium-calcium bentonite represents an intermediate type, with a swelling capacity of approximately 5–7 times their volume. In contrast, calcium bentonite has a lower swelling capacity, expanding only about 2–4 times its volume. It is commonly utilized in cat litter and bleaching agents [27].

In this study, calcium bentonite is used, and the present section provides the information related to the physical and chemical properties of the used soil. The soil is obtained from Eczacıbaşı Esan Industrial Raw Material Company. It is commercially available bentonite supplied in milled and packaged form. The soil has an FS value of 30.41% and is classified as fat clay (CH) according to the unified soil classification system (USCS). The chemical composition of the bentonite is determined by an X-ray fluorescence test at the Earth Science Applications and Research Centre of Ankara University. The particle size distribution and index properties of the bentonite are given in Table 1, while its chemical composition is depicted in Table 2.

Table 1 Physical properties of the bentonite
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Table 2 Chemical composition of the bentonite
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2.2 Glass powder

The disposal of waste materials—such as tires, plastics, and glass—poses a significant challenge for many countries due to climate change. Currently, only a small fraction of these materials is recycled globally, with just 10% of tires, 19.5% of plastics, and 21% of glass being reused. The substantial amount of the remaining waste is typically sent to landfills, contributing to environmental pollution. Consequently, it is crucial to develop new strategies for utilizing these landfill materials more effectively [36]. Given that glass consists the largest percentage among all waste materials, this study focuses on utilizing waste glass and examining its impact on a high plastic clayey soil. The waste glass powder is provided by Boyabat Osmanlı Cam Mozaik glass and ceramics factory. As a sieve analysis is carried out on the material, it is seen that almost all (99.7%) of the material passes through the No. 30 sieve. Therefore, it is decided to use the waste glass powder (WGP) that passes through this sieve. The results of sieve analyses, hydrometer test, and the specific gravity of the WGP are given in Table 3. Additionally, an X-ray fluorescence test is performed at the Earth Science Applications and Research Centre of Ankara University. The results of this test are shown in Table 4.

Table 3 Physical properties of the WGP
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Table 4 Chemical composition of the WGP
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2.3 Expanded polystyrene beads

Expanded polystyrene (EPS) is a petroleum-derived foamy, closed-cell thermoplastic material that is made up of 98% dry air and 2% polystyrene [37]. Even though the material is considered to be environmentally friendly and 100% recyclable, its negative impact on nature should not be ignored. EPS can accumulate in landfills and marine environments, posing risks to wildlife and ecosystems due to its slow decomposition rate and potential for leaching harmful chemicals [37]. In addition to its conventional uses in packaging and insulation, EPS has gained attention in geotechnical engineering for its capacity to improve the mechanical properties of soils. Besides, research has revealed that using EPS as an additive, particularly in bead form (EPSb) as an additive has a positive effect on the FS of expansive soils. In this study, the use of waste EPS in geotechnical applications is investigated by employing EPSb as an additive for the treatment of bentonite. The EPSb used in the tests is obtained from the Istanbul Strafor Factory. The size of the EPSb used in the treatment of soil samples is in the range of 1.00–4.75 mm as shown in Fig. 1.

Fig. 1
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Sizes of EPSb

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3 Testing and results

In this study, the effects of EPSb and WGP on the improvement of fat clay are examined separately. Then, the properties of the treated samples, when both of the waste materials are used as additives, are investigated. The bentonite samples are mixed with additives (EPS and/or WGP) at different percentages by considering the dry weight of the soil. Firstly, the index properties, optimum moisture content (OMC), and the maximum dry density (MDD) of both the treated and the untreated soil samples are determined. Next, the FS and UCS tests are performed on both samples.

3.1 Sample preparation

For the FS and UCS tests, the soil samples and additives are initially mixed with distilled water according to the OMC of related admixtures. Then, they are placed in plastic bags for at least 36 h to allow the material to moist as per ASTM 3080 [38]. The soil samples are compacted at OMC and at MDD by using a standard proctor test apparatus according to ASTM D698 [29]. The samples for the FS and UCS tests are prepared from the standard proctor mold. The abbreviation ‘BGiEj’ is used to represent treated bentonite (B) with i percent waste powdered glass (G) and j percent expanded polystyrene beads (E) as shown in Table 5. The percentage of the additives is determined by considering the dry weight of the bentonite. As an example, the specimen symbol ‘BG2E0.9’ indicates a bentonite sample containing 2% of waste glass powder and 0.9% of EPSb.

Table 5 Abbreviations used to represent the additives of bentonite
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3.2 Atterberg limits

In order to determine the percentage of the stabilizing agents, a common range is selected that varies between 0–6% and 0–2% of the dry weight of the soil sample for WGP and EPSb, respectively [3,4,5,6, 9,10,11, 13, 14, 39].

In the first part of the experiments, only WGP is used as a chemical stabilizer where the fat clay samples are mixed with it at a ratio of 2, 4, and 6% of the dry soil weight. As the percentage of WGP is increased, both liquid limits (LL) and plastic limits (PL) decline, resulting in a drop in the plasticity indexes (PI). Since glass is a non-plastic material, a decrease in the plasticity index could occur, which might be considered as an improvement. In the second part of the experiments, different EPSb percentages (0.3, 0.6, 0.9, and 2.0%) are used as additives. The result is that the PL values with increasing EPSb do not show a definite relation with the percentage values; however, it is observed that as the percentage of EPSb increases, the PI values climb slightly as well. The main consequence of this behavior might be the detachment of EPSb (especially EPSb = 2%) from the samples during the rolling phase of the PL tests (Fig. 2). The test results obtained in [40] are presented in Table 6 and Fig. 3.

Fig. 2
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Examples of detached EPSb during plastic limit tests

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Table 6 Atterberg Limits of the treated and untreated bentonite [40]
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Fig. 3
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Atterberg Limits of treated and untreated fat clay with (a) WGP only, (b) EPSb only, and (c) with WGP and 0.9% EPSb

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3.3 Standard proctor tests

The OMC and MDD values are obtained for untreated and treated bentonite specimens having various percentages of WGP and/or EPSb according to the ASTM D698-12, Method A [29]. The standard proctor, FS, and UCS tests are carried out simultaneously to determine the optimum combination of two wastes, and it is decided to investigate the use of WGP and EPSb percentages as 2, 4, and 6% and 0.3, 0.9, and 2%, respectively.

In view of the results, OMC shows a modest decrease by the addition of WGP, whereas MDD shows a slight increase as depicted in Table 7 and Fig. 4. The addition of WGP slightly reduces OMC, which stays unchanged even upon replacing the fines in the sample with coarse particles. This latter action, in turn, reduces the surface area. As a result, less water is required for compaction; what is more the inclusion of WGP increases MDD which could be attributed to a chemical reaction occurring in WGP-treated samples [13]. These findings are consistent with those of Canakci et al. [9], Fauzi et al. [10], Bilgen [11], Mujtaba, et al. [14], and Olufowobi et al. [41]. As in the studies of Rocco [5], Soundara and Selvakumar [6], and Silveria et al. [42], the OMC values decline slightly with the rise in EPS and remain constant though a decline is observed in the MDD values. The low density and limited moisture absorption of EPSb may explain this fact [42].

Table 7 OMC and MDD values obtained from standard proctor test
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Fig. 4
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Compaction curves for treated and untreated bentonite

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3.4 Free swell

The one-dimensional swell, or free swell (FS), of the treated and untreated samples is determined according to the ASTM D4546-14, Method A [34] since the tests are conducted on reconstructed samples. In the previous studies, it is shown that using glass powder as a stabilizing agent has a positive effect on unconfined compressive strength values and swelling potential, whereas the addition of EPSb reduces the free swell of the soil samples [3, 4, 11, 13, 14]. Due to this reason, firstly, the optimum percentage of EPSb is investigated by treating bentonite with only EPSb, yielding a value of 0.9. Then, the FS of the bentonite is examined for both WGP and WGP-EPSb treatment. The FS test results are shown in Table 8 and Fig. 5.

Table 8 One-dimensional swell test results
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Fig. 5
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Effects of EPSb and WGP on the FS values

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In view of these results, it can be concluded that the reduction in the FS percentage indicates an approximate 27, 20, and 26% of reduction with the usage of 0.9% EPSb, 6% WGP, and 0.9%EPSb + 4%WGP contents, respectively. Soil treated with WGP exhibits less swell, which could be attributed to the change in the water absorption of the mixture. The swelling of clay is generally influenced by the amount of water in the soil and the minerals in the clay that try to absorb water. When a granular material, such as WGP, is used as a stabilizing agent for cohesive soils, the addition of siliceous WGP may lead to tighter packing of clay particles, resulting in a drop in swell potential [14]. It can also be concluded that the amount of FS declines as the ratio of EPSb increases. When EPSb is used for treatment, it serves as an inclusion or barrier to the passage of water into the soil, thereby reducing the swelling potential [4, 6]. It is also observed that, if the bentonite is treated with high EPS (2%) or 6%WGP + 0.9%EPSb contents, the stabilizing agents may not act as a barrier and, as such, may result in a higher swell potential. Finally, it can be concluded that the optimum additive percentages to treat the bentonite used in this study are 4% and 0.9% for WGP and EPSb, respectively.

3.5 Unconfined compressive strength tests

Unconfined compressive strength (UCS) tests on the 0-day cured soil samples are performed according to ASTM D2166-16 [43]. The photographs taken before and after the tests and the test results are given in Figs. 6, 7, and Table 9. All tests are duplicated to ensure that identical samples are tested consistently. From the test results, it can be seen that the UCS increases with the addition of WGP and decreases with the addition of EPSb. These results are in agreement with the previous studies conducted on treated clays [3, 11, 13, 14]. Since the cohesion between EPSb and clay particles is less than that of clay–clay particles, the cohesion of the mixture decreases [3]. Furthermore, the increase in strength with the addition of WGP could be explained by the pozzolanic reaction. The value of UCS increases as more WGP is added until reaching a peak strength value [13]. With further increase in WGP, a decline in UCS is observed in the treated samples. This finding is in line with the previous studies conducted by İbrahim et al. [13], Mujtaba et al. [14] on fat clays, and Canakci et al. [9] on lean clay. Such a declining trend trend could be attributed to a drop in the adhesive strength between waste glass surface and clay soil [9, 13]. Hence, it is noted that the percentage of WGP should be limited to 4% in view of the test results obtained in this study.

Fig. 6
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Clay sample with EPSb before and after UCS test

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Fig. 7
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Effects of EPSb and WGP on the UCS values

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Table 9 Strength parameters of treated and untreated bentonite
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4 Predictive models for FS and UCS

The mechanical characteristics of the specimens (FS and UCS) are predicted using multiple linear regression (MLR) and artificial neural network (ANN) algorithms based on the test results obtained in this study and those reported in previous studies. In this part, the computational details are presented.

4.1 Regression analysis

Regression analysis is a statistical method for determining the relationship between one or more independent variables (predictors) and dependent (outcome) variable [44]. Multiple regression is a type of regression analysis that uses more than one independent variable to predict the dependent variable. The general equation is given in Eq. 1.

$${Y^\prime}=A+{B}_{1}{X}_{1}+ {B}_{2}{X}_{2}+\dots + {B}_{n}{X}_{n}$$
(1)

Here, \({Y^\prime}\) is the dependent variable, \(A\) is the constant value, \({X}_{s}\) are the independent variables, and \({B}_{s}\) are the coefficients assigned to each independent variable. The aim of regression analysis is to obtain the regression coefficients that bring the estimated Y' values as close to the actual Y values as possible. The existence of the statistically significant linear relationship between two variables (predictor and outcome) is controlled by the Pearson correlation coefficient (r) [44]. It is an indicator of a number between −1 and + 1 where an r value of 0.0 shows no relationship and a value that is greater than 0.2 indicates dependency [45]. Additionally, the level of statistical significance of variables is presented using a p value. The relation between the variables can be defined as statistically significant when the values of p are less than 0.05 (i.e., confidence interval at a level of 95%). Furthermore, intercorrelations (especially multicollinearity state) between the independent variables should be evaluated. The variance inflation factor (VIF), an indicator of the effects of independent variables on the regression coefficient standard error, may be used in this evaluation. The VIF values greater than 10 show a high degree of collinearity (or multicollinearity) between these variables [46]. In this study, the relationships between the characteristics of untreated samples and the FS and UCS of the treated soil samples are studied using the MLR model.

In this study, the parameters that have the greatest impact on FS and UCS are identified through statistical analysis. The index properties of the soil, EPSb, and WGP contents in percentage and the FS or UCS of the untreated soil are defined as independent variables. A number of linear regression analyses are performed, and two datasets based on the test results of this study and the results of the relevant studies [3,4,5,6, 11, 13, 14] in the literature are used to predict FS and UCS. In the first dataset, the Atterberg limits and additives are defined as predictors, while the test results of untreated samples in addition to the first dataset are also considered as independent variables in the second dataset. The training datasets consist of 36 and 32 samples in the determination of the FS and UCS, respectively. The effect of predictors on FS and UCS is shown in Table 10. In this table, EPSbp and WGPp represent the percentages of EPSb and WGP, respectively. Depending on the r values, it can be concluded that tests performed on the untreated samples (FSutr and UCSutr) are statistically significant for the prediction of FS and UCS. Then, MLR analyses are carried out. The results of these analyses with high \({R}^{2}\) values are presented in Table 11. The prediction equations are considered as functions of Atterberg limits as well as other independent variables such FSutr and UCSutr. As a result of the assessments, it is concluded that untreated test results significantly affect the FS and UCS of the treated soils. Furthermore, a modest increase in the \({R}^{2}\) values is observed if only the Atterberg limits are used as predictors. The proposed empirical correlations and the \({R}^{2}\) values for training and testing datasets obtained by linear regression analyses are given in Table 12.

Table 10 Effect of predictors on FS and UCS
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Table 11 Model summary of FS and UCS based on the training data
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Table 12 Prediction equations of the FS, UCS, and \({R}^{2}\) values for training and testing datasets
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4.2 The ANN model

Artificial neural network (ANN) is a computing method that is built on artificially simulating the operational principles of the human brain. An ANN could be trained to learn a pattern automatically using input and output data [47]. Artificial multilayer perception (MLP) neural networks are the most commonly utilized ANN model and consist of input, multiple hidden, and output layers [48]. The activation function determines a neuron's output and the output signal (\({o}_{j}\)), which is the outcome of the activation function and can be determined by Eq. 2. In this study, the Sigmoid function is used as the activation function. The input to this function is \({net}_{j}\), which is the total of bias (\({b}_{j}\)) and multiplications of scalar inputs (\({i}_{1}, {i}_{2}, \dots , {i}_{n})\) with weights (\({w}_{1j},{w}_{2j}, \dots , {w}_{nj}\)).

$${o}_{j}=f\left({net}_{j}\right)=f({i}_{1}{w}_{1j}+{i}_{2}{w}_{2j}+\dots + {i}_{n}{w}_{nj}+{b}_{j})$$
(2)

ANN iteratively adjusts the weights of neurons during the training process to reduce prediction errors. The input parameters are processed via hidden and output layers, and the output is estimated. Next, the estimated and real experimental values are compared with each other to calculate errors by using the root mean square of errors (RMSE) given in Eq. 3.

$$RMSE= \frac{\sqrt{{\sum }_{d}{\sum }_{i}\left|{t}_{ip}-{o}_{ip}\right| }}{N}$$
(3)

Here, \({t}_{ip}\) is the real experimental value and \({o}_{ip}\) is the model estimated output over all the data (d). Then, since forward and backward processes are part of the training, the error is fed back into the network and the weights and biases of the neurons are adjusted as a consequence. The effect of the error on these variables is calculated.

Equations 4 and 5 are used to update the weights, \({w}_{ij}\), and biases, \({b}_{i}^{l}\) [49].

$${w}_{i,j}^{l}={w}_{i,j}^{l}-\alpha \frac{\partial E(w,b)}{\partial {w}_{i,j}^{l}}$$
(4)
$${b}_{i}^{l}={b}_{i}^{l}-\alpha \frac{\partial E(w,b)}{\partial {b}_{i}^{l}}$$
(5)

In the above equations, E(w, b) is the error value due to weight, w, and bias, b, and \(\alpha\) is the learning rate. This process is performed numerous times in order to minimize the error and improve the network accuracy, and it is referred to as the iteration number in the training process [50]. In this model, the Levenberg–Marquardt (LM) learning algorithm is utilized which is a type of supervised learning algorithm with different learning rates and without bias terms. The LM algorithm modifies the weights to reduce the errors and obtain the closest results to the experimental outcomes. The maximum and minimum values of each input dataset are used to normalize the data by using Eq. 6.

$$x_{in} = \chi + \beta \frac{{x_i - x_{min_i } }}{{x_{max_i } - x_{min_i } }}$$
( 6)

In this equation, the ith input data are \({x}_{in}\), and the maximum and minimum values of that input data are \({x}_{{max}_{i}}\) and \({x}_{{min}_{i}}\), respectively. The constants \(\chi\) and β are taken as 0.1 and 0.8, respectively, to prevent zero values during the normalization. To achieve the lowest error, several learning rates, iterations, layers, and neurons are tested. The coefficient of correlation (R2), mean squared errors (MSE), and mean absolute percentage error (MAPE) is used to evaluate the performance of models by using Eqs. 7–9.

$${R}^{2}=1-\left(\frac{{\sum }_{i}{\left|{t}_{i}-{o}_{i}\right|}^{2}}{{\sum }_{i}{\left({t}_{i}-\overline{t }\right)}^{2}}\right)$$
(7)
$$MAPE= \frac{100}{p}{\sum }_{i}\left|\frac{{t}_{i}-{o}_{i}}{{o}_{i}}\right|$$
(8)
$$MSE=\frac{1}{p}\sum {\left|{t}_{i}-{o}_{i}\right|}^{2}$$
(9)

In Eqs. 7–9, the target value achieved by the experiments is \({t}_{i}\), the model output is \({o}_{i}\), the mean value of experimental real data is \(\overline{t },\) and the total amount of data available in the datasets is shown by p.

The satisfactory performance of ANN models makes them a popular and efficient tool for overcoming various engineering challenges. As stated earlier, the FS and UCS of treated soil samples are estimated based on index properties and test results of untreated soil samples using a previously developed ANN algorithm [15]. The flow chart of the algorithm is provided in Fig. 8.

Fig. 8
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Flow chart of the ANN algorithm

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4.2.1 Results of ANN model

Two different datasets that consist of 45 and 40 data are used to predict the FS and UCS, respectively. The datasets are generated based on the data obtained from both the literature and this experimental study. Subsequently, the data are randomly divided into training and testing datasets. The sources from which the randomly selected data are sourced and the numbers of the data points in both the testing and training datasets are as follows.

In the case of the FS dataset, comprising a total of 45 data points, ten data points are obtained from this experimental study, with eight designated for training and two for testing (Table 8). Additionally, six data points are obtained by İbrahim et al. [13], with five allocated for training and one for testing. Similarly, eight data points are sourced from Mujtaba et al. [14], with six for training and two for testing. Five data points are collected from Soundara and Selvakumar [6], with four for training and one for testing, while four data points are gathered from Rocco [5], with three for training and one for testing. Finally, twelve data points are obtained from İlluri [4], with ten for training and two for testing.

For predicting unconfined compressive strength (UCS), a dataset of 40 data points is employed, with subsets selected based on the following studies:

From the present experimental study, nine datasets are obtained, with seven for training and two for testing (Table 9). Additionally, twelve datasets are taken from Bilgen [11], with ten for training and two for testing, whereas six datasets are provided by İbrahim et al. [13], with four for training and two for testing. Eight datasets are collected from Mujtaba et al. [14], with seven for training and one for testing. Lastly, five datasets are supplied by Shirazi et al. [3], with four for training and one for testing.

For training the ANN model, 36 and 32 data out of 45 and 40—that is, 80% of the datasets—are selected for training, and the remaining 20% of the data is chosen for testing purposes. The FS and UCS of the treated soils are estimated using three different combinations of inputs:

  • Three inputs including WGPp (waste glass powder percentage), EPSbp (waste EPSb percentage), and test results performed on untreated samples (FSutr or UCSutr).

  • Five inputs including WGPp (waste glass powder percentage), EPSbp (waste EPSb percentage), LL (liquid limit), PL (plastic limit), and PI (plasticity index) of untreated samples.

  • Six inputs including WGPp (waste glass powder percentage), EPSbp (waste EPSb percentage), LL (liquid limit), PL (plastic limit), PI (plasticity index), and test results performed on untreated samples (FSutr or UCSutr).

Various combinations of input parameters, such as LL (liquid limit), PL (plastic limit), PI (plasticity index), test results conducted on untreated samples (FSutr or UCSutr), and the proportions of waste glass powder (WGPp) and waste EPS beads (EPSbp), are considered as model input parameters. Meanwhile, the model outputs, represented by FS and UCS of the treated soils, are illustrated in Fig. 9.

Fig. 9
figure 9

ANN structure used in this research

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It should be noted that the LL, PL, and PI of the soil can be determined quickly and economically, making sample supply more easily since they are the tests performed on disturbed samples. Whereas, to determine FSutr or UCSutr, undisturbed samples should be collected from the site with special equipment and, for this reason, the duration of these tests may be relatively long depending on the type of clay (e.g., FS tests duration for the bentonite used in this study is approximately 15 days).

Firstly, numerous analyses are carried out to determine the best network structure for various input combinations based on statistical performance metrics (\({R}^{2}, MSE, and MAPE\)). The best results are obtained from runs with a learning rate of 0.2 for 3 hidden layers with 5, 10, and 5 neurons. The 35,000 iterations designated for the UCS analysis involving six variables and 50,000 iterations allocated for all other analyses. The process times, testing and training the \({R}^{2}\) values mean squared errors (MSE), and mean absolute percentage error (MAPE with selected learning rates) for different iterations (ranging from 5000 to 50,000) are presented in Tables 13 and 14 for FS and UCS, respectively.

Table 13 Comparison between the ANN models of FS with six, five, and three inputs with different iteration numbers
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Table 14 Comparison between the ANN models of UCS with six, five, and three inputs with different iteration numbers
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Figures 10, 11, 12, 13, 14, 15, 16, 17, 18,19, 20, and 21 and Tables 15, 16, and 17 show the results of the ANN models with three, five, and six input variables for training and testing processes. The real experimental data and estimated values for FS and UCS are shown in Figs. 10a, 11, 12, 13, 14, and 15a and 10d, 11, 12, 13, 14, and 15d for the training process, respectively. Besides, the residual plots (Figs. 10b, 11, 12, 13, 14, and 15b and 10d, 11, 12, 13, 14, and 15d) indicate that the residuals are distributed randomly and unbiasedly throughout the training pattern in all cases. The histogram of residuals for all ANN models is produced to check the variance distribution. These findings are shown in Figs. 10c, 11, 12, 13, 14, and 15c. It is seen that the inaccuracy of estimations for all models is regularly distributed and all the histograms have symmetric bell-shaped forms, which is indicative of an accurate normality assumption [51]. When the training \({R}^{2}\) values for models with three, five, and six input variables are assessed, it can be concluded that all models are successfully trained with \({R}^{2}\ge 0.965\) (Tables 15, 16, 17).

Fig. 10
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Training results for the three-input ANN model of FS tests (Inputs: WGPp, EPSbp, and FSutr)

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Fig. 11
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Training results for the five-input ANN model of FS tests (Inputs: WGPp, EPSbp, LL, PL, and PI)

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Fig. 12
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Training results for the six-input ANN model of FS tests (Inputs: WGPp, EPSbp, LL, PL, PI, and FSutr)

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Fig. 13
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Training results for the three-input ANN model of UCS tests (Inputs: WGPp, EPSbp, and UCSutr)

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Fig. 14
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Training results for the five-input ANN model of UCS tests (Inputs: WGPp, EPSbp, LL, PL, and PI)

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Fig. 15
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Training results for the six-input ANN model of UCS tests (Inputs: WGPp, EPSbp, LL, PL, PI, and UCSutr)

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Fig. 16
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Testing results for the three-input ANN model of FS tests (Inputs: WGPp, EPSbp, and FSutr)

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Fig. 17
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Testing results for the five-input ANN model of FS tests (Inputs: WGPp, EPSbp, LL, PL, and PI)

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Fig. 18
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Testing results for the six-input ANN model of FS tests (Inputs: WGPp, EPSbp, LL, PL, PI, and FSutr)

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Fig. 19
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Testing results for the three-input ANN model of UCS tests (Inputs: WGPp, EPSbp, and UCSutr)

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Fig. 20
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Testing results for the five-input ANN model of UCS tests (Inputs: WGPp, EPSbp, LL, PL, and PI)

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Fig. 21
figure 21

Testing results for the six-input ANN model of UCS tests (Inputs: WGPp, EPSbp, LL, PL, PI, and UCSutr)

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Table 15 Statistical data obtained for training and testing processes with three-input parameters
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Table 16 Statistical data obtained for training and testing processes with five-input parameters
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Table 17 Statistical data obtained for training and testing processes with six-input parameters
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Figures 16, 17, 18, 19, 20, and 21 show a comparison between the model estimation for FS, UCS, and the obtained values from the tests in the form of fitted plot lines for testing procedures. The \({R}^{2}\) values for the testing models of three, five, and six inputs are 0.998, 0.978, and 0.995 for FS and 0.930, 0.935, and 0.934 for UCS, respectively. These results can be regarded as very satisfactory.

4.2.2 K-fold cross-validation

The performance of data-based methods may be sensitive to variations in data. Therefore, it is necessary to engender variations in experiments by selecting multiple test datasets. In order to check if the ANN estimations are dependent or not on pre-defined training and testing datasets, a threefold cross-validation technique is used. The supplied dataset is stratified and partitioned into three equal folds for this purpose. Stratification is used to obtain folds that are good representatives of the whole data. The training and testing process is repeated three times, with one of the three folds chosen as the test data and the remaining two as training data for each run [52]. The performance metrics (MSE, MAPE, and R2) for the training and testing processes are calculated for each run, and the model's predictive performance is calculated by averaging the performance values of three runs. The average performance metrics derived from threefold cross-validation are shown in Table 18. This procedure is done for each output (FS and UCS) as well as for each ANN model individually. As seen from the results, the changing training and testing data have no effect on the performance of the model, and the model is not dependent on a fixed dataset.

Table 18 Threefold cross-validation results
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4.3 Results and discussion

The prediction of the FS and UCS of the treated samples is crucial in ground improvement applications. In order to predict the empirical equations, datasets are formed based on the test result of this study and the relevant studies in the literature. Considering the results of multiple linear regression analyses, it is found that all the FSs and UCSs of the treated soils can be evaluated comprehensively when the relevant test result (\({FS}_{utr}\) or \({UCS}_{utr}\)) performed on the undisturbed samples is the only independent variable of the prediction equation. However, slightly better results can be achieved if a second variable, such as the plastic limit (PL) of the untreated sample, is added to the prediction function to estimate UCS. Finally, it can be concluded that, if the FS and UCS values of the untreated soil samples (\({FS}_{utr}\) or \({UCS}_{utr}\)) are available, the FS and UCS values of the treated soils could be predicted by using Eqs. 10 and 11 given below where the \({R}^{2}\) values are more than 0.926.

$$FS=1.591-5.965EPSbp-0.180WGPp+{0.844FS}_{utr}$$
(10)
$$UCS=-58.460-102.948EPSbp+5.889WGPp+1.203{UCS}_{utr}$$
(11)

A developed ANN model is also used to estimate the FS and UCS of treated soil samples, which can be used with or without FSutr and UCSutr depending on available data. The results show that the model slightly outperforms linear regression in the prediction of FS and UCS by using additives, Atterberg Limits, FSutr and, UCSutr (\({R}^{2}=0.996\,\,{\text{and}}\,\,0.975\)). Besides, this alternative strategy produces similar results in the estimation of FS and UCS (\({R}^{2}=0.987 {\text{and}} 0.964)\) by using limited input data, such as additives and the Atterberg Limits. Similar datasets are employed for training and testing purposes in order to compare the results obtained from the MLR and ANN algorithms. Using the testing data, the performance of the proposed equations and the trained ANN model is tested. Figures 22 and 23 show the experimentally obtained FS and UCS values from this and prior studies, as well as the results of the prediction equations (Eqs. 10 and 11) and the ANN method.

Fig. 22
figure 22

Comparison of the estimated FS values with the testing data obtained from experimental results of soils in this study and previous researches

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Fig. 23
figure 23

Comparison of the estimated UCS values with the testing data obtained from experimental results of soils in this study and previous researches

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As stated earlier, the first part of this study investigates both the individual and combined effects of WGP and/or EPSb on the engineering properties of high plasticity clay through a series of experiments. In the second part, the study focuses on utilizing the effectiveness of ANN methods in estimating the mechanical properties of the clay. It is shown that the algorithm satisfactorily predicts the FS and UCS of the treated soil samples. On top of this, empirical equations are also derived through MLR analyses. This study not only examines the individual effects of waste glass powder (WGP) and expanded polystyrene beads (EPSb) on high plasticity clay, but also investigates their combined impact—a novel approach not previously explored in the literature. While individual regression analyses and ANN models are already available specifically for each waste material in the literature, this research focuses on the effects of these methods and models when used in a combined form. Therefore, the regression and ANN models are developed in accordance with this approach, thus representing a meaningful contribution to the field.

5 Conclusion

In this study, the change in the index and the mechanical properties of treated soil samples with waste materials are studied. In the experimental part, the fat clay sample is treated with different percentages of additives (waste glass powder and/or EPS beads), and the Atterberg limits, optimum moisture content, maximum dry density, free swell, and unconfined compressive strength values are obtained. Both the liquid limits (LL) and the plastic limits (PL) decrease as the percentage of WGP addition increases, resulting in a decline in the plasticity indexes (PI). The OMC and MDD values for untreated and treated bentonite with various percentages of WGP and/or EPSb are determined. The soil mixtures are compacted at these values by using a standard proctor test apparatus; for the FS and UCS tests, the samples are extracted from the proctor mold. It is observed that the addition of EPSb reduces the FS and UCS of the soil samples; however, employing WGP as a stabilizing agent has a significant effect on the FS and UCS. In conclusion, the ideal percentage of additives is decided as 4% WGP and 0.9% EPSb for the fat clay used in this study.

In the computational part, two datasets containing 45 and 40 sets of experimental test results obtained by this study as well as reported findings in the literature are used to develop and verify the models for the estimation of FS and UCS of the treated samples, respectively. As much as 80% of this data is used for training, whereas the remaining 20% is used for testing purposes. Based on these datasets, the empirical prediction equations are developed by using MLR. The ANN models are trained to estimate the FS and UCS of the treated clays. The performances of the MLR and ANN models are also compared by using the same datasets and input variables. The results indicate that both models have the ability to predict the FS and UCS of the treated soil samples with an acceptable level of accuracy in cases where the properties of the untreated soil samples together with waste material percentage (with three inputs- EPSbp, WGPp, FSutr or UCSutr) are used. The obtained \({R}^{2}\) values of FS and UCS using the ANN model with three inputs are 0.984 and 0.965 for training, and 0.998 and 0.930 for testing, respectively. These values are reasonably close to those of MLR for FS and UCS, namely the \({R}^{2}\) values of 0.936 and 0.926 for training, as well as 0.981 and 0.928 for testing, respectively. Furthermore, the ANN models with five and six inputs are successfully implemented. It is observed that, if there are limited known input parameters, the ANN model with five inputs should be used. Furthermore, the model with six inputs can also be used in case the test results carried out on untreated soil samples are available. The numerical results indicate that the training \({R}^{2}\) values for FS and UCS are 0.968 and 0.995 and and 0.974 and 0.972 for five and six inputs, respectively. On the other hand, the testing \({R}^{2}\) values for these tests are 0.978 and 0.935 and 0.995 and 0.934, respectively. Based on these findings, it can be concluded that the performance of the ANN models in all the scenarios investigated is promising, and that this method can be used to predict FS and UCS if only limited parameters are available.

Finally, in order to assess the independency of the ANN models on the selection of datasets, a threefold cross-validation technique is used to validate the performance of the models. The results of this validation reveal that the ANN models used in this study to achieve similar results over all the employed datasets. To conclude, this methodology can be adapted to various datasets without overfitting, and both of the proposed MLR and ANN methods can successfully predict the FS and UCS of improved soils.