Predicting shear strength of quartzite using portable index tests: a machine learning approach

Abstract

Shear strength of rock, characterized by the internal friction angle (φ) and cohesion (c), is the basis for the design and stability evaluation of foundation, rock slope and underground excavation. Obtaining φ and c through traditional methods such as laboratory or on-site tests is usually time-consuming and costly. This study adopts three algorithm-optimized machine learning models: Particle Swarm optimized (PSO) Support Vector Machine (SVM), Dung Beetle optimized (DBO) Random Forest (RF) and Adaptive Moment estimation optimized (ADAM) Feedforward Neural Network (FNN) to predict φ and c. The inputs of the model are three easily obtainable indices: point load strength index I_s(50), Schmidt hammer rebound number S_rn, and ultrasonic S-wave pulse velocity V_s, and the outputs are φ and c. The training data of the model comes from the laboratory tests of 63 groups of quartzite specimens in Malaysia. This study also uses R² and four error indicators - mean absolute error (MAE), average absolute percentage error (MAPE), root mean square error (RMSE) and relative percentage error (REP) - to evaluate the performance of the model. The results show that the R² of the three models is more than 0.85, and the relative error of φ and c is basically less than 10%. Among them, the PSO-SVM model has the best predictive performance, with the highest R² and the lowest error index (within 5% relative error). DBO-RF and ADAM-FNN models also have good prediction accuracy. This shows that all three models can predict the rock shear strength under a limited amount of data based on three portable indices, saving costs for engineering projects.

1 Introduction

Rock shear strength is crucial to the stability and safety of engineering structures such as tunnels, foundations and underground openings (Shinoda et al., 2019; Zhang et al., 2022; Ma, Zuo et al., 2023; Ouyang et al., 2025). Shear strength is characterized by two parameters: the internal friction angle (φ), representing the frictional resistance between rock particles, and cohesion (c), reflecting the inherent bonding strength among particles (Singh et al., 2020). The methods for determining φ and c include direct shear tests, triaxial compression tests, and in-situ testing techniques (Muralha et al., 2014). However, these methods are often time-consuming and costly (Karev et al., 2021; Bai et al., 2022; Zhao et al., 2023).

In some emergency engineering projects, such as assessing the risk of rock explosion during underground excavation and carrying out support design, there is an urgent need for the estimation method of φ and c that can be quickly implemented on site (Dai et al., 2024a; 2024b). Therefore, researchers have turned to alternative methods such as point load tests, Schmidt hammer rebound tests, and ultrasonic pulse velocity tests. These portable tests correspond to three parameters: point load strength index I_s(50), Schmidt hammer rebound number S_rn, and ultrasonic S-wave pulse velocity V_s. These have been widely used to estimate uniaxial compressive strength (UCS) (Çobanoğlu and Çelik, 2008; Kahraman et al., 2009; Kahraman, 2014; Şahin et al., 2020; Jamshidi, 2022), Brazillian tensile strength (BTS) (Heidari et al., 2012; Rastegar et al., 2020; Munemo, 2021), and shear strength (Hajdarwish and Shakoor, 2006; Karaman et al., 2015; Zhang et al., 2020).

Machine learning techniques have been widely used to predict rock mechanical properties. Early studies mainly focused on the prediction of UCS and BTS (Beiki et al., 2013; Wang and Aladejare, 2016; Kurtulus et al., 2018; Moussas and Diamantis, 2021), while recent studies have gradually expanded to modeling φ and c (Jahed Armaghani et al., 2014; Kainthola et al., 2015; Khandelwal et al., 2018; Shen and Jimenez, 2018; Gao, Nait Amar et al., 2020; Hiba et al., 2021; Choi et al., 2022; Fathipour-Azar, 2022; Mahmoodzadeh et al., 2022; Shahani et al., 2022; Lei et al., 2025). Among various modeling methods, support vector machines (SVM), random forests (RF), and feedforward neural networks (FNN) are widely used because they can capture complex linear and nonlinear relationships (Jahed Armaghani et al., 2014; Shen and Jimenez, 2018; Hiba et al., 2021; Huang et al., 2021; Fathipour-Azar, 2022; Shahani et al., 2022; Mohammadi Behboud et al., 2023; Siddig et al., 2023).

To further improve the predictive performance of the models, this study uses three algorithms to optimize the model, including Particle Swarm optimization (PSO), Dung Beetle optimization (DBO) and Adaptive Moment estimation optimization (ADAM) (Jahed Armaghani et al., 2014; Khandelwal et al., 2018; Mohammadi Behboud et al., 2023). The effectiveness of these optimization strategies has been verified in many studies. For example, Mahmoodzadeh, Mohammadi et al. (2022) applied six metaheuristic algorithms, including PSO, to optimize the Long Short-Term Memory (LSTM) model and found that the PSO-LSTM combination achieved the best performance. Similarly, Han and Xue (2024) showed that the DBO-optimized RF model (DBO-RF) outperformed other methods in predicting φ and c. These results highlight the potential of PSO and DBO in improving the predictive ability of the models, while the efficiency of ADAM in neural network training has also been verified in related studies (Zhang, 2018).

However, most existing studies rely on large datasets, and investigations of machine learning performance under small-sample conditions remain limited. To address this gap, this study systematically develops and compares three algorithm-optimized hybrid models—PSO-SVM, DBO-RF, and ADAM-FNN—for predicting φ and c of quartzite using easily obtainable indices obtainable through portable equipment (I_s(50), S_rn, and V_s) as input. Predictive accuracy, fitting capability, and statistical performance of each model are evaluated on a small-sample dataset, providing a reliable methodology and practical guidance for rapid assessment of rock shear strength under limited-data conditions.

2 Database

2.1 Data collection and analysis

A total of 63 groups of quartzite samples were collected from metro and foundation projects in the Klang Valley, Malaysia, where Paleozoic–Mesozoic quartzite formations with local granite intrusions predominate. To ensure lithological consistency, each group was defined such that all specimens originated from the same borehole core or from adjacent boreholes at similar depths. Each group consisted of eight specimens, including one for UCS test, one for BTS test, three for direct shear tests, and three for point load index tests, on which the nondestructive tests (UPV and Schmidt hammer tests) were also performed. The test results are summarized in Table 1, and an overview of the dataset is presented in Table 2.

Prior to model construction, the linear correlations among the seven parameters were evaluated to identify appropriate combinations of input features. As illustrated by the correlation matrix in Figure 1, relatively weak linear relationships were found between φ and c with UCS and BTS (R² ranging from 0.64 to 0.70). Considering that UCS and BTS are not easily measured under field conditions, the study selected I_s(50), S_rn, and V_s as the input parameters for prediction modeling.

FIGURE 1

2.2 Data preprocessing

Given the limited sample size (N = 63), outlier values were likely to exert substantial influence on model accuracy. To address this, outliers were identified using Tukey’s interquartile range (IQR) method (Tukey, 1977). A data point was classified as an outlier if it met the condition specified in Equation 1:where x represents an independent observation, Q₁ and Q₃ denote the first and third quartiles, and IQR = Q₃ − Q₁ is the interquartile range.

A total of 3 outliers (4.76% of the dataset) were identified: I_s(50) in Group 8, c in Group 29, and S_rn in sample 55, and these outliers were replaced using the moving median imputation method. (Tukey, 1977). This method can smooth noise data and reduce the impact of abnormal data. To enhance the model’s Ability to capture nonlinear relationships, This study expands the three input characteristics (I_s(50), S_rn, and V_s) to ten, including three original features, respective square terms (I_s(50))², (S_rn)², and (V_s)², three interactive terms (I_s(50) × S_rn, I_s(50) × V_s, S_rn × V_s), and the sample median.

Subsequently, this study used a cross-verification method (LOO-CV) to divide the data set. (Hastie, 2009). Leave a cross-validation is a special form of k-fold cross-validation, and its number k is equal to the total number of samples N. The method takes a single sample as a test set, and the remaining N-1 samples participate in model training and train N times to find the best model. The hyperparameter optimization of the models in this study was nested within each LOO-CV training fold.

Following the data division, the standardization method was adopted based on the absolute deviation median (MAD) to standardize the ten characteristics (Leys et al., 2013). This method is applicable to small data sets with non-normal distribution, which can effectively reduce the impact of extreme values on feature normalization. The standardized value can be calculated using Equation 2.where Median(X) is the median of the data set, and MAD(X) is the median of absolute deviation. The data preprocessing process is shown in Figure 2.

FIGURE 2

3 Prediction models for φ and c

3.1 Selection of prediction models

This study selects three representative machine learning models: SVM, RF and FNN. These models have been widely used in geotechnical parameter prediction (Jahed Armaghani et al., 2014; Hiba et al., 2021; Mahmoodzadeh et al., 2022; Han and Xue, 2024).

SVM is a supervised learning method that can transform low-dimensional nonlinear problems into linear problems in high-dimensional space with the help of kernel functions such as radial basis function (RBF), which is especially suitable for multi-input parameter data sets. The model has a simple structure and a small number of hyperparameters, which is easy to combine with other optimization algorithms to improve the prediction performance.

RF is an integrated learning method based on the principle of decision tree and bagging (Breiman, 1996). It improves the stability and generalization ability of the model by integrating the decision tree of multiple subset training. RF is suitable for input-output relationships or problems that are difficult to clarify. To further optimize RF, this study adopts the DBO to adjust the core hyperparameters of the decision tree.

FNN is a fully connected neural network including input layer, hidden layer and output layer. It has strong nonlinear approximation ability but is more sensitive to small sample training sets. Therefore, this study adopts the ADAM optimization algorithm to adaptively adjust the learning rate during the training process, which not only improves the convergence speed but also reduces the risk of over-mitting.

The three models correspond to statistical learning, integrated learning and deep learning in the field of machine learning respectively. And three optimization algorithms were matched: PSO, DBO, and ADAM to improve the prediction performance of the model. All processes are implemented based on MATLAB software.

3.2 Establishment of prediction models

3.2.1 PSO-SVM model

The SVM model maps the input data (I_s(50), S_rn, and V_s) to a high-dimensional space through the radial basis function (RBF). The performance of SVM is mainly affected by two core hyperparameters: the penalty parameter C and the kernel width γ. Among them, C is used to balance the relationship between training error minimization and classification interval maximization, and γ determines the influence range of a single training sample in high-dimensional space.

To optimize C and γ, this study integrates the PSO algorithm into the SVM model. In PSO, each particle represents a group of candidate solutions (i.e., a group of C and γ), and the algorithm minimizes the cross-verification error by iterating the position of the particle in the search space. In this study, the particle swarm scale is set to 30 and the number of iterations is set to 50. The workflow of the PSO-SVM model is shown in Figure 3, and the details of the optimized parameters are listed in Table 3.

FIGURE 3

3.2.2 DBO-RF model

Key RF hyperparameters include the number of sampled features m, the number of trees n, and the minimum number of samples per leaf node. Selecting appropriate values for these parameters is crucial for maximizing model performance (Chen and Xue, 2022).

The optimized DBO-RF parameters are shown in Table 4, and the modeling workflow is illustrated in Figure 4.

FIGURE 4

3.2.3 ADAM-FNN model

The FNN used in this study comprised an input layer of 10 nodes, two hidden layers with 64 and 32 neurons, and an output layer with a single neuron corresponding to either φ or c. The Rectified Linear Unit (ReLU) activation function was employed in the hidden layers to enhance nonlinear learning capability, while a linear activation function was applied in the output layer for regression purposes.

This study uses the He initialization method to initialize the weight and bias (He et al., 2015), which scales the weight variance according to the number of input neurons. The ADAM algorithm integrates the advantages of the momentum method and the RMSProp algorithm, and adjusts the learning rate by using the first-order moment and second-order moment of the gradient (Kingma and Ba, 2014). The parameter update mechanism of ADAM is shown in Equations 3–6.

First-moment estimate (mean):

Second-moment estimate (variance):

Bias correction:

Parameter update:where t is the time step, g_t represents the gradient at t, η is the learning rate, and β₁ and β₂ are the exponential decay rates of the first-order moment estimate and the second-order moment estimate respectively. In this study, η = 0.001, β₁ = 0.9, and β₂ = 0.999 were adopted following the default settings recommended by Kingma and Ba (Kingma and Ba, 2014), which have been validated to ensure stable and efficient convergence. The small constant ϵ = 10^–8 is introduced in the formula to prevent division by zero errors, and θ represents the model parameters (weights and biases) to be optimized.

To prevent overfitting, early stopping is used during training: training is terminated if the validation error does not improve for 20 consecutive training epochs. In addition, a random Dropout technique is applied at a ratio of 0.3 during training to improve the model’s generalization ability. Table 5 lists the final optimization parameters used in the ADAM-FNN model, and the overall modeling process is shown in Figure 5.

FIGURE 5

4 Accuracy assessment of the models

4.1 Statistical evaluation metrics

To evaluate the predictive performance of the model, this study uses four commonly used statistical indicators: coefficient of determination (R²), mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). Their mathematical expressions are shown in Equations 7–10.where y_i and are the measured and predicted values of the ith observation, respectively; is the mean of the measured values; and n = 63 represents the total number of data observations in the dataset.

R² can explain the degree of variation of the model. MAE is the average value of the absolute value of the sample prediction error, which can reflect the prediction accuracy of the model. MAPE is the percentage of the real value of the error, which is convenient for error comparison between data sets of different scales. RMSE gives a higher weight to the large error, and can sensitively identify significant prediction deviations. The above indicators comprehensively evaluate the performance of the model from the two dimensions of absolute error and relative error. The closer r2 is to 1, the closer MAE, MAPE and RMSE are to 0, the higher the prediction accuracy and reliability of the model.

4.2 Error analysis of predictive models

4.2.1 PSO-SVM model

The model’s predictive performance can also be evaluated using the percentage relative error (REP), as shown in Figure 6. For both φ and c, the predicted values closely follow the trends of the actual values, and all relative errors are below 10%. Specifically, the REP for φ ranges from 0.34% to 2.69%, and the REP for c ranges from 0.57% to 4.68%, indicating that the model has excellent predictive accuracy and generalization ability.

FIGURE 6

Figure 7 shows the relationship between the measured values of φ and c and the predicted values of the PSO-SVM model. The R² for φ reaches 0.9476, and the R² for c reaches 0.9632, confirming a good correlation between the parameters and the model’s good fit. The narrow 95% confidence interval and the predicted interval shown in the figure are attributed to the limited range of variation in the data itself, as well as the high consistency between the measured and predicted values, further confirming the model’s stability.

FIGURE 7

4.2.2 DBO-RF model

Figure 8 presents the relative error results for φ and c predictions using the DBO-RF model. Relative errors for φ ranged from 0.43% to 4.03%, and for c from 1.05% to 9.68%. Though these are slightly higher than those achieved by PSO-SVM, all values remained under 10%, reflecting acceptable prediction accuracy.

FIGURE 8

Figure 9 shows the scatter plots of actual versus predicted values for φ and c. The R² values were 0.8979 for φ and 0.9107 for c, indicating strong, though slightly weaker, model performance compared to PSO-SVM. Most data points fell within the 95% confidence intervals, confirming reliable model fitting.

FIGURE 9

4.2.3 ADAM-FNN model

Figure 10 presents the relative error distributions for φ and c predictions. For φ, relative errors ranged from 0.36% to 3.36%, while those for c ranged from 1.32% to 9.55%. Although these errors are slightly higher than those of the PSO-SVM model, they are comparable to the DBO-RF model and remain within acceptable limits.

FIGURE 10

Figure 11 shows the actual versus predicted scatter plots for both parameters. The R² value for φ was 0.9176, and for c it was 0.9012, indicating strong predictive ability and a well-fitted model. The narrow prediction intervals and tight clustering of points along the diagonal line demonstrate the model’s reliability in capturing the relationship between input and output variables.

FIGURE 11

4.3 Comparison of model evaluation metrics

Table 6 lists the statistical metrics for each model. Figure 12A shows the results for φ. In terms of R², the PSO-SVM model has the best prediction performance, followed by the DBO-RF and ADAM-FNN models. All three have R² values exceeding 0.89, indicating good model fit. Regarding MAE, the PSO-SVM model performs best (0.6765), followed by the ADAM-FNN model (0.8466) and the DBO-RF model (0.9449), highlighting the advantage of PSO-SVM in absolute error control. PSO-SVM also outperforms the other two models in MAPE and RMSE.

FIGURE 12

Figure 12B shows the results for c. The PSO-SVM model again performs best, with the R² of 0.9632, significantly higher than the DBO-RF model (0.9107) and the ADAM-FNN model (0.9012). The DBO-RF model is slightly better than the ADAM-FNN model in MAE, MAPE and RMSE. These results indicate that the PSO-SVM model is not only better in terms of error values, but also statistically more reliable, which further supports its superiority.

During model training, I_s(50) was found to exert a significant influence on the model, while S_rn and V_s had a relatively minor impact. This is likely because the point load test involves a combined failure mechanism of tension and shear, where the tensile component is correlated with c and the shear component with φ, thus forming an intrinsic link with shear strength. In contrast, S_rn (surface hardness) and V_s (elastic wave velocity/compactness) have a weaker correlation with the shear failure mechanism, resulting in their lower influence.

To further evaluate the differences between the models, paired t-tests were conducted based on the relative errors of each sample for both φ and cohesion c, where p is the significance level, indicating the probability that the difference between models is caused by random variation. The significance levels of 0.05 and 0.01 were used. For both φ and c, the results show that the PSO-SVM model performs significantly better than the other models (p < 0.01).

The excellent performance of the PSO-SVM model is mainly attributed to two core factors: first, the SVM model has a simple structure and good robustness on small sample data; second, the PSO algorithm can efficiently complete parameter optimization in the medium-dimensional search space. These characteristics make the model achieve a good balance between generalization ability and fitting accuracy. Due to the characteristics of self-discrete splitting, the RF model may have limitations in capturing subtle nonlinear relationships; under the condition of limited sample size, the FNN model may still have a fitting phenomenon even if it is optimized by the ADAM algorithm. In general, all three models can give reliable prediction results.

In previous studies, Mahmoodzadeh et al. (2022) developed a PSO-LSTM model with R² values of 0.9983 (c) and 0.9831 (φ) using 244 sandstone datasets, while Han and Xue (2024) achieved optimal performance with the DBO-RF model (R² = 0.9989 for c, R² = 0.9952 for φ) based on 199 multi-lithology samples. Although the proposed models in this study exhibit slightly lower prediction accuracy (R² = 0.9632 for c, R² = 0.9476 for φ), they still provide valuable region-specific empirical insights for shear strength prediction of quartzite in the Klang Valley area of Malaysia, supplementing existing research focused on different rock types and geological contexts.

For engineering applications, the PSO-SVM model is suitable for projects requiring high overall precision, such as the support design of high-risk slopes. The DBO-RF model is slightly superior to the ADAM-FNN model in predicting cohesion c, while the ADAM-FNN model outperforms the DBO-RF model slightly in predicting friction angle φ. For projects with lower precision requirements, such as foundation pit engineering, the model can be selected according to specific needs.

This study has certain limitations. The dataset size is relatively small, and the models were only validated by LOO-CV without an independent external test dataset, which may compromise the persuasiveness of the models. In addition, the established models are developed for quartzite in Malaysia, thus their applicability to other rock types and geological regions outside the study area is limited. Nevertheless, these models still provide valuable experience for predicting the shear strength of quartzite within the study area. Future research will expand the sample database with more independent datasets covering diverse lithologies and geological settings, and adopt rigorous external validation methods to further enhance the applicability of the models.

5 Conclusion

References

• 1

  BaiJ.DiaoY.JiaC.LiuC.ZhangM.WangC. (2022). A review of advances in triaxial tests: instruments, test techniques and prospects. KSCE J. Civ. Eng.26 (8), 3325–3341. 10.1007/s12205-022-1345-1

  • CrossRef
  • Google Scholar
  • View reference in article

• 2

  BeikiM.MajdiA.GivshadA. D. (2013). Application of genetic programming to predict the uniaxial compressive strength and elastic modulus of carbonate rocks. Int. J. Rock Mech. Min. Sci.63, 159–169. 10.1016/j.ijrmms.2013.08.004

  • CrossRef
  • Google Scholar

• 3

  BreimanL. (1996). Bagging predictors. Mach. Learn.24 (2), 123–140. 10.1007/BF00058655

  • CrossRef
  • Google Scholar
  • View reference in article

• 4

  BreimanL. (2001). Random forests. Mach. Learn.45 (1), 5–32. 10.1023/A:1010933404324

  • CrossRef
  • Google Scholar
  • View reference in article

• 5

  ChenC.XueX. (2022). A novel hybrid intelligent model for the prediction of creep coefficients based on random forest and support vector machine. Ocean. Eng.266, 113191. 10.1016/j.oceaneng.2022.113191

  • CrossRef
  • Google Scholar
  • View reference in article

• 6

  ChoiS.JeongH.CheonD.-S. (2022). Prediction of mohr-coulomb constants of selected Korean rocks based on extreme gradient boosting method and its evaluation. KSCE J. Civ. Eng.26 (5), 2468–2477. 10.1007/s12205-022-1388-3

  • CrossRef
  • Google Scholar
  • View reference in article

• 7

  Çobanoğluİ.ÇelikS. B. (2008). Estimation of uniaxial compressive strength from point load strength, schmidt hardness and P-wave velocity. Bull. Eng. Geol. Environ.67 (4), 491–498. 10.1007/s10064-008-0158-x

  • CrossRef
  • Google Scholar
  • View reference in article

• 8

  DaiJ.GongF.HeZ.XuL. (2024a). Quantitative estimation method for the excavation-induced weakening effect of rock mass parameters in deep tunnels. Eng. Geol.330, 107416. 10.1016/j.enggeo.2024.107416

  • CrossRef
  • Google Scholar
  • View reference in article

• 9

  DaiJ.GongF.XuL. (2024b). Rockburst criterion and evaluation method for potential rockburst pit depth considering excavation damage effect. J. Rock Mech. Geotechnical Eng.16 (5), 1649–1666. 10.1016/j.jrmge.2023.08.016

  • CrossRef
  • Google Scholar
  • View reference in article

• 10

  Fathipour-AzarH. (2022). Data-oriented prediction of rocks’ mohr–Coulomb parameters. Archive Appl. Mech.92 (8), 2483–2494. 10.1007/s00419-022-02190-6

  • CrossRef
  • Google Scholar
  • View reference in article

• 11

  GaoJ.Nait AmarM.MotahariM. R.HasanipanahM.Jahed ArmaghaniD. (2020). Two novel combined systems for predicting the peak shear strength using RBFNN and meta-heuristic computing paradigms. Eng. Comput.38, 129–140. 10.1007/s00366-020-01059-y

  • CrossRef
  • Google Scholar
  • View reference in article

• 12

  HajdarwishA.ShakoorA. (2006). Predicting the Shear Strength Parameters of Mudrocks.

  • Google Scholar
  • View reference in article

• 13

  HanD.XueX. (2024). Machine learning-based prediction of shear strength parameters of rock materials. Rock Mech. Rock Eng.57 (10), 8795–8819. 10.1007/s00603-024-04012-3

  • CrossRef
  • Google Scholar
  • View reference in article

• 14

  HastieT. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer. 10.1007/978-0-387-84858-7

  • CrossRef
  • Google Scholar
  • View reference in article

• 15

  HeK.ZhangX.RenS.SunJ. (2015). Delving Deep into Rectifiers: Surpassing Human-Level Performance on Imagenet Classification. 2015 IEEE International Conference on Computer Vision ICCV. 10.1109/ICCV.2015.123

  • CrossRef
  • Google Scholar
  • View reference in article

• 16

  HeidariM.KhanlariG. R.Torabi KavehM.KargarianS. (2012). Predicting the uniaxial compressive and tensile strengths of gypsum rock by point load testing. Rock Mech. Rock Eng.45 (2), 265–273. 10.1007/s00603-011-0196-8

  • CrossRef
  • Google Scholar
  • View reference in article

• 17

  HibaM.IbrahimA. F.ElkatatnyS.AliA. (2021). Prediction of cohesion and friction angle from well-logging data using decision tree and random forest. Arabian J. Geosciences15 (1), 26. 10.1007/s12517-021-09154-0

  • CrossRef
  • Google Scholar
  • View reference in article

• 18

  HuangM.HongC.ChenJ.MaC.LiC.HuangY. (2021). Prediction of peak shear strength of rock joints based on back-propagation neural network. Int. J. Geomechanics21 (6), 04021085. 10.1061/(ASCE)GM.1943-5622.0002033

  • CrossRef
  • Google Scholar
  • View reference in article

• 19

  Jahed ArmaghaniD.HajihassaniM.Yazdani BejarbanehB.MartoA.TonnizamE. (2014). Indirect measure of shale shear strength parameters by means of rock index tests through an optimized artificial neural network. Measurement55, 487–498. 10.1016/j.measurement.2014.06.001

  • CrossRef
  • Google Scholar
  • View reference in article

• 20

  JamshidiA. (2022). A comparative study of point load index test procedures in predicting the uniaxial compressive strength of sandstones. Rock Mech. Rock Eng.55 (7), 4507–4516. 10.1007/s00603-022-02877-w

  • CrossRef
  • Google Scholar
  • View reference in article

• 21

  KahramanS. (2014). The determination of uniaxial compressive strength from point load strength for pyroclastic rocks. Eng. Geol.170, 33–42. 10.1016/j.enggeo.2013.12.009

  • CrossRef
  • Google Scholar
  • View reference in article

• 22

  KahramanS.GunaydinO.AlberM.FenerM. (2009). Evaluating the strength and deformability properties of Misis fault breccia using artificial neural networks. Expert Syst. Appl.36 (3, Part 2), 6874–6878. 10.1016/j.eswa.2008.08.002

  • CrossRef
  • Google Scholar

• 23

  KaintholaA.SinghP. K.VermaD.SinghR.SarkarK.SinghT. N. (2015). Prediction of strength parameters of himalayan rocks: a statistical and ANFIS approach. Geotechnical Geol. Eng.33 (5), 1255–1278. 10.1007/s10706-015-9899-z

  • CrossRef
  • Google Scholar
  • View reference in article

• 24

  KaramanK.CihangirF.ErcikdiB.KesimalA.DemirelS. (2015). Utilization of the Brazilian test for estimating the uniaxial compressive strength and shear strength parameters. J. South. Afr. Inst. Min. Metallurgy115 (3), 185–192. 10.17159/2411-9717/2015/v115n3a3

  • CrossRef
  • Google Scholar
  • View reference in article

• 25

  KarevV. I.KhimuliaV. V.ShevtsovN. I. (2021). Experimental studies of the deformation, destruction and filtration in rocks: a review. Mech. Solids56 (5), 613–630. 10.3103/S0025654421050125

  • CrossRef
  • Google Scholar
  • View reference in article

• 26

  KhandelwalM.MartoA.FatemiS. A.GhoroqiM.ArmaghaniD. J.SinghT. N.et al (2018). Implementing an ANN model optimized by genetic algorithm for estimating cohesion of limestone samples. Eng. Comput.34 (2), 307–317. 10.1007/s00366-017-0541-y

  • CrossRef
  • Google Scholar
  • View reference in article

• 27

  KingmaD. P.BaJ. (2014). Adam: a method for stochastic optimization. Corr. abs/1412.6980. 10.48550/arXiv.1412.6980

  • CrossRef
  • Google Scholar
  • View reference in article

• 28

  KurtulusC.SertçelikF.Sertcelikİ. (2018). Estimation of unconfined uniaxial compressive strength using schmidt hardness and ultrasonic pulse velocity. Teh. Vjesn.25, 1569–1574. 10.17559/TV-20170217110722

  • CrossRef
  • Google Scholar
  • View reference in article

• 29

  LeiD.ZhangY.LuZ.WangG.LaiZ.LinM.et al (2025). A machine learning framework for predicting shear strength properties of rock materials. Sci. Rep.15 (1), 8748. 10.1038/s41598-025-91436-8

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 30

  LeysC.LeyC.KleinO.BernardP.LicataL. (2013). Detecting outliers: do not use standard deviation around the mean, use absolute deviation around the median. J. Exp. Soc. Psychol.49 (4), 764–766. 10.1016/j.jesp.2013.03.013

  • CrossRef
  • Google Scholar
  • View reference in article

• 31

  MaZ.ZuoJ.ZhuF.LiuH.XuC. (2023). Non-orthogonal failure behavior of roadway surrounding rock subjected to deep complicated stress. Rock Mech. Rock Eng.56 (9), 6261–6283. 10.1007/s00603-023-03397-x

  • CrossRef
  • Google Scholar
  • View reference in article

• 32

  MahmoodzadehA.MohammadiM.Ghafoor SalimS.Farid Hama AliH.Hashim IbrahimH.Nariman AbdulhamidS.et al (2022). Machine learning techniques to predict rock strength parameters. Rock Mech. Rock Eng.55 (3), 1721–1741. 10.1007/s00603-021-02747-x

  • CrossRef
  • Google Scholar
  • View reference in article

• 33

  Mohammadi BehboudM.RamezanzadehA.TokhmechiB.MehradM.DavoodiS. (2023). Estimation of geomechanical rock characteristics from specific energy data using combination of wavelet transform with ANFIS-PSO algorithm. J. Petroleum Explor. Prod. Technol.13 (8), 1715–1740. 10.1007/s13202-023-01644-z

  • CrossRef
  • Google Scholar
  • View reference in article

• 34

  MoussasV. C.DiamantisK. (2021). Predicting uniaxial compressive strength of serpentinites through physical, dynamic and mechanical properties using neural networks. J. Rock Mech. Geotechnical Eng.13 (1), 167–175. 10.1016/j.jrmge.2020.10.001

  • CrossRef
  • Google Scholar
  • View reference in article

• 35

  MunemoP. K. a. P.MunemoP. (2021). Investigating the correlations between point load strength index, uniaxial compressive strength and Brazilian tensile strength of sandstones. A case study of QwaQwa sandstone deposit. Int. J. Min. Mineral Eng.12 (1), 67–83. 10.1504/ijmme.2021.114915

  • CrossRef
  • Google Scholar
  • View reference in article

• 36

  MuralhaJ.GrasselliG.TatoneB.BlümelM.ChryssanthakisP.YujingJ. (2014). ISRM suggested method for laboratory determination of the shear strength of rock joints: revised version. Rock Mech. Rock Eng.47, 291–302. 10.1007/s00603-013-0519-z

  • CrossRef
  • Google Scholar
  • View reference in article

• 37

  OuyangS.ZhangX.MaY.YaoC.YangJ.YeZ.et al (2025). Shear failure behavior and a shear strength prediction model of unbonded rock–cement interface. Int. J. Geomechanics25 (6), 04025082. 10.1061/IJGNAI.GMENG-10809

  • CrossRef
  • Google Scholar
  • View reference in article

• 38

  RastegarF.NejatiH. R.GhazvinianA.HadeiM. R.NazerigiviA. (2020). On applicability of some indirect tests for estimation of tensile strength of anisotropic rocks. J. Min. Environ.11 (3), 711–720. 10.22044/jme.2020.9220.1813

  • CrossRef
  • Google Scholar
  • View reference in article

• 39

  ŞahinM.UlusayR.KarakulH. (2020). Point load strength index of half-cut core specimens and correlation with uniaxial compressive strength. Rock Mech. Rock Eng.53, 3745–3760. 10.1007/s00603-020-02137-9

  • CrossRef
  • Google Scholar
  • View reference in article

• 40

  ShahaniN. M.UllahB.ShahK. S.HassanF. U.AliR.ElkotbM. A.et al (2022). Predicting angle of internal friction and cohesion of rocks based on machine learning algorithms. Mathematics10 (20), 3875. 10.3390/math10203875

  • CrossRef
  • Google Scholar
  • View reference in article

• 41

  ShenJ.JimenezR. (2018). Predicting the shear strength parameters of sandstone using genetic programming. Bull. Eng. Geol. Environ.77 (4), 1647–1662. 10.1007/s10064-017-1023-6

  • CrossRef
  • Google Scholar
  • View reference in article

• 42

  ShinodaM.MiyataY.KurokawaU.KondoK. (2019). Regional landslide susceptibility following the 2016 Kumamoto earthquake using back-calculated geomaterial strength parameters. Landslides16 (8), 1497–1516. 10.1007/s10346-019-01171-1

  • CrossRef
  • Google Scholar
  • View reference in article

• 43

  SiddigO.IbrahimA. F.ElkatatnyS. (2023). Estimation of rocks’ failure parameters from drilling data by using artificial neural network. Sci. Rep.13 (1), 3146. 10.1038/s41598-023-30092-2

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 44

  SinghA.AyothiramanR.RaoK. S. (2020). Failure criteria for isotropic rocks using a smooth approximation of modified mohr–coulomb failure function. Geotechnical Geol. Eng.38 (4), 4385–4404. 10.1007/s10706-020-01287-5

  • CrossRef
  • Google Scholar

• 45

  TukeyJ. W. (1977). Exploratory Data Analysis. Springer. 10.1007/978-0-387-32833-1_136

  • CrossRef
  • Google Scholar

• 46

  WangY.AladejareA. E. (2016). Bayesian characterization of correlation between uniaxial compressive strength and Young's modulus of rock. Int. J. Rock Mech. Min. Sci.85, 10–19. 10.1016/j.ijrmms.2016.02.010

  • CrossRef
  • Google Scholar

• 47

  ZhangZ. (2018). “Improved adam optimizer for deep neural networks,” in 2018 IEEE/ACM 26th International Symposium on Quality of Service (Iwqos). 10.1109/IWQoS.2018.8624183

  • CrossRef
  • Google Scholar

• 48

  ZhangZ.ShengQ.FuX.ZhouY.HuangJ.DuY. (2020). An approach to predicting the shear strength of soil-rock mixture based on rock block proportion. Bull. Eng. Geol. Environ.79, 2423–2437. 10.1007/s10064-019-01658-0

  • CrossRef
  • Google Scholar

• 49

  ZhangX.ZhuX.YaoC.OuyangS.YangJ.JiangQ. (2022). Investigation on the shear behavior of rough rock joints prepared by three-dimensional engraving technique. J. Test. Eval.50 (3), 1425–1440. 10.1520/JTE20210282

  • CrossRef
  • Google Scholar

• 50

  ZhaoJ.HuL.FengX.XiaoY.GuoY. (2023). Shear failure mechanisms of sandstone subjected to direct, true triaxial and confining shear test conditions. Rock Mech. Rock Eng.56 (9), 6889–6903. 10.1007/s00603-023-03410-3

  • CrossRef
  • Google Scholar